proceedings of the focused research program on spectral theory and boundary problems ·...
TRANSCRIPT
Distribution Category:Mathematics and Computer
Science (UC-405)
ANL-87-26, Vol. 4 ANL--87-26-Vol.4
DE90 000920
ARGONNE NATIONAL LABORATORY9700 South Cass Avenue
Argonne, Illinois 60439-4801
PROCEEDINGS OF THE FOCUSED RESEARCH PROGRAM ONSPECTRAL THEORY AND BOUNDARY VALUE PROBLEMS
VOL. 4: NONLINEAR DIFFERENTIAL EQUATIONS
Hans G. Kaper, Man Kam Kwong, and Anton Zettl, organizers
Gail W. Pieper, technical editor
Mathematics and Computer Science Division
August 1989
This work was supported in part by the Applied Mathematical Sciences subprogram of the Officeof Energy Research, U. S. Department of Energy, under Contract W-31-109-Eng-38.
A major purpose of the Techni-cal Information Center is to providethe broadest dissemination possi-ble of information contained inDOE's Research and DevelopmentReports to business, industry, theacademic community, and federal,state and local governments.
Although a small portion of thisreport is not reproducible, it isbeing made available to expeditethe availability of information on theresearch discussed herein.
Contents
Preface..........................................................................................................................................viiList of Participants and Visitors.....................................................................................................ixSchedule of Talks .......................................................................................................................... xi
Self-Intersecting Solutions of the Prescribed Mean Curvature Equation - F. V. Atkinson
Abstract..............................................................................................................................11. Introduction...............................................................................................................12. Basic Hypotheses and Initial Value Problem...............................................................33. Continuation of Solutions ............................................................................................ 54. Monotone Properties....................................................................................................65. Asym ptotics (1): Conditions for u(s) -+ o................................................................76. Further Asym ptotics (2)........................................................................... 97. Further Asymptotics (3).............................................................................................108. Estimates of Logarithmic Type .................................................................................. 119. A Lem m a on Asym ptotics of Integrals.......................................................................1310. Improvement of Theorems 2 and 3................................1411. Asymptotic Formulae for F(s), r(s).........................................................................16Acknowledgments............................................................................................................17References........................................................................................................................18Appendix..........................................................................................................................19
Bounds for Vertical Points of Solutions of Prescribed Mean Curvature Type Equations, I -F. V. Atkinson and L. A. Peletier
Abstract............................................................................................................................211. Introduction................................................................................................................212. Basic Hypotheses.......................................................................................................233. The Initial Value Problem..........................................................................................244. Estimates for Continuation.........................................................................................265. The Main Result.........................................................................................................276. Corollaries...............................................................................................................287. The Reversed Sign Case.............................................................................................308. Asymptotics: First Approximations .......................................................................... 329. A Second Approximation to U...................................................................................3410. A Second Approximation to R .................................................................................. 37Acknowledgments............................................................................................................40References........................................................................................................................40Appendix: Som e Num erical Constants...................................................................... ..42
ii
Large Solutions of Elliptic Equations Involving Critical Exponents -F. V. Atkinson and L. A. Peletier
Abstract............................................................................................................................431. Introduction................................................................................................................432. Preliminary Estimates ................................................................................................ 483. The Case 1 5 q < k -2................................................................................................544. The Case q z k -2 ...................................................................................................... 63
Acknowledgment ............................................................................................................. 69References........................................................................................................................70Appendix..........................................................................................................................71
Non-Negative Solutions for a Class of Radially Symmetric Non-Positone Problems -Alfonso Castro and R. Shivaji
Abstract............................................................................................................................731. Introduction................................................................................................................732. Preliminaries and Notations.......................................................................................743. Main Lemmas and Proof of Theorem 1.1..................................................................764. Remarks ..................................................................................................................... 79
References........................................................................................................................79
Quenching for Semilinear Singular Parabolic Problems - C. Y. Chan and Hans G. Kaper
Abstract............................................................................................................................811. Introduction................................................................................................................812. Quenching..................................................................................................................833. Existence of a Critical Length....................................................................................864. Quenching Tim e..................................................................................................... 885. Determination of Critical Length...............................................................................896. Example.....................................................................................................................91
Acknowledgment.. .......................... ............................... 92Refereces............................ ............... 92References ........................................................................................................................ 92
Existence Results of Steady States of Semilinear Reaction-Diffusion Equations and Their Appli-cations - C. Y. Chan and Man Kam Kwong
Abstract............................................................................................................................931. Introduction................................................................................................................932. Nagumo's Lemma......................................................................................................943. Steady-State Solutions...............................................................................................954. Quenching................................................................................................................1005. Minimal and Maximal Solutions..............................................................................1026. Asymptotic Decay and Blow-Up..............................................................................105
References.......................................................................................................................109
iv
Uniqueness Results for Some Nonlinear Initial and Boundary Value Problems -Hans G. Kaper and Man Kam Kwong
Abstract..........................................................................................................................1111. Statement of Results.................................................................................................1112. Proof of Theorem 1 ............................................................................................. .l3. Proof of Theorem 2..................................................................................................1174. Mean Curvature Equation ....................................................................................... 121Acknowledgments..........................................................................................................122References......................................................................................................................122
Uniqueness of Non-Negative Solutions of a Class of Semilinear Elliptic Equations -Hans G. Kaper and Man Kam Kwong
Abstract.........................................................................................................................1251. Introduction..............................................................................................................1252. General Properties of Solutions................................................................................1283. Distinct Solutions Must Intersect.............................................................................1324. Intersecting Solutions Are Identical.........................................................................1355. Conclusion and Discussion ...................................................................................... 138Acknowledgments..........................................................................................................138References......................................................................................................................139
A Non-Oscillation Theorem for the Emden-Fowler Equation; Ground States for SemilinearElliptic Equations with Critical Exponents - Hans G. Kaper and Man Kam Kwong
Abstract..........................................................................................................................1411. Introduction and Statement of Results.....................................................................1412. Proof of Theorem 1..................................................................................................144
Acknowledgments..........................................................................................................164References......................................................................................................................164Appendix: A Generalized Sturm Comparison Theorem...............................................166
Concavity and Monotonicity Properties of Solutions of Emden-Fowler Equations -Hans G. Kaper and Man Kam Kwong
A bstract........................................................................................................... ......... 1691. Introduction..............................................................................................................1692. Regular Coefficients.................................................................................................1713. Singular Coefficients................................................................................................174
References......................................................................................................................179Appendix: Auxiliary Results.........................................................................................180
V
Uniqueness for a Class of Non-Linear Initial Value Problems -Hans G. Kaper and Man Kam Kwong
Abstract .......................................................................................................................... 1851. Statement of the Theorem ........................................................................................ 1852. Proofof the Theorem ............................................................................................... 1863. Remarks ................................................................................................................... 189
References......................................................................................................................191
Oscillation of Emden-Fowler Systems - Man Kam Kwong and James S. W. Wong
Abstract .......................................................................................................................... 1931. Introduction..............................................................................................................1932. Superlinear Case: Atkinson's Theorem...................................................................1943. Sublinear Case: Belohorec's Theorem .................................................................... 1964. Generalization of Waltman's Theorem.....................................................................1985. A Counterexample and Remarks..............................................................................200
References.......................................................................................................................202
Local and Global Properties of Solutions of Quasilinear Elliptic Equations -Mohammed Guedda and Laurent Veron
Abstract .......................................................................................................................... 2051. Introduction..............................................................................................................2052. The Subcritical Case ................................................................................................ 2073. Global Solutions ...................................................................................................... 2134. The Supercritical Case ............................................................................................. 2175. The Critical Case......................................................................................................224
References......................................................................................................................228Appendix: Local Existence Results..............................................................................230
vi
Preface
This is the fourth and last volume of a series of reports containing the proceedings of theFocused Research Program on "Spectral Theory and Boundary Value Problems," which washeld at Argonne National Laboratory during the period 1986-1987. The program was organ-ized by the Mathematics and Computer Science (MCS) Division as part of its activities inapplied analysis. Members of the organizing committee were F. V. Atkinson, H. G. Kaper(chairman), M. K. Kwong, A. M. Krall, and A. Zettl.
The objective of the program was to provide an opportunity for research and exchange ofviews, problems, and ideas in three main areas of investigation: (1) the theory of singularSturm-Liouville equations, (2) the asymptotic analysis of the Titchmarsh-Weyl m(X)-coefficient, and (3) the qualitative theory of nonlinear differential equations. The program hadfive full-time participants, who were joined by five more participants for periods of severalmonths. Twenty-four mathematicians from the United States, Canada, and Europe visited forshorter periods for seminars and technical discussions. These proceedings are the permanentrecord of the research stimulated by the year-long program.
The MCS Division generously supported the activities of the Focused Research Pro-gram. A grant for the visitors program was provided by the Argonne Universities AssociationTrust Fund.
Following this preface is a list of all participants and visitors with their currentaffiliations and addresses. Also included is a schedule of the talks presented as part of theresearch program. We express our gratitude to our colleagues and especially to those whocontributed manuscripts to the proceedings.
Hans G. KaperMan Kam Kwong
Anton Zettl
vi ". .
Argonne National LaboratoryMathematics and Computer Science Division
1986-87 Focused Research Program"Spectral Theory and Boundary Value Problems"
Participants
Part-timeFull-time
F. V. AtkinsonDepartment of MathematicsUniversity of TorontoToronto MSS IA1, OntarioCanadaOctober 1986 - July 1987
Hans G. KaperMathematics and Computer Science Div.Argonne National Laboratory9700 South Cass AvenueArgonne, IL 604394844September 1986 - September 1987
Allan M. KraelDepartment of MathematicsPennsylvania State University215 McAllister BuildingUniversity Park, PA 16802September 1986 - May 1987
Man Kam KwongMathematics and Computer Science DivisionArgonne National LaboratoryArgonne, IL 60439-4844September 1986 - September 1987
Anton ZettiDepartment of Mathematical SciencesNorthern Illinois UniversityDeKalb, IL 60115-2888September 1986 - June 1987
W. AllegrettoDepartment of MathematicsUniversity of AlbertaEdmonton, Alberta T6G 2G1CanadaDates of visit: April 28-30, 1987
Paul B. BaileyNumerical Mathematics DivisionSandia National LaboratcriesAlbuquerque, NM 87185Dates of visit: April 20-24, 1987
Alfonso CastroDepartment of MathematicsNorth Texas State UniversityDenton, TX 76203-5116May -July 1987
C. Y. ChanDepartment of MathematicsUniversity of Southwestern LouisianaLafayette, LA 70504-1010May - July 1987
Charles T. FultonDepartment of Applied MathematicsFlorida Institute of TechnologyMelbourne, FL 32901April - June 1987
Marc GarbeyDepartment of MathematicsU. de ValenciennesLe Mont Houy59326 ValenciennesFranceJune - July 1987
Eduardo SocolovskyDepartment of MathematicsUniversity of PittsburghPittsburgh, PA 15260June - September 1987
Visitors
Chr. BennewltzDepartment of MathematicsUniversity of UppsalaSwedenDates of visit: March 17-31, 1987
H. BeizingerDepartment of MathematicsUniversity of illinois273 Altgeld HallUrbana, IL 61801Dates of visit: March 16-17, 1987
ix
R. C. BrownMathematics DepartmentUniversity of AlabamaTuscaloosa. AL 35487-1416Dates of visit April 13-18, 1987
S. ChenDepartment of MathematicsShandong UniversityJinan, ShandongPeople's Republic of ChinaDates of visit March 17-20, 1987
P. Concus50A-2129Lawrence Berkeley LaboratoryBerkeley, CA 94720Dates of visit January 30-31, 1987
L. ErbeDepartment of MathematicsUniversity of AlbertaEdmonton, Alberta T6G 2G1CanadaDates of visit April 27-30, 1987
W. N. EverittDepartment of MathematicsThe University of BirminghamP. O. Box 363Birmingham B15 2TUnited KingdomDates of visit April 16-30, 1987
J. GoldsteinDepartment of MathematicsTulane UniversityNew Orleans, LA 70118Dates of visit May 13-14, 1987
G. HalvorsenInstitute for Energy TechnologyDepartment KRS, Box 402007 KjellerNorwayDates of visit May 18-26, 1987
B. J. HarrisDepartment of Mathematical SciencesNorthern Illinois UniversityDeKalb, IL 60115-2888Dates of visit June-August, 1987
D. HintonDepartment of MathematicsUniversity of TennesseeKnoxville, TN 37996-1300Dates of visit April 13-18, 1987
V. JurdJevicDepartment of MathematicsUniversity of TorontoToronto, Ontario MSS IAlCanadaDate of visit July 30, 1987
A. B. MingareillDepartment of MathematicsUniversity of Ottawa585 King EdwardOttawa KIN 6N5CanadaDates of visit: 3/1-7, 4/27-30, 5/22-23, 1987
J. NeubergerDepartment of MathematicsNorth Texas State UniversityP. O. Box 5116Denton, TX 76203-5116Dates of visit: April 1-4, 1987
S. PruessMathematics DepartmentColorado School of MinesGolden, CO 80401Dates of visit: June 15-19, 1987
T. ReadDepartment of MathematicsWestern Washington UniversityBellingham, Washington 98225Dates of visit: April 13-18, 1987
J. RidenhourDepartment of MathematicsUtah State UniversityLogan, UT 84322Dazes of visit: May 13-20, 1987
Bernd SchultzeUniversitaet Gesmathochschule EssenFachbereich 6, MathematikPostfach 103 7644300 Essen IWest GermanyDates of visit: May 31-June 8, 1987
G. SellInst. for Mathematics and Its ApplicationsUniversity of Minnesota206 Church StreetMinneapolis, MN 55455Date of visit April 23, 1987
J. SerrinDepartment of MathematicsUniversity of Minnesota206 Church StreetMinneapolis, MN 55455Date of visit June 18, 1987
J. K. ShawDepartment of MathematicsVirginia Polytechnic Institute
and State UniversityBlacksburg, VA 24061Dates of visit: April 13-18, 1987
x
Argonne National LaboratoryMathematics and Computer Science Division
1986-87 Focused Research Program"Spectral Theory and Boundary Value Problems"
Schedule of Talks
October 15
October 22
October 28
November 7
November 13
January 14
January 15
January 16
January 21
January 30
March 17
March 18
April 1
April 2
April 14
April 15
April 15
April 16
April 16
April 17
April 17
Allan Krall, "Orthogonal Polynomials and Boundary Value Problems"
Allan Krall, "Orthogonal PolynomiaLs and Boundary Value Problems"
Allan Krall, "M ()-Theory for Singular Hamiltonian Systems"
Derick Atkinson, "Pruefer Transformation for Systems of Second-OrderDifferential Equations"
Derick Atkinson, "Pruefer Transformations for Systems of Second-OrderDifferential Equations, II"
Allan Krall, "Singular Hamiltonian Systems"
Allan Krall, "The Titchmarsh-Wevl M-Function for Singular Hamiltonian Systems"
Allan Krall, "The Titchmarsh-Weyl M-Function for Singular Hamiltonian Systems, II"
Derick Atkinson, "Asymptotics of the Titchmarsh-Weyl M-Function for SingularHamiltonian Systems"
Bert Peletier, "The Initial Development of Dead Core in a Reacuon Diffusion Equation"
Hal Benzinger, "Chaotic Dynamical Systems"
Shaozhu Chen, "Asymptotic Linearity of the Solutions of Second-order LinearDifferential Equations"
John Neuberger, "Numerical Computation of Eigenvalues of the Schroedinger Equation"
Michael Jolly, "The Geometry of the Global Attractor for a Reaction-Diffusion Equation"
Derick Atkinson, R. C. Brown, C. T. Fulton, D. Hinton, H. G. Kaper, A. KrallG. K. Leaf, Minkoff, T. Read, J. Shaw, A. Zettl, general discussion
Allan Krall, "Characterization of Singular Boundary Conditions"
Tony Zettl, "Norm Inequalities for Differential and Difference Operators"
Don Hinton, "One Variable Weighted Interpolation Inequalities"
Ken Shaw, "Extensions of Levinson's Theorem to Dirac Systems"
Tom Read, "Sturm-Liouville Problems with Large Leading Coefficients"
Hans Kaper, "Spectral Analysis of a Singular Fourth-Order Differential Operator Arisingin Combustion"
xi
April 20 Paul Bailey, "Computation of Eigenvalues and Eigenfunctions of Sturm-Liouville Equationsusing SLEIGN"
April 21 Paul Bailey, "Computation of Eigenvalues and Eigenfunctions of Sturm-Liouville Equationsusing SLEIGN, II"
April 22 Nome Everitt, "The Laplace Tidal Wave Equation"
April 23 George Sell, "The Principle of Spatial Averaging and Inertial Manifolds"
April 23 Paul Bailey, "Computation of Eigenvalues and Eigenfunctions of Sturm-Liouville Equationsusing SLEIGN, III"
April 28 Lynn Erbe, "Oscillation Theory for Systems of Second-Order Differential Equatior."
April 29 Walter Allegretto, "Spectral Analysis of Second-Order Boundary Problems withIndefinite Weight Functions"
April 30 Charles Fulton, "Asymptotics of m (X) for Singular Potentials"
May 14 Jerry Goldstein, "Recent Developments in Thomas-Fermi Theory"
May 15 Charles Fulton, "Singular Hamiltonian Systems"
May 18 Jerry Ridenhour, "Zeros of Solutions of n-th Order Differential Equations"
May 20 Charles Fulton, "The Bessel-squared Operator in the lim-2, lim-3, and lim-4 Cases"
May 21 Goiskalk Halvorsen, "Oscillation Results for Second-Order Equations"
May 22 Derick Atkinson, "Estimation of m ()) in a Case with an Oscillating Leading Coefficient"
May 27 Hans Kaper, "A Non-oscillation Theorem for an Emden-Fowler Equation"
June 1 C. Y. Chan, "A Generalization of the Thomas-Fermi Equation"
June 3 Bernd Schultze, "Spectral Properties of Nonselfadjoint Differential Operators"
June 5 Alfonso Castro, "Superlinear Boundary Value Problems"
June 9 Man Kam Kwong, "Concavity of Solutions of Certain Emden-Fowler Equations"
June 15 Charles Fulton, "Convergence of Spectral Functions"
June 16 Bernie Matkowsky, "Introduction to Bifurcation Theory"
June 18 James Serrin, "Asymptotics of the Emden-Fowler Equation"
June 19 Steve Pruess, "SPDNSF: A Code to Compute the SPectral DeNSity Function"
June 25 Bernie Matkowsky, "Stability Analysis and Bifurcation Theory"
July 17 Marc Garbey, "A Quasilinear Prabolic-hyperbolic Singular Perturbation Problem"
July 30 Val Jurdjevic, "Differential Equations of Control Theory"
July 31 Bernie Harris, "Asymptotics of the Titchmarsh-Weyl m ()-coefficient"
xii
SELF-INTERSECTING SOLUTIONSOF THE PRESCRIBED MEAN CURVATURE EQUATION
F. V. Atkinson*Department of Mathematics
University of TorontoToronto M5S 1A1, Ontario
Canada
Abstract
Inequalities and asymptotic estimates are obtained for solutions of the system
dr/ds = cos yi, du/ds = sin yr, dr/ds = f (u) - (N-1)r-1 sin yi
in a group of cases in which f (u) is positive, increasing, and unbounded. Thissystem is the intrinsic version of the radially symmetric prescribed mean curva-ture equation, and permits the study of certain generalized solutions of this equa-tion.
1. Introduction
The present paper is devoted to the asymptotic behavior as u - ., r -+ co of certain gen-eralized solutions of the problem
(d/dr)(rN-Iq (u) = rN-1lf(W, (1.1)
where q(v) has the form
q(v) = v/(l+ v 2) , (1.2)
and N > 1. In the case of positive integral N, (1.1) arises from the search for radially symmetricsolutions u = a (r) of
div{(V u)/(1 + $V u 12)1) = f (u) . (1.3)
This can be interpreted to mean that f (u) is to be, apart from a numerical factor, the mean curva-ture of an associated surface of revolution. The equation is also of interest in capilarity theory.
From the point of view of differential equation theory, the salient feature of (1.1)-(1.2) isperhaps that the function q(v) has only a finite range, namely, (-1,1). This has the consequencethat for an initial value problem, in which u(r0 ), i'(ro) are prescribed, continuation of the solu-tion for r > ro may terminate in a "vertical point" at r = r 1 , say, with u(r) tending to a finitelimit u 1 as r -+r 1 -0, but u'(r) becoming unbounded; the graph of u(r) in the (r,u)-plane
Senior Mathematician Emeritus at the Mathematics and Computer Science Division, Argonne NationalLaboratory, October 1, 1986 - July 17, 1987.
1
exhibits a finite point (r1 , u1 ) with vertical tangent. Estimates for such points are of some
interest; refer in particular to [Atkinson, Peletier, and Serrin 1988], [Atkinson and Peletier 1988],
and [Concus and Finn 1979].
The occurrence of a vertical point need not mean the end of the investigation. The con-
tinuation of solutions beyond such a point may take place, locally at least, by interchanging the
roles of r and u. The solution r = r (u) may then be continued up to a point, if any, where r'(u)
becomes unbounded, at which point one may revert to the form u = u (r), and so on. An alterna-
tive procedure employed here is to use a parametric version of the equation, as in [Concus and
Finn 1979].
Naturally, the details and results of such a procedure depend heavily, both quantitatively
and qualitatively, on the choice of the function f (u). In this paper we confine attention to the
situation where u > 0 and where, fcr such u, f (u) is positive and increasing. Furthermore, we
consider mainly initial data (in the u = u (r)version of the problem) of the form
u(ro)= y>0, u'(ro)=0, (1.4)
for some r o >0. The graph, in the (r,u)-plane, then takes the form of an ascending series of over-
lapping loops, as illustrated in Figure 1 of the Appendix, in which u, r are well-defined functions
of the arc-length s. Our aim is to discuss the asymptotic behavior of u (s), r (s) as s -+ c. We are
thus concerned with the asymptotic behavior of the solution of the three-dimensional first-order
system
dr/ds = cos r , (1.5)
du/ds = sin yi, (1.6)
dy/ds = f (u) - (N-1)r~ sin y , (1.7)
with initial data (now with u = u (s), etc.)
u(0) ='y, (1.8)
r(0) = r0o, (1.9)
4(0) = 0 . (1.10)
Here yi is the angle between the tangent and the r-axis, (s, fi) being the usual "intrinsic" coordi-
nates. We assume that f e C '[y,oo), with f > 0, f'> 0. The solution then, as it happens, exists
for all s > 0, with dti/ds >0 and y(s) -+oo as s -+0oo.
Sections 2 through 5 are devoted to preliminaries. In Sections 2 and 3 we deal with the
initial-value problem and introduce basic functionals; these functionals yield inequalities govern-
ing u (s) and the u.. Sections 4 and 5 are devoted to the existence of the first and subsequent
loops and further inequalities, existence of solutions for all s > 0 being then established.
2
Sections 6 through 9 deal with asymptotic behavior, under progressively narrower
hypotheses; however, fer example, any positive polynomial with positive derivative would be
admissible. Dealing first with the question of boundedness, we show in Section 6 that u (s) -+ oowith s. At this stage we know that r (s) > r3 > 0 for s > s3 , but do not show that r(s) -* oo; we
can, however, say that the horizontal fluctuations in r (s) tend to 0 as s -+ oo. In Section 7 we
derive an asymptotic formula for F (u (s)), where F denotes as usual an integral off ; this formula
is of an implicit nature, as it involves the unknown function r(s). In Section 8 we derive an
asymptotic formula for r(s), again implicit since it involves f (u(s)), and show that r(s) -+ o. In
Section 9 we bring these together and obtain asymptotic relations connecting the logarithms of
F (u(s)), r (s), and s.
It is then possible to feed the result of Section 9 back into the argument of Sections 7 and 8
to obtain asymptotic results for F and r, rather than just for their logarithms. This is the subject
of Sections 10 and 11.
We do not discuss here other types of solution, arising from different hypotheses concerning
f. If, as before, we have u > 0, f > 0, and f'> 0, but f is replaced by-fin (1.1), the graph takes
the form of a descending series of horizontal oscillations without intersections, as illustrated in
Figure 2 of the Appendix and discussed in [Atkinson, Peletier, and Serrin 1988], in which r is a
well-defined function of u with a series of maxima and minima; this gives rise to the study of
"ground states," in which u > 0, u -+0 as r -+ o . Another possibility is that a trajectory might
pass from the self-intersecting pattern to the zigzag pattern, if f (u) changes sign. Another area of
interest is given by "singular solutions," in which I u (r) I-+ - - as r -40. A variety of graphs
may be seen in [Concus and Finn 1979], [Eells 1987], [Evers 1985], and [Levine 1988]. Prob-
lems associated with numerical solution of the system (1.5)-(1.7) are discussed in [Evers 1985]
and [Levine 1988].
In the case of constant f, exact solutions are possible in terms of the roulettes of conics; this
case has a long history, for which we cite the survey paper by Eells [1987], where graphs showing
the various possibilities may be seen.
2. Basic Hypotheses and Initial Value Problem
We make throughout the assumptions that
fe C'[y,oo) (2.1)
and that
f (u) > 0, f '(u) > 0, (2.2)
for u y. We assume also that N > 1; in the ODE formulation it need not be integral.
3
More detailed assumptions will be needed from time to time, in particular the following:
f(u)-*+oo as u ->oo , (2.3)
F(u)f '(u)/f2(u)--4 K E (0,1), (2.4)
as u -+ cc, where
F(u) = f(v)dv , (2.5)0
and finally,
f E C [y,o)f''(u)f (u) = Of' 2(u)) , (2.6)
as u -* o.
The initial value problem (1.8)-(1.10) for the system (1.5)-(1.7) presents no difficulty in the
case where r0 > 0. In the case where r0 = 0, the initial value problem has been dealt with in
[Atkinson, Peletier, and Serrin 1988] and [Atkinson and Peletier 1988], in the formulation with r
as an independent variable. In sketching the details, we use the notation U(r) for u as a function
of r, we need this notation only in the present section. Assuming the solution to exist for small r,
we write
1(r) = r JtN-1f (U ()d: (2.7)0
and then have
U'(r) = I(rX1 - 1 2(r))-" (2.8)
and finally
U(r) = y- j(t)(l - 12(t))"'dt. (2.9)0
This may be used to establish the existence of U (r) for small r.
We can then define yi,s as functions of r for small r by
r
tan y = U'(r) , s = Jl + U' 2 (t))"dt . (2.10)0
As r -+ 0, we shall have by (2.7)
1(r) ~rf(y)/N , (2.11)
so that
U'(r) -rf (y)/N , U(r) - y~(1/2)r2f (y)/N . (2.12)
4
From (2.4) we then have
W ~rf (y)/N , s - r , (2.13)
and so
u(s) - y - (1/2)s2f (y)/N , (s) ~ sf (y)/N . (2.14)
The last result can be sharpened slightly. We have, as s -4 0,
dy/ds -+f (y)/N . (2.15)
This follows in using the first of (2.8) in the last of (1.4). In particular, we have di/ds > 0 for
small s > 0; in fact, this is true for all s > 0.
3. Continuation of Solutions
We have just shown that the problem (1.5)-(1.10) has a solution for small s > 0, such that
r (s)> 0 , (3.1)
d19/ds > 0. (3.2)
We now show that this solution can be continued for all s > 0, with (3.1) holding throughout.
For this purpose, we use two standard functionals. In any interval [0,s] in which the solution
exists, we define
E0 (s) = F(u)+ cos W , (3.3)
E1 (s) = rN/N - rN-1sin i/f (u). (3.4)
Using (1.5)-(1.7), we have
dEo/ds = (N - 1)r-1sin 2W4, (3.5)
dE 1 /ds = rN-1sin 24 f'(u)f 2(u). (3.6)
It follows from (3.5) that in any such interval (0,s'], we have
F(u(s))+ cos W(s) > F (y)+lI , (3.7)
and, in particular,
u(s)> y. (3.8)
From (3.6), we see that E1 () is nondecreasing, and certainly strictly increasing in some interval
(0,E). Since E1 (0) z 0, we have that E1 (s) will have a positive lower bound for s > E. In view of
(3.8), we see that
5
r N/N + rN-1/f(Y)
will have a positive lower bound for s > e, and so also r itself. Since r is positive, the solution of
(1.5)-(1.7) can be continued indefinitely, with (3.1) holding throughout.
To get (3.2), we use again the fact that E 1 (s) > 0 for s > 0. We have
rN/N > rN-1 sin Nlf(),
and so
r~1 sin y<f(u)/N.
It thus follows from (1.7) that
dyl/ds > f (u)/N > f (y)/N , s >0. (3.9)
If, of course, sin W 50, we have from (1.7) the stronger result that dye/ds z f (Y).
In particular, it follows that
W(s) > sf (Y)/N , s > 0 , (3.10)
and
w(s) "- oo as s -+ c . (3.11)
4. Monotone Properties
In describing the graphs in the (r,u)-plane, we need to identify the points at which the
tangent is horizontal or vertical. Accordingly, we denote by so,s 1 ,s2,... the points at which
y, = nn/2 , n = 0,1,.... (4.1)
The corresponding values of r,u will be denoted by r.,u.. The values given by (4.1) exist and are
unique, in view of (3.9)-(3.11).
It follows from (3.5) that the sequence
E0 (s4 ) , n = 0,1,2,... , (4.2)
is monotone increasing. From this we deduce the properties
Y = uo <u4 < ug- -- , (4.3)
u2 < 6 < ut<'"'-"(4.4)
u1 <u3 <U5 < - . (4.5)
By similar arguments, we have that
6
u(s)> u4, for s >s4. , n =0,1,...,
as already noted in (3.9) in the case n = 0.
Next we collect consequences of (3.6). Since we assume f ' to be strictly positive, we have
that the sequence
E i(s,) , n = 0,1,... , (4.7)
is strictly increasing. An important conclusion is that
r3 < r7 < r 1l < -"-" (4.8)
The first of these inequalities is typical. We have that E1 (s3 ) < E1 (s7 ), or explicitly
r/N + r -1/f (u3 ) <r/N + r- 1 /f(u7).
Since f (u7) >f (u3), by (4.5), we must have r7 > r3 , as asserted.
We deduce that
r(s) > r4.._1 , s > s41_1 , n = 1,2,... (4.9)
For ex-mple, in the case n = 1, we have (by (1.5)) that in the interval (s3 ,ss) r(s) increases from
r3 to r5 , and likewise that in the interval (55,57), it decreases from r5 to r7 (> r3 ). Hence,r (s) > r3 for s in the interval (s,s7 ). A similar argument applies if n > 1.
We remark that the sequence ro,r2 ,r4 ,... is monotone increasing. However, this does not
seem to be true for the sequence r1 ,r5 ,... .
A further conclusion is that r(s) either remains bounded as s -+ or else tends to 00. This
follows from the corresponding statement for E 1 (s).
5. Asymptotics (1): Conditions for u (s) -+
In the first round of asymptotic analysis we ask whether u(s),r(s) remain bounded as
s -+ oc or tend to infinity with s; these are the only possibilities. In this section, we deal with the
simpler case of u(s). We prove first the following theorem.
THEOREM 1. Let (2.1)-(2.2) hold. Then
u(s) -+oo as s -+ ,. (5.1)
If also
f(u) -+oo as u--+oo , (5.2)
then
7
(4.6)
as n -4 oo.
Proof. Since E0 (s) is monotone, it either is 'ounded or tends to oo, and the same holds for F(u).
Thus, u(s) either tends to oo or is bounded. Thus, if (5.1) is false, E0 (s) must be bounded. We
show that this statement leads to a contradiction.
We apply (3.5) over the intervals (p,a.), n = 1,2,..., where
V(p) = (n + 1/4)c, yi(a.) = (n + 1/2) , (5.4)
so that an = s2n+ 1 . We must have, if F(u) is bounded,
ea.
Ifr-1 sin2y ds < 00 (5.5)P.
We show that this is impossible.
Let C denote some upper bound for u(s). In (p,a) we have
dyds = f (u) - (N-1)r~1 sin W < f (C) ,
and so, by (5.4),
a~ - p~ > it/4f (C )) . (5.6)
In the same interval we have sin2 y N 1/2, and also
r(s) < s s2,+1 < N(n + 1/2)/f (y).
by (3.9). Thus, the sum in (5.5) exceeds
If (y)/{8f (C)N(2n+1)} ,
which is infinite. This gives a contradiction and so proves (5.1).
For the second part, we integrate (1.5) over (s.,s.+1) and get
I r~+1 - r I < s~+1 - s . (5.7)
We have also that
x/2 = y(s.+1) - r(s.) = (s+ 1 - s.)4'(a), (5.8)
for some a E (s.,s~+ 1). Since y'(s) -+ oo as s -+ a*, by (1.7) and (5.1)-(5.2) we have from (5.8)
that
which by (5.7) proves (5.3).
8
rn+1 -r, -4 0 (5.3)
Thus the oscillations of the graph in the horizontal direction tend to zero a the graph
extends upwards, and the graph converges (in a sense) to a curve. The asymptotics of this curve
form our subject in the next section.
6. Further Asymptotics (2)
Under more restrictive hypotheses we obtain fairly precise estimates of the growth ofF (u (s)), r (s), it(s) as s -+ co, showing that they behave as powers of s. In the first of a series of
stages of the argument, Sections 7 through 9, we develop such approximations in a weaker loga-
rithmic version.
In addition to our previous hypotheses that f e [0,oo), f (u)> 0, f '(u) > 0, we now require
that f e C "[0, oo) and that
f '('s)F(u)/f2(u) -K e (0,1) , (6.1)
f '(u)f (u) = Off" 2(u)), (6.2)
as u -+ oo. For example, if f (u) is a polynomial of degree m, we shall have K = m/(m+l). Con-
versely, it follows easily from (6.1) that
log F (u) ~ (I - K)f1log u , log f (u)-K (1- K)~'log u , (6.3)
and also that
f(u)/f2 (u) -+ 0 , (6.4)
as u - oo.
In the first step we relate F (u) to an integral involving r (u). This is followed in Section 7
by an integral relation in the reverse direction. These two relations are brought together in Sec-
tion 8 to yield explicit asymptotic formulae of logarithmic type in terms of s. We have the fol-
lowing theorem.
THEOREM 2. As s -4 o,
F(u) ~-(1/2)(N - 1)Jds/r . (6.5)
We remark first that the right is unbounded as s -+ o, in view of the fact that, by (1.5),
r (s):5 s . (6.6)
Proof. For this we use the identity
9
(d/ds){F(u) + cos V + (N - 1)(sin 2N)/(4rf)} = (N - 1)/(2r)
- (N - 1)(4r 2f-1{2(N - 1)sin AV cos 21y + sin 2yj cos yr)
- (N - 1)(4rf2)-'f 'sin 241 sin y. (6.7)
This may be verified directly, using (1.5)-(1.7). Here the right-hand side may be written in the
form
(N - 1)/(2r) + 0 (1/(r2f)) + 0 (l/(rF)),
where we have used (6.1). We then get the result by integrating over (1,s), noting that r is
bounded from zero and that f,F -+coc with s by Theorem 1 and (6.3).
At this point, we do not have specific asymptotic information on r(s) which would yield an
explicit estimate for F (u (s)). We can, however, say that
F(u)C= (s), (6.8)
lim inf (log s)~1F(u) > 0 . (6.9)
The first of these follows immediately fror' (3.5), together with the fact that r(s)> r3s for s > s3 .
The second follows from Theorem 2, together with the estimate r (s) = O(s) as s -+ ..
7. Further Asymptotics (3)
We now consider the behavior of r (s) as s -+ « and prove the following theorem.
THEOREM 3. Under the conditions of Section 6, we have
r(s)~ (I/2)Jf'(u)/f2(u)ds , (7.1)
as s -4 co.
Proof. This is less direct than that of Theorem 2. We start by recalling from (3.6) that
(d/ds)(rN/N - rN-tf sin iy) = rN-1I 2i(ff2 ). (7.2)
By an argument involving (1.5)-(1.7) and partial integration, we are led to replace this by
10
(d/ds)(rN/N - rN- 1 f 1 sin 1 - rN-1f sin 2 3 /(4f3))
= rN-1f /(2f2) -(N - 1)rN-2f '(2 cos 21r sin 1 - sin 2'1 Cos'4) / (4f3)
+ rN-1 sin 241sin '1ff/(4f3)) (7.3)
= rN-1f /(2f2 ) +0 (rN- 2 f f 3) + O (rNl(-ff 3 )7. (74)
We then integrate over (1,s). The contributions of the last two terms in (7.4) are of lower
order than that of the first. In the case of the second term, this is so since r is bounded from zero
and since f (u (s)) -+ c. In the case of the last term, we use (6.2),(6.4). We deduce that
s
rN/N-~ (1/2)JrN-1(f-/f2)ds . (7.5)1
We now deduce (7.1) by arguments of Gronwall-Bihdri type. We rewrite (7.5) in the form
-1+1/N
rN-1( l 2) rN-1(f -f2)ds ~ (N/2)1-1/N f 2,(76
so that, integrating, we have
/N{ rN -1(f - 2)ds ~ -(N/2)1-1/N (f -f2) s(7.7)
1
Hence,
N
'rN-1(f-/f2)ds ~-(2/N)2(-N S(,,f2
On substituting into (7.5), we get the required result (7.1).
We have in particular that r(s) -4oo as s -+ o. This follows from (7.1) since
f '(u)/f2(u)-~ K/F(u) and
F(u(s)) = O(s) (7.8)
by (3.5).
8. Estimates of Logarithmic Type
In the next stage of the argument, we obtain asymptotic estimates for log F (u) and log r in
terms of log s. We then use these results to sharpen the reasoning in Sections 6 and 7, to getfirst-term asymptotics for F (u) and r themselves. We continue to make the assumptions (6.1)-(6.2).
11
The results (6.5) and (7.1) can be assembled in the form
F (u)~ (1/2)(N - 1)Jds /r , (8.1)
r ~ (K12)Jds/F , (8.2)1
where in the last case we have used (6.1). We thus obtain a system of asymptotic integral equa-
tions for the unknown functions F (u(s)), r (s), which we can solve in a certain logarithmic sense.
We note that both integrals on the right diverge as s - c, by (6.6) and (7.8).
We introduce the function
G(s)_= .d (dt/F (u (T))}, s> 1. (8.3)
Then
-1
G'(s)= fd/F(u(t))J
and so
- G "(s)/G'2(s) = 1./F(u(s)). (8.4)
Substituting (8.2) into (8.1), we have
F-(u(s))-K'(N -1)G(s), (8.5)
and so (8.4) shows that
G'(s)G (s)/G'2(s)--+ -K(N - 1) 1 , (8.6)
ass -+ .
To integrate this, we put G (s) = exp H (s), which leads to
(H"+ H'2)/H'2 -+-K(N - 1)-i,
so that, as s - ,
H'IH2 -- A ,
say, where A = 1 + K /(N - 1). Integrating, we get H'~ 1/(As), so that H -A ~log s. Since
H = log G, it follows from (8.5) that
log F(u(s)) -A~'logs .
In a similar way, we can eliminate F from (8.1)-(8.2) to get
12
r (s) - K(N - 1)~1 da { dt/r (t) ,
which leads to
log r(s)-A '~logs
where A'= Il+ (N - 1)/K.
We sum up this preliminary result as the following theorem.
THEOREM 4. Let (6.1)-(6.2) hold. Then, as s -+ 00,
log F (u(s))-(N - 1){N - I + K} 1 log s , (8.7)
logr(s)~-K(N-lI +K)ilogs . (8.8)
We list some consequences. It follows from (6.1) that
log f (u) ~ K log F (u) (8.9)
as u,s -* o, and so from (8.7) we have that
logf (u) -K(N - 1)(N - 1 +K)~ 1 logs . (8.10)
We use this in (1.6) to deduce that
log W(s) - (N - 1 +KN)(N - 1 +K)~'logs . (8.11)
A further conclusion from (8.7)-(8.8) is that
log(r (s)F(u(s))) ~ log s, (8.12)
which is noteworthy in being independent of N and K.
9. A Lemma on Asymptotics of Integrals
We make repeated use of the following simple remark.
LEMMA 1. Let A (s),B(s) be continuous functions on [1,oo) to (0,oo) such that, as s -+ oo,
log A(s)/log s -+*a, (9.1)
lim sup log B(s)/log s = 1, (9.2)
where
a>-1, a> . (9.3)
Then, as s -+ co,
13
SB(t)dt = 0 (sA (t)dt , (9.4)
for any E E (0, a-3).
It follows from (9.1) that for any 85> 0, some s' and s > s' we have
A (s) > s (Q-8 , (9.5)
so that
(A (t)dt > A OS (+a4) (9.6)
for some A 0 > 0. A similar argument shows that as s -+ *,
S B(t)dt = o {'t()dtJ. (9.7)
Ifa < -1 we can choose 8> 0 so that the right is O(1) as s -4co, while 1 + a -58> 0, so that
(9.4) will hold with the choice = 1 + a - S. If on the other hand10 -.-1, the right of (9.7) is
and we get the result on choosing, say, 8 = (a -13 - e)/3.
10. Improvement ci Theorems 2 and 3
We now review the proofs of these theorems with the aid of Theorem 4 and cite the results
as Lemmas 2 and 3. The new proofs will, of course, be quite close to the previous ones. In the
following, e will denote generically some positive number which may be replaced by any smaller
such number, and so may be taken to be the same in all of a finite number of occurrences. We go
back first to the argument of Section 6 and get the following lemma.
LEMMA 2. As s -4 c,
F (u) = ((N - 1)/2+ 0(s))Jds/r . (10.1)
Proof. Integrating the identity (6.7) over (1,s), we obtain
F(u)+ 0(1) = ((N - 1)/2)Jds/r +JO(1/(r2f))ds +JO(/(rF))ds . (10.2)I 1I
We now apply Lemma I and use Theorem 4 to give the corresponding values of "a" and "13". In
the present case for the main term on the right of (10.2), the first integral, we have from (8.8)
14
a =-K/(N- 1 +K), (10.3)
so that a > -1, as required by (9.3). For the second integral in (10.2) one has. by (8.8)-(8.9),
$ = -(K + 2K(N-1)}/(N - 1 + K), (10.4)
so that 3 < a, as required by (9.3). For the last integral in (10.2) we have
1=-1, (10.5)
as already noted in (8.12), so that 13< a in this case also. We thus get (10.1) on the basis ofLemma 1.
For the analogue of (10.1) with r,F interchanged, we need a slight strengthening of (6.1).
We assume that
f '(u)F (u)/f2(u) = K + 0([f (u)]-)} , (10.6)
as u -+ c. We then have the following lemma.
LEMMA 3. As s -+ oo,
r(s) = (K/2)(1 +O(s))}Jds/F(u) . (10.7)1
As in Section 7, we prove first an exponentiated version (cf. (7.5)), namely,
rN/N = (K/2 + O(se))frN-1F1ds. (10.8)
For this we integrate (7.3) over (1,s) and get
rN/N(1 + O(r~ 1f)) = (K/2)rN-1F(1 + O(f~ +r 1F-1 + F-2 ))ds ,
where we have used the fact that
(f '/f 3)'= f''f3 - 3(f'/f2)2 =2((f7f2 )2) = O(1/F2 ).
It follows that, as s -+ co,
rN/N(1+ O(s~E)) = (K/2)rN-1F1( + O(s-))ds, (10.9)1
by the results of Theorem 4, and (10.8) follows on use of Lemma 1.
Following the lines of (7.6)-(7.7), we deduce from (10.8) that
-1+1/N
r N-1 F -1 r N-1 F -1 ds = {KN /2 + 0 (s~-c)1-1/NF~-,
15
as s -* oo. Integrating, we get
1/N s
N (rN-1F_1ds +0(1) = (KN/2+0(s~')}1-/NJds/F ,1
and here the O(1) term may be absorbed into the error-term on the right. Hence
s N
JrN-1F-lds = (KN/2 + O(s~))N-1 N-N(ssiFJ
We then substitute this on the right of (10.8) and obtain (10.7) on taking N-th roots. This proves
Lemma 3.
11. Asymptotic Formulae for F (s), r (s)
Retaining the new hypothesis (10.6). we now strengthen the results (8.7)-(8.8) of Theorem 4
and replace the asymptotic formula , for logF and logr by similar formulae for F and r them-
selves. Roughly speaking, we go though the argument of Section 8 and replace the relation "- ...as s -+ co" by the relation "= (1 + J (S-E)) x ...". Again, e is a suitably small positive number.
We prove the following theorem.
THEOREM 5. As s -4 co, for some constant D, we have
F(u(s)) = Ds(I -Y(K+N-1)( + O(s-E)}(11.1)
r (s) = ((K+N-1)/(2D))sK/(K+N-1)(l + 0(su)} . (11.2)
We use the construction (8.3) and derive (8.4). In place of (8.5) we have, using (10.1),
(10.7),
s -1
F(u(s)) = ((N-1)/2)+ O(s~1))J(K/2+ O(aE))-1'dt/F(u(T)) da
= ((N-1)/K)(1 + O(s~)4l3{+ 0(at)} [.dt/F (u@t))]dt . (11.3)
We now use Lemma I with
A (s) = 1/Jdt/F (u (t)), B (s) = s-A (s),1
together with (8.7), and deduce from (11.1) that
16
F (u(s)) = ((N -1)/K){[1 + O(s-)}j [1dt/F (u Ct))] dt
= ((N-1)/K)( + O(s-E))G (s). (11.4)
Hence, by (8.4),
G'(s)G (s)/G'2(s) = (N-1)/K + O(se).
The substitution H (s) = logG (s) now leads to
H'(s)/H'2 (s) = - (K+N-1)/(N-1)+ O(s-E)
and so on integration to
1/H'(s) = (K+N-1)s/(N-1)+ 0 (s1-).
Hence
H'(s) = (N-1)/((K+N-1)s} + 0(s~1-E
and so
logG (s) = ((N-1)/(K+N-1))logs + C + O(s-e), (11.5)
ass -+ c, for some constant C. We then get (11.1) with the aid of (11.4).
The companion result (11.2) can now be deduced on substituting (11.1) in (10.7).
We remark that we obtain an improvement of (8.12), namely,
r(s)F(u(s)) = (K+N--1)s1 + O(s-))
Acknowledgments
This paper incorporates work initiated during a visit to the Mathematics and Computer Sci-
ence Division, Argonne National Laboratory, September 1986 - July 1987 (host: Dr. H. G.
Kaper). Appreciation is expressed for this support and for numerous discussions with Dr. Kaper.
The paper develops complementary aspects of the papers by Atkinson, Peletier, and Serrin [1988]
and Atkinson and Peletier [1988]. The author is indebted to Professors Peletier and Serrin for dis-
cussions. Appreciation is also expressed for the continuing support of the National Sciences and
Engineering Research Council of Canada under Grant No. A-3979.
17
References
F. V. Atkinson and L. A. Peletier 1988. "Bounds for vertical points for prescribed mean curva-ture equations," Proc. 1986-87 Focused Research Program on "Spectral Theory and Boun-dary Value Problems," ANL-87-26, Vol. 4, Hans G. Kaper, Man Kam Kwong, and AntonZeal (eds.), Argonne National Laboratory, Argonne, Illinois.
F. V. Atkinson, L. A. Peletier, and J. Serrin 1988. "Ground states for the prescribed mean curva-ture equation: the supercritical case," Proceedings of the Microprogram on NonlinearDiffusion Equations and their Equilibrium States, Springer-Verlag, New York, 51-74.
M.-F. Bidaut-Veron 1986. "Global existence and uniqueness results for singular solutions of thecapillarity equation," Pacific J. Math. 125, no. 2, 317-333.
M.-F. Bidaut-Veron 1987. Private communication.
P. Concus and R. Finn 1975. "A singular solution of the capillarity equations, II: Uniqueness,"Invent. Math. 29, 149-160.
P. Concus and R. Finn 1979. "The shape of a pendant liquid drop," Phil. Trans. Roy. Soc. Lon-don, A, Math. and Phys. Sci. 292, 307-340.
J. Eells 1987. "The surfaces of Delaunay," Mathematical Intelligencer 9, 53-57.
T. K. Evers 1985. "Numerical search for ground state solutions of a modified capillary equa-tion," M. Sc. thesis, Dept. of Mathematics, Iowa State University.
H. A. Levine 1988. "Numerical searches for ground state solutions of a modified capillary equa-tions and for solutions of the charge balance equation," Proceedings of the Microprogram onNonlinear Diffusion Equations and their Equilibrium States, Springer-Verlag, New York.
18
Appendix
The following two figures illustrate the main (but not the only) type of graph encountered in
the theory of the equations (1.5)-(1.7). They are based respectively on the choices
f(u)=u3 , f(u)=-u3 ,
in both cases with N = 2. The first case (Figure 1) is that of the present paper, and the second
case is that of the paper by Atkinson, Peletier, and Serrin [1988], along with many other papers.
Other references are made at the end of the Introduction.
U
r
Figure 1. The Case f(u) >0, f'(u)>O0
19
r
Figure 2. The Case f(u) <0, f (u) < 0
20
BOUNDS FOR VERTICAL POINTS OF SOLUTIONS OF PRESCRIBEDMEAN CURVATURE TYPE EQUATIONS, I
F. V. AtkinsonDepartment of Mathematics
University of TorontoToronto M5S 1A1, Ontario
Canada
L. A. PeletierDept. of Mathematics and Computer Science
University of LeidenP.O. Box 9512
2300 RA LeidenNetherlands
Abstract
Conditions are obtained under which radial solutions of generalized capillary-type equations exhibit a first vertical point, together with bounds and asymptoticestimates for these points.
1. Introduction
There has been much interest in the equation
div Vu_ 2_+f(u)=0 (1.1)4l+ IVu 2
in RN, N 2, and particularly in radial solutions
u = u(r), r = lx I . (1.2)
For such solutions (1.1) may be replaced by the ODE
N-1 + r -
91+u '2 1+ f()=0 13
Historically, most attention has been given to the case in which f (u) is linear, which arises in
capillarity theory [Concus 1968, Concus and Finn 1979, and Finn 1986]. However, partly
inspired by studies on
Au+f(u)=0, (1.4)
and recent results on "critical exponents" (see, e.g., [Atkinson and Peletier 1986a, 1986b, 19881,
Senior Mathematician Emeritus at the Mathematics and Computer Science Division, Argonne
National Laboratory, October 1, 1986 - July 17, 1987.
21
and references given there), work has been initiated on (1.1-1.3) with more general functions
f (u); here f may be seen as prescribing the "mean curvature" for a surface of revolution. In this
paper we consider solutions regular at r = 0, though other types of solution are also of great
interest [Bidaut-Veron 1987, Vazquez and Vmron 1980].
The continuability and global behavior of solutions of (1.3) are affected not only by the non-
linearity in the term f (u), but also, and more crucially, by the nature of the nonlinearity in the
second-order term. This has the feat':e that the function of u' concerned, namely,
u'/(1+u'2)11 ,
can take values only in the interval (-1,1). The fact that this function has a bounded range, and an
unbounded inverse, gives rise to the phenomenon of interest in this paper, the occurrence of a
"vertical point" in the solution of the initial value problem
u(0) = y, u'(0) = 0. (1.5)
Under certain conditions, and for certain initial data y, u (r) can be continued as a C' -solution of
(1.3) only in a half-open interval [0,R) and will have the property that
u(r)-+U , I u'(r) l-+ o (1.6)
as r -+ R -0. Here U and R are finite.
Our aim is to study the existence and location of such points. During much of this paper,
we will actually consider a slightly more general equation than (1.3).
The phenomenon of a vertical point can occur in two senses:
(i) f (u) > 0 in the relevant neighborhood of y, the solution-graph (r,u(r)) bends down to a
vertical point, and u'(r) - -oo as r -+ R -0,
(ii) f (u) <0 near u = y (or f (u)> 0 and f (u) is to be replaced by -f(u) in (1.3)), the graph
bends upwards to a vertical point, so that u'(r) - +00 as r -4 R -0.
Here we give a unified treatment, applicable to both cases, obtaining sufficient conditions for a
vertical point to occur, and bounds for R and U.
The occurrence of vertical points is, of course, incompatible with the concept of a continu-
ously differentiable solution, and so can legitimately be interpreted to mean simply that the solu-
tion cannot be continued. Thus in one type of investigation one seeks the broadest criteria, on the
equation or on the initial height y, ensuring thai solutions can be continued without such points.
An alternative and perhaps more fruitful point of view is to ask for ways of continuing the
solution beyond a vertical point, possibly by means of some reparametrization. In at least two
possible forms of this continuation, the trajectory exhibits a succession of vertical points. In one
of these forms r emerges as a well-defined function of u with a sequence of maxima and minima,
whereas in the other the trajectory is self-intersecting, and r and u are well-defined functions of
22
the arc length (see [Concus and Finn 1979] for a detailed study of both forms in the case
f (u) = ku). The former type is the subject of the forthcoming paper [Atkinson, Peletier, and Ser-
rin 1988] on "ground states."
The present paper is confined to estimates of the first vertical point; these are naturally basic
to the study of the subsequent behavior of the solution; some such estimates were presented and
used in [Atkinson, Peletier, and Serrin 1988].
In Section 2 we list basic hypotheses regarding the equation, which appears in generalized
form in (2.1). Sections 3 and 4 are devoted to the initial value problem; further studies of such
problems have been given recently in [Kaper and Kwong 1988]. Sections 5 and 6 give estimates
for the first vertical point, along with sufficient conditions for its existence in the case of (1.3)
(slightly generalized) with f (u) > 0; Section 7 deals with this in the case when f (u) < 0. Sec-
tions 8-10 are devoted to asymptotics with large initial height y, with progressive specialization to
the original equation (1.3).
2. Basic Hypotheses
We shall view (1.3) as a special case of an equation
(p (r)q(u'(r)))'+p (r)f (u) = 0 , (2.1)
where, in the case of (1.3),
p(r) = rN-1 , q(v) = v/(1+v 2)%. (2.2-3)
Somewhat more general equations have been investigated in [Atkinson and Peletier 1988] from
the point of view of global behavior, more precisely regarding existence of "ground states." Here
our focus is more on the behavior up to a first vertical point.
We are concerned with an arc of the solution curve on which u (r) is decreasing, and so need
consider the behavior of q (v) only when v 5 0. For simplicity we assume q odd. We make the
hypotheses
Hq 1. q E C 1((--o,0)(0,o)) , q (0) = 0 ,
Hq2. q (v) = -q(-v) , q(v) > 0 for all v # 0 ,
Hq3. q(v)- 1 as v -+o,
Hq4. vq'(v) e L (0,oo) .
We shall write
C = Jvq'(v)dv0
23
Restriction to the value "1" in Hq3 does not involve a loss of generality; departure from the
oddness hypothesis for q would involve only minor complications. The hypotheses Hql-4 are,
for example, satisfied in the case
q(v) = v Iv Im-1(l+v 2 )-f 2 (2.4)
if m > I; the case m = 1 is that of (2.3).
Concerning p we assume
Hpl. p e C1 [O,oo), p(0)=0,Hp2. p'(r)>0 for r>0,
Hp3. For constants K, L, with 0 < K 5 L < 1, we have for all r 0
r
Kp 2(r):5 p'(r)fp(s)ds 5Lp 2(r) . (2.5)0
For example, in the case (2.2) we have
K = L = 1--. (2.6)N
Our hypotheses for f (u) are limited to a u-interval E within which the vertical point is
expected to occur. We write
T = Y[0,] ,(2.7)
where yo remains to be chosen, as a function of'y, and assume
Hfl. f>OonE,
Hf2. f satisfies a uniform Lipschitz condition on Z.
The behavior of f outside E will be irrelevant.
3. The Initial Value Problem
We consider first the existence of a solution of (1.5), (2.1) in a neighborhood of r = 0. As
usual, we start by assuming such a solution to exist, derive an integral equation, and claim that
the latter has a solution. Integrating (2.1) formally we get
qJ(u'(r)) =- 1rp (s)f (u(s))ds , (3.1)
at which point we must introduce the inverse function Q of q. We have
Q: (-1,l)-+ (-o,o), Qe C(-l,1l)r C1((-1,O)u(0,1)). (3.2)
24
We introduce the notation
J(r) = l p(s)f(u(s))ds (3.3)p(r) o
and then have from (3.1) that
u'(r) = -Q (l(r)) , (3.4)
provided that I(r) is in the domain of Q, that is to say,
-1 < 1(r) < 1 . (3.5)
Integrating (3.4), we get
u(r) = y_-JQ (I(s))ds . (3.6)0
Assuming, as indicated above, that f satisfies a Lipschitz condition in a left neighborhood of y, wehave that (3.6) defines a contraction mapping over a suitably small interval, with as consequences
the existence and uniqueness of a solution; if, of course, we assumed only the continuity of f, we
could deduce existence without uniqueness.
We can then use (3.4) to continue the solution, with the aid of an analog of (3.6), or by stan-
dard results on initial value problems, so long as (3.5) remains in force and u(r) remains in E.
In view of Hq2 we shall have
0 > u'(r) > - (3.7)
so long as (3.5) holds, along with our hypotheses on f. We denote by r0 e (0,ooJ the largest
number such that on (0,r0 ) we have
u'(r)> -oo, u(r) E E . (3.8)
Here u must necessarily tend to a limit as r - r0 , which we denote by u0. The situation must
fall into one of the following three cases:
Case !. r0 = oo.
Here (3.7) is in force for all r > 0, and u (r) decreases monotonely to some limit in [y0,yJ.
Case ll. r o0< oo and uo= yo.
Here the trajectory (r,u (r)) terminates at the lower boundary of the half-strip
[0,oo x [YoY],(3.9)
possibly with a vertical tangent.
25
Case!!!. r0o<co anduo>yo.
Here the trajectory terminates in the interior of (3.9). In this case, that of main interest in
this paper, we must necessarily have
lim inf u'(r) = (3.10)r-sr,
or equivalently
lim sup!(r)= 1 . (3.11)r-4Tg
4. Estimates for Continuation
We now investigate (3.3), (3.6) in more detail. We need the additional notation
fm = min f (u), fm = max f (u) (4.1)
over u e ,and
r
J(r)= !'(r)= f (u(r)) - p'() Jp(s)f(u(s))ds. (4.2)p2 (r)o
We note that, so long as u(r)E EE,
J(r) 2f.- -IMfM, (4.3)
J(r)Sfu -Kfm. (4.4)
As a preliminary to the main result we note the following lemma.
LEMMA 1. Let
fm-LfM>0. (4.5)
Then Case I is excluded.
Proof. It follows from (2.4) and (4.2) that, so long as u (r) e
J(r) f,, -LfM,
and so
I(r)Her(fe-JIM).
Hence
26
(4.6)
A slightly stronger condition will eliminate Case II. We need the notation
1 G
C = JQ(y)dy = Jxq'(x)dx. (4.7)0 0
For example, in the prescribed mean curvature case one has
Q (Y) _= Y ,i -y 2
and so
C = 1 . (4.8)
We have then the following lemma.
LEMMA 2. Let
(Y-Yo)(fm-LfM) C . (4.9)
Then only Case III can hold.
Proof. Since (4.9) implies (4.5), Case I is excluded by Lemma 1, so that it remains to exclude
Case II. Supposing Case II to hold, we have on making r -+ ro in (3.6) that
~0 C CY-Yo = JQ ((s))ds < m f, C , (4.10)
0 mini fg - LfM
since
I' = J 2 fm - Lf,,, Z0.
This contradicts (4.9), and so proves Lemma 2.
5. The Main Result
It follows from Lemma 2 that if (4.9) holds, then there are numbers
Re (0,oo), U e (Yo,y) (5.1)
such that the solution of (1.5), (2.1) satisfies
0 > u(r) > -co , (5.2)
Y > u(r)> Yo (5.3)
27
for 0 < r < R, and
u'(r) - -co, u(r) -+ u0 E ( 0 ,y) as r -+ R . (5.4-5)
Our main result provides bounds for these numbers.
THEOREM 1. Let (4.9) hold. Then there exist numbers R and U as in (5.1)-(5.5). They satisfy the
inequalities
1 R 5 , (5.6-7)fM - Kfm fm - LfM
C CY-mCf 5U5 Y- .f (5.8-9)
fm - LIM fm - Kfm
Proof. The existence of R and U having been established by Lemmas 1 and 2, we go to the proof
of (5.6)-(5.7). We have now I(R) = 1, and
I'(r) fm - Kfm . (5.10)
Since 1(0) = 0, this proves (5.6). The proof of (5.7) is similar and has, in essence, been given in
the proof of Lemma 1. To prove (5.8)-(5.9), we note that
R 1y - U = JQ (I(s))ds = JQ (1)/. (5.11)
0 0
We then get the result on using the bounds (4.3)-(4.4) forJ = I'.
6. Corollaries
Though the condition (4.9) for the trajectory to have a vertical point is reasonably simple, it
seems desirable to derive more explicit versions, which are only slightly more restrictive. These
are based on the remark that
f -fm5IIf 'ds,
with equality in the case that f is monotone. We have then the following theorem.
THEOREM 2. For some a > 0 take
E= - , Y (6.1)f (Y)
and let
28
(1-L- -)f(Y) 2fIf'Ids . (6.2)a
Then there exist numbers R and U satisfying (5.1)-(5.5) and also
IS RS a ,(6.3)(2-K -L -C/a)f (y) C(y)
Y- a C U - . (6.4)f(U) (2-K-L-C/a)f (y)
Proof. We have in this case
Yo =Y- O , (6.5)f ()
and
f- L = (1-L)fM -(fM-fm)
z(1-L)f(y)-JfIf 'Ids
CZ --- f (Y),(6.6)
by (6.2). From (6.5)-(6.6) we conclude that (49) is satisfied.
We get the upper bound for R and the lower bound for U from (5.7)-(5.8) and (6.6). For the
remaining bounds we observe that
fM - Kfm = (1-K)fm + (fMfm)
S (1-K)f (Y)+ f If 'Ids
5 (2-K- L - )f (y) (6.7)
and use (5.6) and (5.9).
Specializing furher, we have
JIf'IdsS a sup If'IE f (Y)
and so we may replace (6.2) as a sufficient condition by
asup If '1 5(1-L--)f 2(Y). (6.8)a
Two distinct ways of achieving this are (i) to assume that
29
0:5f '(u)5fl -L- )f 2 (u)' (6.9)a a
for u e E, or (ii) to assume that (6.9) holds for u = y, along with
f'>0, f"''0foruE . (6.10)
In one way of using (6.8) or (6.9), we choose a so ,s to get the weakest restriction on
f '(y)/f 2(y) which ensures, together with our other assumptions, the existence of a vertical point.
In particular, this would enable us to catch as many cases as possible of the noncontinuability of a
solution in the classical sense. This is given by choosing a so as to maximize the coefficient on
the right of (6.9), which is to say by choosing
2C (6.11)a -L'
in which case (6.8) is equivalent to
sup I f ' 1 1L) f2 (y), (6.12)4C
where the supremum is taken over the interval
Y - 2C , Y .(6.13)
(1-L)f (y)
For example, in the case (2.2)-(2.3), (6.12) takes the form
sup If 'I 12 f 2(y). (6.14)4N
A rather similar condition appears in [Serrin 1987] as a sufficient condition for noncontinuability.
However, it is more in the spirit of this paper to choose a so as to narrow the gap between
the upper and lower limits on R and U in (6.3)-(6.4) by taking a close to C/(1-L). We may do
this in an asymptotic sense if, as is commonly the case,
f(y)/f 2 (y)_-+0 as y-+o0. (6.15)
We pursue this approach in Sections 8-10.
7. The Reversed Sign Case
For want of a better name, we use the term "reversed sign" case to refer to the behavior of
solutions of
(p(r)q(u'(r))}'-p(r)g(u) = 0, u(0) = y, u'(0) = 0, (7.1)
where p and q are as previously, and g is positive in a neighborhood of y. This can be brought
30
within the scope of the previous analysis by the substitutions
v=2Y-u, f(v)=g(u). (7.2)
This leads to the problem
(p(r)q(v'(r))}'+p(r)f (v) = 0, v(0) = y, v'(0) = 0, (7.3)
where f (v) is positive in a neighborhood of v = Y.
Thus, translating Theorem 2 into this setting, we assume that for some a> 0, g (u) is posi-
tive and uniformly Lipschitz in the interval
and that
Jgds 5(1 - L- -)g (y) . (7.5)E. a
Then the solution of (7.1) can be continued in (0,R) with u'> 0 and with u'(r) -+ , u(r) -+ U
as r -> R - 0, where
a 15R 5 (7.6)Cg (y) (2-K-L--C/a)g (y)
and
a CY+ a U SY+ . (7.7)
g(Y) (2-K-L-C/a)g(y)
This is the situation referred to in (ii) of Section 1, in which the trajectory (r, u (r)) "bends
upwards" to a vertical point.
In cases of current interest, we have for some c that
g ErC' [c, oo) , g (u) > 0, g'(u) 2!0 for u Z c . (7.8)
For this situation a simpler result holds, in which the existence of a first vertical point does not
depend on additional quantitative hypotheses. This involves a modification of the arguments
used for Theorems 1 and 2.
For any y c, we consider the initial value problem (7.1) and apply the transformation (7.2),
obtaining the problem (7.3). In this case we now have
f (v)>-0, f '(v)50, for v <y. (7.9)
While Theorems I and 2 are available in this situation, the argument can be improved. Since
f (v) is nondecreasing as v decreases, we can replace (4.3) by
J(r) (l-L)f (v (r)).(l-L)f (y) , (7.10)
31
so long as v'(r) remains finite, and replace the expression f,-LfM by (1-L)f (y) in the sequel. In
particular, the condition (4.9) of Lemma 2 may be replaced by
(t-o)(1-L)f(y) C, (7.11)
and this is certainly true for fixed y if 'Y-Yo is sufficiently large, so that the solution has a vertical
point.
Making corresponding modifications in the inequalities of Theorem I and translating the
results back into terms of the problem (7.1), we obtain the estimates given by the following
theorem.
THEoREM 3. Let (7.8) hold, and let y c. Write
1Ti =1Y+ .(7.12)
(1-L)g (y)
Then the solution of (7.1) reaches a vertical point (R,U), where
__ __ __1
1 5 R 5 , (7.13-14)g (Y1)-Kg (y) (1-L)g (y)
C C7+ CK USY7+ .(7.15-16)
g (Y1)-Kg (Y) (1-L)g (y)
These bounds correspond to (5.6)-(5.9) of Theorem 1.
8. Asymptotics: First Approximations
We now use the estimates of Sections 5 and 6 to obtain asymptotic approximations to
R and U with a partial specialization of the problem (2.1). Existence of R and U will now depend
on the initial height y being "sufficiently large."
We set p (r) = rN- 1 as in (1.3), but retain the general form of q (u'), the resulting problem
being
(rN-1 q(u'))'+rN-1f(u)=O, u()=Y, u()=. (8.1)
In accordance with (2.5) we take
K = L = (N-1)/N. (8.2)
With the full specialization to (1.3), we would also take C = 1.
We shall consider also the problem (7.1), in which the solution curves bend upwards.
For the case of (8.1) we need to assume that, for u c say,
32
f>0, f">0, f "O (
and that
f '(u)/f 2(u) -40 as u -+oo. (8.4)
With these hypotheses we have for this case the following theorem.
THEOREM 4. For large y, R and U exist and satisfy, as y -+ 0o,
R =N 1N J ,(8.5)f (Y) f 3(Y)
U= y- C f() + 0 .((8.6)
Proof. We use the sufficient condition (6.8), in the formulation that (6.9) should hold for
u = y > c, along with (6.10). We assume y so large that
S'( ) 1 1 (8.7)f 2(y) 4CN2
and also so large that
y-2C N > c . (8.8)f (Y)
Putting u = y and L = 1 - (1/N) in (6.9), we may rewrite the result as
a2f (y)2 -- N+C< 0. (8.9)
Solving this for equality, on the basis of (8.7), we take
a=2C N1 + (1-4N 2Cf'(y)/f 2(,Y))"%(8.10)
In particular, we have a < 2CN, so that (8.8) will ensure that (8.3) is available in the relevantinterval E. Thus the requirements (8.3), (8.7), and (8.8) ensure the existence R and U.
It remains to prove the estimates (8.5)-(8.6), which we deduce from (6.3)-(6.4). From
(8.10) we have
a=CN+0 ( as y-+ oo (8.11)f 2at)
and also in this case
33
(8.3)
2-K-L - +Oas y-+oo, (8.12)
and these yield (8.5)-(8.6). This completes the proof.
We now consider from the asymptotic point of view the "reversed sign" case of (7.1) with
g (u) > 0, g'(u) 0 , g'(u) z O, for u > c , (8.13)
and y c. Here the existence of R and U is ensured by Theorem 3, and the corresponding bounds
are
N <-R 4 N , (8.14)g(y)+N(g(y 1)-g(y)) g(y)
N Ny+ C 5U S Y+ C , (8.15)g (Y)+ N(g (yj) - g()) g (Y)
where yi = y + N /g (Y). Assuming then that
'(u)-+O0as u -+ o , (8.16)
92(u)
we deduce from (8.14-15) the following theorem.
THEOREM 5. Under the hypotheses (8.13), (8.16), we have as y -4 c,
R = N (1+0(1)), (8.17)g(7)
NU = y+ C (1+o(1)). (8.18)
g (Y)
The error term here could be reduced if (8.16) were strengthened.
9. A Second Approximation to U
Higher approximations to the first vertical point appear to be needed for the investigation of
the second such point. We develop this aspect for the case (1.3), so that we now take C = 1, in
addition to (8.2). We assume the conditions of Theorem 4, so that R and U exist, and satisfy the
asymptotic formulae (8.5)-(8.6). We make the further assumption that
f() -+ 0 as u -+ . (9.1)f (u)f '(u)
The improved version of (8.6) is the following theorem.
THEOREM 6. Let (9.1) hold, as well as (8.3)-(8.4). Then, as y -+ ,
34
Y-U =NANL{3+1+o(1)), (9.2)f(Y) f 3(y)
where
AN = N3 12- (N-1) f -N-t 1N+1_2)- (9.3)
We remark that it is easily shown that AN > 0.
Proof. We have, with the notation (4.2),
1_-_U==_ds(=! .(9.4)2
-1 0 r112
In the present case p (r) = rN- 1, and (4.2) takes the form
r
J(r) = f (u(r)) - (N-1)r -N N-f (u(s))ds . (9.5)0
In particular,
J(r)-+ .as r-+0. (9.6)N
Integrating by parts in (9.4), we get
R
Y - U N= f I-ds . (9.7)f(Y) 0 J
Next we calculate 1(r) and insert the result in (9.7). We have
r
J'(r) = f '(u (r))u'(r) - (N-1)r-'f (u (r)) + N (N-1)r 1 JsN-1f (u (s))ds . (9.8)0
By an integration by parts, we can replace this by
r
J'(r) = f '(u(r))u'(r) - (N-1)rJ-Nf-N (s))u(s)ds.(9.9)0
Substituting in (9.7) and using the fact that
we have
35
R R s
y- U - N f'J-2ds - (N-1) J-24 Ts-N-1ds JNf'1 dt (9.10)(Y) 0 o o
= S -S2,
say. We proceed to estimate S1 and 82 separately; both of them will have the asymptotic form
const. 1 j(1+o(1)) . (9.11)f 3(Y)
By Section 6, for large y the u-values concerned lie in
Yo:=Y- 2 f u Sy, (9.12)
and we claim that for such u
f (u) = f (Y)(1+o(1)), (9.13)
f '(u) = f '(y)(1+o(1)) . (9.14)
The first follows from (8.4). We have
log f (Y) - log f (Y0) = if '1duYo
= o JfduI =o(1),Y
by (8.4). We deduce (9.14) in a similar way from (9.1). It then follows from (9.5) that
J(r) = 4.1(1+o(1)) . (9.15)
Hence
S1 = Jf J-1 dl_~ N 3.f .(9.16)S2 3 ((Y)
In the iterated integral in 62 we make the changes of variable
a =1(s), 'r=I(t). (9.17)
This gives
36
S2 = (N-1)fJJ -3 I-\ 2sN-1-daJ1 Nf- dt.(9.18)0 0 I1-12
We then use (9.13)-(9.-15), together with the consequence that
a-L s, t-f t. (9.19)N N
Substituting these results in (9.18), we get
$2 f(N-1)N3 ( ,(9.20)
where SN denotes the definite integral appearing in (9.3), that is to say
=SN = 5a-N1(g_g2)d N+1(1 _t2)-%d
00
This completes the proof of Theorem 6.
10. A Second Approximation to R
Continuing with the case of a solution "bending downwards" to a vertical point, we givethe companion result to Theorem 6.
THEOREM 7. Let (8.3)-(8.-4), (9.1) hold. Then, as y4 oc,
R=( + NBN 01+0(1)), (10.1)f (Y) f 3(Y)
where
BN = N2 (i - oN+1(l 2-'daJ. (0.2)
As in the case of AN we remark that BN > 0.
Proof. Putting r = R in (3.1) and changing the sign throughout, we get
R
1 = R1~NsN-1f(u(s))ds (10.3)0
R R 1-NR=W--f (U)- sNf (u)u'ds.
Since u' = -J(1_- 2 ', this may be rewritten in the form
37
R N _RlMN 4R= N -- R f'u) I d.(10.4)f(U) Jf(U) f u
We first replace U by y in the term N/f(U), and proceed to estimate the resulting error.
Since
U = (- N 1+o (1)), (10.5)f (Y)
by (8.6), we have by (9.13)
f (U) = f (y) -N ( )1+o(1)). (10.6)f (Y)
Hence
SN + N2 N (1+0(1)) . (10.7)f (U) f (Y) f 3(y)
It remains to find an approximation for the last term in (10.4). We make the change of vari-
able I(s) = a, and use the asymptotic results
Rf~ N f(U) -f(Y), f '(u)~ f'(Y), ~ ,f( (10.8)f(Y) da f(Y)
already discussed in Section 9. These yield, together with (10.7), the result of Theorem 7.
For the problem discussed by Concus and Finn [1979], in which
N=2, f(u)=u,
Theorems 6 and 7 yield
U_=y- 2K- 4+8iog21 1( {+o(1)},
2 4R = -+-{1+o(1)),Y 3?
as y -* oo. This is consistent with the results obtained in [Concus 1968] by formal methods.
Very similar results to Theorem 6 and 7 hold for the case (7.1) of solutions bending up to a
vertical point. We assume now that
g E C2 [c,oo), g >0, g'> 0, g"0, (10.9)
and also
-2 -0 , -+0 (10.10)g (gg)
as u -+ oo. The existence of U and R is then assured, and we have the following theorem.
38
THEOREM 8. Asy-+oo,
U ='y+ N +AN (1 +0(1)), (10.11)ge(y) +Ag3()(lOl)
R = - BN (1 +()).(10.12)g(y) Ng3(y)
Instead of using a change of variables to transform this to the previous case, it seems
simpler to repeat the argument with the appropriate minor modifications, indicated briefly in what
follows. Thus we now write
I1(r) = r1 JsN-1g(u(s))ds . (10.13)0
We are concerned with a maximal interval [0,R) in which
01 1(r) < 1 (10.14)
and have there
, 1u= 1 ,2 (10.15)
so that
R 1
U-Y=J ds . (10.16)
To obtain the result corresponding to Theorem 6 we use the argument of (9.4)-(9.20) to
(10.16), with the aid of the function
r
J1 (r) = I1 '(r) = g(u(r))-(N-1)rNJsN-1g(u(s))ds. (10.17)0
In analogy to (9.15) one has
J1(r) = -- (1+0(1)) (10.18)
in the relevant interval as y -+ 'o. Also, in analogy to (9.9),
J 1'= g'(u)u'- N-1 r-N-1sNgu(s))u(sjs. (10.19)N 0
We then integrate by parts on the right of (10.16), inserting factors J1 , 1/JI, as in (9.4), (9.7), and
insert the expression (10.19) for Ji ', to obtain the analog of (9.10). We get as a result
39
U = y+ N +AN .3 (1+o(1)) as y -+oo . (10.20)g(Y) g (y)
Here AN has the same meaning as in (9.3).
In the case of R we have in place of (10.4) the result
R=N R -N -R = N + R1U JsNg(u)I(1)'ds . (10.21)g(U) g o(U)
Here we use (10.20) (or (8.18)) in approximating to U on the right of (10.21), and use analogs of
(10.8) in approximating to the integral. We get
R = g)-BN83(Y)+ ()) .(10.22)g (y) - g3(y) 1o1)
For cases of these asymptotics when f (u), g (u) are linear we refer to [Concus 1968,
Concus and Finn 1979, and Finn 1986].
Acknowledgments
The present paper, written at Argonne National Laboratory, develops a special aspect of thejoint paper with Serrin [Atkinson, Peletier, and Serrin 1987] and owes much to numerous conver-
sations with Prof. J. Serrin and Dr. H. G. Kaper. Appreciation is expressed for the support and
facilities of Argonne and also for the continuing support of the National Sciences and Engineer-
ing Research Council of Canada under Grant No. A3979.
References
F. V. Atkinson and L. A. Peletier 1986a. "Emden-Fowler equations involving criticalexponents," Nonlinear Analysis, Theorem, Methods and Applications 10, 755-776.
F. V. Atkinson and L. A. Peletier 1986b. "Ground states of -Au = f (u) and the Emden-Fowlerequation," Arch. Rat. Mech. Anal. 93, 103-127.
F. V. Atkinson and L. A. Peletier 1988. "On non-existence of ground states," Quart. of Math.39, 1-20.
F. V. Atkinson, L. A. Peletier, and J. Serrin 1988. "Ground states for the prescribed mean curva-ture equation: the supercritical case," to appear in Nonlinear Diffusion Equations and TheirEquilibrium States, ed. W.-M. Ni, L. A. Peletier, and J. Serrin, MSRI Conf. Proc., Springer-Verlag, New York.
40
Marie-Francoise Bidaut-Vron 1987. "A singular solution of a quasilinear partial differentialequation," preprint.
P. Concus 1968. "Static menisci in a vertical right-circular cylinder," J. Fluid Mech. 34, 481-495.
P. Concus and R. Finn 1979. "The shape of a pendent liquid drop," Phil. Trans. Roy. Soc. Lon-don A., Math. and Phys. Sci. 292, 307-340.
R. Finn 1986. Equilibrium Capillary Surfaces, Springer-Verlag, New Yodk.
H. G. Kaper and M. K. Kwong 1988. "Uniqueness results for some nonlinear initial and boun-dary value problems," Arch. Rat. Mech. Anal. 102, 45-56.
J. Serrin 1987. "Positive solutions of a prescribed mean curvature problem," Lecture Notes inMathematics, to appear.
J.-L. Vazquez and L. Vdron 1980. "Removable singularities of some strongly nonlinear ellipticequations," Manuscripta Math. 33, 129-144.
41
Appendix: Some Numerical Constants
For convenience we list here the values of the constants AN and BN which feature in
Theorems 6, 7, and 8.
42
N AN BN
2 4(1 + 21og 2)/3 4/3
3 9 9(1- 3,x /16)
4 16(3+ 41og 2)/5 112/15
5 4825/144 25(1 - 5 i /32)
LARGE SOLUTIONS OF ELLIPTIC EQUATIONSINVOLVING CRITICAL EXPONENTS
F. V. Atkinson*Department of Mathematics
University of TorontoToronto M5S 1A1, Ontario
Canada
L. A. PeletierMathematical InstituteUniversity of Leiden
P. O. Box 95122300 RA Leiden
Netherlands
Abstract
We consider positive radial solutions of the equationN+2
-Au=XuQ+uN-2 1 N+2
in the unit ball B in RN, which vanish on the boundary. For c > 0, let X(c) andu(-,c) denote the eigenvalue and eigenfunction with the property that u(0,c) = c.In this paper we obtain precise estimates for X(c) and u (-,c) as c tends to infinity.
1. Introduction
We consider the problem
-Au = XU q + Up in S2
(P) u>0 in(Gu=0 on3(i2,
where 0 is a bounded domain in RN (N > 2) with smooth boundary Al, X E R and 1 < q <p.
If p (N+2)/(N-2) and ? 5 0, Problem P fails to have a solution if i2 is star shaped [Poho-
zaev 1965]. In this paper we shall study the critical borderline case
N+2N-2
It was shown by Brezis and Nirenberg [1983] that in this case, Problem P does have a solution for
certain positive values of X. Specifically they prove the following results:
Senior Mathematician Emeritus at the Mathematics and Computer Science Division, Argonne National
Laboratory, October 1, 1986 - July 17, 1987.
43
Case I: q = 1. If N z 4,Problem P has a solution if and only if .e (0, X), but if N = 3, it has a
solution if and only if I.E (I*,X). Here 10 denotes the principal eigenvalue of -A with Dirichet
boundary conditions in f, and X* some number in (0, 0 ), which equals I/4 if 0 is a ball.
Case II: q > 1. If N -4, Problem P has a solution for every a > 0, but if N = 3, then
(a) if 3 < q < 5, there exists a solution for every a,> 0;
(b) if 1 <q q 3, there exists a solution if I is sufficiently large.
Subsequently, in [Atkinson and Peletier 1986], Problem P was studied in further detail
when (z is a ball BR, centered at the origin and with radius R, and hence solutions have radial
symmetry [Gidas, Ni, and Nirenberg 1979]. In this situation there was no need to restrict N to
integer values. It was shown that if
6-N2<N<4 and 1<q<6 ,N-2
there exists a constant A > 0 such that if A.> 5, Problem P has at least two solutions, and if A <
Problem P has no solutions.
This result was established as a corollary of a detailed study of the properties of the bifurca-
tion diagram associated with Problem P for different values of q and N. Indeed, let X(c) be a
value of the eigenvalue A for which Problem P has a solution in the ball BR, such that u (0) = c.
Then the following properties of X(c) were established.
PROPOSITION 1. Suppose q = 1. Then there exist positive constants I and I, (o > XI) such
that
(i) X(c) <X4 for all c >0 and X(c) -4 0 as c -+0;
(ii) if 2 < N < 4, then X(c) >Xa,, for all c > 0 and X(c) -+ ,, as c -4 c;(iii) if N ? 4, then A(c) -+0 as c -oo.
REMARK. The number 0 is the principal eigenvalue of -A with Dirichet boundary conditions; if
N = 3, Al is the principal eigenvalue of the problem
-Au = Au and u > 0 , in BR-(0)
u=0 onaBR
u(x) -(1/ Ix I) -+ 0 as x -+ 0 .
44
In fact, if R = 1,X0 =x2 and i=x2/4 in this case.
PROPOSTION 2. Suppose I < q < p = (N+2)/(N-2). Then,
(i) if 1 < q < 2/(N-2), X(c) ~ c q~1 as c - 00;
(ii) if q = 2/(N -2), X(c) ~=c-(N-4 Y(N-2)logc as c -+ 00;
as c -c 0.
Here we mean by f (x) ~ g (x) as x -+ oo that the quotient f (x)/g (x) is bounded above and below
by positive constants if x is large enough. The results of Propositions I and 2 are sketched in Fig.
I and Fig. 2.
I I
I IN4 I
I II N=3I I
I II II II II I
.y7 /4 2x
Figure 1. q = 1.
C
q=4 q=3
q=2
x
Figure2. q> 1, N=3.
For the case N = 3 some of these results were independently obtained by McLeod [19861.
The object of this paper is to obtain precise estimates for the behavior of the eigenvalue
X(c) and the solution u(-,c) of Problem P as c -+ 00. This study was partly motivated by the fol-
lowing conjecture of Brezis [1986b]:
45
(iii) if 2/(N -2) < q < p, X(c) ~ c p-q-2
Suppose N = 3 and q = 3. Then
lim (c)= 81c - _ R
where R is the radius of the ball.
Our main results are formulated in the following two theorems. A special role is played by
the solution ( ,4) of the eigenvalue problem
A$+ $ =0, $0> 0 in B 1-0)(E) $= 0 on B 1
fix - CN as x -40
Ix) N-2 --*0
where
CN = {N (N -2))(N-2Y2
If 1 q < 2/(N-2), this problem has a unique radially symmetric solution.
THEOREM A. Let ((c),u(-,c)) be a solution of Problem P in which S = B1 , such that
u(0,c) = c.
(a) If 1 <q < 21(N -2), then
c 1-q X(c) -+ sas c -+oo,
where p. is the eigenvalue in Problem E.
(b) If q = 2/(N-2), then
c(N -4)/(N -2)logC X(c) - N-2 cN as c -4 o.
(c) If 2/(N -2) < q < p, then
c4P+ 2X(c) -+ L(N,q) as c -+ ,
where
r ( +1 N22L(N,q) = N -
S p-qI qN --2 J22
Here p = (N+2)/(N-2), and r denotes the gamma function.
The next theorem provides a description of u(-,c) as c - oo.
46
THEOREM B. Let (&(c),u (-,c)) be a solution of Problem P in which f2 = B 1 , such that
u (0,c) = c.
(a) If 1 < q < 2/(N -2) then for every x e B 1 -[0},
cu(x,c) - $(x) as c -+ o,
where 0 is the eigenfunction of Problem E. The convergence is uniform on annuli
{c I-1/(N-2)<IXI<1.
(b) If 2/(N-2):5 q < p, then for every x e B 1-(0},
cu(x,c) -+ cN(II x -(N-2)-l) as c -+ ,
uniformly on annuli ((1 + a c-2- )~/-~))-l( < 2) < lxI<1), a> 0.
An important ingredient in the proof of Theorems A and B is an explicit upper bound forsolutions of Problem P. Since it can claim some interest in its own right, we formulate it below.
Consider the problem
r-Au = f (u), u > 0 in BRU = 0 on aBR
in which f E C ([0,oo)) n C1(0,oo), f (0) = 0 and f (s) > 0 when s > 0.
LEMMA C. Let u be a solution of Problem P* and f have the property
N+2-f (s ) - sf '(s )>-0for all s z0.
N -2
Then
u (x) <-w (x,u(0)) in BR ,
where
w(x,c)=c 11+ cf4(c)I }12 2N (N-2)
It is easily verified that the function w is the solution of the problem
Aw+c-pf(c)w'=0, w >0 in RN
w(x)-4c as x-+0,
in which p = (N+2)/(N-2).
REMARK. The method we use in this paper to discuss the properties of solutions of Problem P
47
can also be used in connection with the problem
-Au = Ix S(u1+3P), u > 0 in BR
u = 0 on aBR ,
provided s < 2. Then the critical power is
N+2-2sN-2
and the values of q that correspond respectively to 2/(N-2) and (6-N)/(N-2) become
2-s 6-N -2sand.
N-2 N-2
(See also [Ni 1982] and [Atkinson and Peletier 1988].)
2. Preliminary Estimates
Let s2 be the unit ball B1. Then solutions of Problem P are radially symmetric, and we may
substitute u = u (r), r = Ix 1. This yields the two-point boundary value problem
u'+ N-1 u'+ku +u =0, 0<r<1r
(I) >00 r<1u'(0) = 0, u(l)= 0,
primes denoting differentiation with respect to r. We can eliminate ?, by appropriately scaling r
and u. Setting
p = %(P-'y2(P-q)r = k-2y(-q)r (2.1)
u(r) = ( -q)v(p), (2.2)
we obtain
N-2.v'+N v'+v4 +vP =0, 0 < p < R
v >0 0<-p <R (2.3)v'(0)= 0, v(R) = 0,
where
R - X(p-1IY2(p-q) - L(k-2Y(p-q) (2.4)
and primes now denote differentiation with respect to p.
A second transformation turns (2.3) into an Emden-Fowler type equation. Introducing the
variables
48
N-2
t= N-2t=,y()=v(P) , (2.5)
we arrive at the problem
()y'+t-kf(y)=0, y >0 on (T,oo)Y(T)=0, y'( )=0,
where
k = 2 N-1N-2
f(y) =y +yP (2.6)
and
]N-2
T = (N-2 = (N-2)N- 2X-2(p-).(2.7)
As in [Atkinson and Peletier 1986a, 1986b] we study Problem II by means of a shooting
technique. Replacing the boundary condition at t = T by one at infinity, we shall study time prob-
lem
( r11) y''+ tkf (y) = 0 (2.8)y (t) -+ y as t -+oo , (2.9)
assuming that k > 2, y > 0 and that f satisfies the hypothesis
(H) f E C([0,oo)) n C'(0,oo) , f (0) = 0 and f (u) > 0 if u > 0.
The assumption that k > 2 guarantees that there exists for every y > 0 a unique solution,which we denote by y (t, y), provided t is large enough. It decreases as t decreases and reaches
zero at some nonnegative value T of t
T(y) = inft > 0: y (-,y) > 0 on (t, o)}.
Both y (t, y) and T (y) depend continuously on 1.
The main thrust of this paper will consist of deriving asymptotic estimates for T(y) and
y (t, y) as y -+ * . The identities (2.2) and (2.7) then enable us to translate these estimates into
corresponding ones for X(c) and u (x,c) as c -+ o.A central role is played by the following upper bound for y (t, y), proved in [Atkinson and
Peletier 1986b].
LEMMA 2.1. Let y (t, y) be the solution of Problem III in which f satisfies (H) and has the pro-
perty
49
sf'(s):5(2k-3)f1(s) for all s > 0 .
Then
y(t,y)<z(t,y) for allit T(y),
where
1-i(-2)
z (t, y) = y t 1tk2 + (2.11)k -1 Y
REMARK 1. The function f defined by (2.6) satisfies (2.10).
REMARK 2. The function z is the solution of the problem
z''+ t-k~-2kf (y)z2k- 3 = 0 0 < t <0oo0, (2.12)
z(t,y)>'0 0<t<oo,
lim z(t,) = Y.(2.13)
A number of properties of the function z are listed and proved in the Appendix.
REMARK 3. It will be convenient to introduce the number
Tk(Y) = -i 1 (k-2)
k-1 Y
At t = To the two terms in the denominator of z are equal, and
z(To,Y) = 2~1(-2).
Observe that because of the specific form (2.6) of f,
To(y) = k Y2 [1 I+0(1)] as y-+ oo .
The upper bound of Lemma 2.1 can be used to derive a lower bound for y. This is done in
the next lemma.
LEMMA 2.2. Lety (t, y) be the solution of Problem III in which f is given by
f (y) =y ;+ y 2-3
(a) If 1< q < k-2, then
1 zq-1(t Y)t2-y (t,y) > z (t,y) [ 1 - ' .tyt~
1=-2k+ =(k-1-q)(k-2-q)
(b) If q = k -2, then
so
(2.10)
y (t, y) > z(t, y) - k- #2-log(l+(To/t)k-2}]
(c) If k-2. < q < 2k -3, then
Y (t, y) > 2qz (t, y) - k-k12 2 +
Proof. If we integrate (2.8) twice, and use (2.9), we obtain for t > T,
y(t) = y- J(s-t)s-f(y (s))ds
> y - J(s-t)skf(z(s))dsI
= y- J(s-t)s-kz-3(s)ds0
- J(s-t)s-kzq(s)ds . (2.14)r
If we integrate (2.12) twice and use (2.13), we obtain
z (t) = y - (1+ ~2k+q)f (s -t)z2k-3(s)ds . (2.15)
r
Using this in (2.14) to eliminate the first integral, we arrive at
y (t) > I z t) - f(s -t)s-kz q(s)ds . (2.16)
To estimate the remaining integral, which we denote by I, we distinguish three cases:
1 <q <k-2, q = k-2, k-2 <q <2k-3 .
(a) 1 5 q < k-2. As in [Atkinson and Peletier 1986b] we use the bound
z(s) _z(t)(s/t) for T(y)5t 5 s <o.
This yields
Zq(t)t z-kI <(k-1-q)(k-2-q)
and the desired bound follows from (2.16).
51
(b) q = k-2. We now use the explicit expression for z to estimate I. Thus we obtain, using
Lemma A2 from the Appendix,
I Js1-kzk-2(s)ds
= 1 _2 2 k f dx
k-2 , x(1+x)
1= (/To)k~ 2 log(1+fl),
where T = (t/To)k-2 . This yields, upon substitution in (2.16), the desired inequality.
(c) k-2 < q <2k-3. As in the previous case we use the explicit expression for z to bound I.
This yields, when we use Lemma A2 again,
I J s-kz (s)ds
Sk--2 f (y)
k-2
where t = (t/To)k- 2 and B is the incomplete beta function [Abramowitz and Stegun 1965]:
0
B(r,a,b) = xa-1 (1 +x)Y0-- dx .
Since a = (q/(k -2)) - 1 > 0 the integral converges down tot = 0, whence
1<k-1 +-1/*B 0,q4- 1,1
k-2 f (Y) k-2
k-1 fq-k+2 f (y)
from which the lower bound for y follows.
The following simple lower bound, valid only for t > aTo (a > 0) will be useful.
LEMMA 2.3. Let y (t, y) be the solution of Problem III in which f is given by (2.6). Then for every
a > 0 there exists a positive constant L, which depends only on a,k, and q, such that
y (t,'y)> z(t,y)[1 - L-y-2k+q]
for t > aTo and y large enough.
52
Proof
(i) Suppose 1 q < k -2. Then, because z (t) S k 1 t/Y, Lemma 2.2 yields
1 _(k1/ )-1tI-_*_ _
y(t,y) > z(t,y){ 2k+q - t1+- ~ ~(k -1-q )(k -2-q )
For t > aTo the required inequality follows.
(ii) Suppose q = k-2. Then, by Lemmas 2.2, Al and A2,
y (t, Y) > 1 z (t,Y) - k-i 1-klog(1 + ai1) .ceaz(t,) 1 - k2 k Y]]k1+Y 2k-
when y is large enough.
(iii) If k-2 < q < 2k-3, we proceed as in (ii), using Lemma 2.2 and Lemma A2. We omit the
details.
Sometimes it will be convenient to extend the definition off (y) to negative values of y by
setting
f(= yq-~1y + ly I'-y. (2.17)
This allows us then to continue the solution y (t, y) to values of t < T(Y). In fact we can extend
y (t, y) to a function defined for all t > 0, as the following lemma shows.
LEMMA 2.4. Let y (t,y) be defined on (to,oo). Then for allit to:
(a) Iy(t,y)I <y;
(b) Iy'(t,y)I < 2F(y)t-kfor all t to.
Proof. Consider the energy function
E(t) = - t y' 2 (t, y)+ F(y (t, y)),2
where
Y
F(y) = ff(z)dz.0
Since E (t) -+ F (y) as t -*oo [Atkinson and Peletier 1986a] and E'(t) > 0, it follows that
F(y(t,y)) <F(y) for t 2to ,
from which the two assertions follow.
53
3. The Case1:5 q < k-2
To describe the asymptotic behavior of y (t, y) as y -+ 00 in the parameter range 1 5 q < k -2,we need to rescale y and t appropriately. Thus we introduce the new variables
t (s)= y(t), s = -1t, (3.1)
where
k -2k -q -1
We shall show that for every s > 0
(s) -13(s)as y-+oo,
where 13is the solution of the problem
P'+ s-k 1 1 I9~11$ = 0 0 <s <oo (3.2)
$(s)-ks -+0 as s -+coo. (3.3)
The existence and uniqueness of $3 is ensured, precisely for the parameter values considered
[Atkinson and Peletier 1986b].
Specifically, we shall prove the following basic asymptotic result.
PROPOSITION 3.1. Let 11 ands be defied by (3.1) and $ by (3.2) and (3.3). Then for every s* > 0
and xK> 0, there exist constants K 1 > 0 and K 2 > 0, such that
Ir(s)-13(s)I K1 f-+2 for s* s5 icy (3.4a)
I1'(s)-1$'(s)I 15K2 "(-+1 for s* <-s icy. (3.4b)
Define
So =inf(s>0:3>0on(s,oo)). (3.5)
Because $ is concave on [So,oo) and converges to k 1 s as s -+ 00, it follows that So > 0 and that
[ '(So) > k1. Thus, the following theorem is an immediate corollary of Proposition 3.1.
THEOREM 3.2. Let y(t,y) be the solution of Problem Ill in which f is given by (2.6) and
1!5q <K-2. Then
(a) y- T(y) -* So as y-+ o ,;
(b) For every s * > 0 there exists a constant K > 0 such that
54
YV I y (t, y) -b(t, y)I <Kyf-k+2 for Yl-vs* 'st S --,
where
b(t,y) = y~( t) .
We shall first prove Proposition 3.1 and then exploit Theorem 3.2 to prove part (a) of
Theorems A and B.
Proof of Proposition 3.1. Introducing the new variables T and s, we find that
1 ''+ s- (94 + ~49') = 0 s > 0 , (3.6)
where
S=(p-q)v>0.
Henceforth it will be tacitly understood that if x e R- and a E R, then x = Ix a-Il
We shall proceed in two steps. First we shall prove the propositions for values of s for
which both i(s) and n(s) are nonnegative. Thus, writing S(y) = y-' T(y), we shall assume here
that s 2 s 1 = max(S(y), So). In the second step we shall extend the bounds (3.4a,b) to values of
S <S 1 .
Let s1 5 s < a. Then, if we integrate (3.6) twice, we obtain the integral equation for a:
a
71(s) = 1(a) - '(a)(a-s) - f(r -s)rk(f1 + y4 t')dr .S
For p, we arrive at a similar equation
$(s) = p(a) - 3'(a)(a-s) - J(r-s)r-IYdr.S
Thus, writing w = ,-1, we obtain
w (s) = w (a) - w'(a)(a-s) - J(r-s)r-k(lq- 4q)drS
- J(r-s)r-ky-rldr,S
or
Iw(a)l sA(a)+JB(r)Iw(r)Idr, (3.7)S
where
55
A (a) = Iw (a)I + al w'(a)I +y Jr 1-kAP(r)dr (3.8)S
and
B (r) = rlk _1 )k - q(r)
rg(r)- (r)
Recall that by construction O(r) <_k 1 r and by Lemmas 2.1 and Al, ii(r)5k ir. Hence, by the
Mean Value Theorem
B (r) 5 qkj-1 ri~- .(3.9)
We now conclude from Gronwall's lemma that
Iw (s)I -A (a)exp sk~-ki. (3.10)k -q -1
Thus, it remains to estimate the coefficient A (a) and then choose a appropriately.
Write
A (a) = A 1(a) +A 2(a) +A 3(a),
where
A 1 (a) = Iw (a)I , A 2 (a) = a I w'(a) I,
and
A 3 (a) = yTJr1k*P(r)dr .S
We shall estimate A 1, A 2, and A 3 in turn. In doing so, we shall introduce a generic constant C,
which depends only on k and q.
A 1. Define the function
c(s) = TzV(YV-vs) ,
where z is the function introduced in Lemma 2.1. Then we can estimate A1 (a) by
A 1(a)S I1(a)-((a)I + I (a)-k 1 a I + Ik 1 a-(3(a) I.
Proceeding as in [Atkinson and Peletier 1986b], we find that
Ik1 a-13(a)I SCae-k+ 2 ,
and inspection of the expression (2.11) for z yields
I( (a) -k ia I 5 C(of -P + /ak~1) .
56
Finally, by Lemma 2.1
11(a) <tc(a)
and by Lemma 2.3
n(a) > ((a)(1+y')-1 -CaQ-k+2 ,
whence
11(a) - C(a) I < ((a)rP -+ CaQ-k+2
< k 1a ' p+CaoQ-k+ 2
Thus
A 1(a)5C(yf~p + aq-k+ 2 +y40k-1)
A 2. We proceed as withA 1:
A 2 (a)a (a)-('(a)I + a I '(a)-k1 I +aIk 1 -$'(a)I.
Following [Atkinson and Peletier 1986b] again, we obtain
I k 1-a'(a) 15 C aQ-k+1
and the explicit expression for z yields
I ('(a) -k 1I C(y' + ,4ak-2)
To estimate I rl'-c'1 observe that
W'(s) = yy'() , ('(s) = Yy'(t)
and so
I r'(a)-C'(a)I = YIy'(t)-z'(T)I , (3.11)
where t = y-"a. Integration of Equation (2.8) for y over (t, oo) yields, in view of (2.9),
y'(t) = Jsk (Y(s)+yP(s)}ds
<Js-k[z (s) + z (s))ds
by Lemma 2.1. Integration of (2.12) yields
z'(t) = (1+y-P)Js-zP(s)ds
and hence
57
y'(t) < z'(t)+ Js-kz(s)ds (3.12)
< z'(t )+ r tq~-**,
k-1-q
where we have used Lemma Al.
On the other hand, by Lemma 2.2 and Lemma Al
y (t) c (t)z(t) ,
where
c (t) = (l+y'~p)- - C -qt~-k+l
and hence
y'(t) > cP(t)Jsk-zP(s)ds (3.13)
=- Pt z'(t).1+~
If we now set t = ,tin (3.12) and (3.13), and return to (3.11), we obtain
1 '(Q)-C'(Q) I < C (~p + aq ~k+1).
Thus, we obtain for A 2 (a):
A 2(a) 5 C (0f~p + Qq-k+2 +-6k-1)
A 3. Since n(s) < ((s) < k 1s and p = 2k-3,
A3(c) < kiy4Jrk-2dr = Cy4 ak~.0
Finally, putting the estimates for A 1 , A 2 , and A 3 together, we obtain
A (a) 5 C(y~P + aq-k+2 + y-4 ak') .
At this stage we set a = icy. Remembering that k > 2, we observe that
1+q-p = q-2k+4 < q-k+2
and
58
-8+k -I = -(p -q)v+k-1 _ q-p+k-1 = q-k+2.
Thus, we conclude that
A (a) &Cy-k+2 ,
which completes the proof of (3.4a) for s s1.
To prove (3.4b), we integrate (3.2) and (3.6) over (s,a) and subtract. This yields
I '(s) - $'(s) I Ii'(a) - $'(a) I +Jr-A l ?(r) -p(r) I dr + yJr-ngP(r)dr .S s
Using the estimates for A 2(CY), A 3(), and Il - I established above, we arrive at the desired
bound.
Next, we suppose that s* < s1. It follows from [Atkinson and Peletier 1986b] that S(y) is
uniformly bounded for y large; i.e., there exists a number s* 1> So such that S(y) < s* 1 for all y
large.
Observe that on [s*, s ], i and 0 satisfy the same differential equation with bounded
coefficient s-k, except for the term y4skll(s). Since 11 need not be nonnegative, we can no
longer use Lemma 2.1 to bound it. Instead, we shall use a variant of Lemma 2.4. Consider the
energy function
E0(s) = 1- s'2(s)+ +yi .2q+1 p+1
Ats = S(y), I(s) = 0and
Eo(S(y)) = kS(y) '2(S(y))
By [Atkinson and Peletier 1986b] and the previous bounds, E0 (S(y)) is uniformly bounded for y
large. Because E0'(s) > 0 and the terms in E0 are all nonnegative, it follows that Ti(s) is uni-
formly bounded on [s*, S(y)j. In fact, we find, using the bounds for S(y) and r '(S(y)) for large y,
that
lim sup 1(s)I S ( Sok)1(+0 for 0 < s 5 So .y-+- 2
Thus, for s* s <s1,
yas~-*rlp(s) = O(y6) as y -+0o .
Remembering that -S <q - k + 2, we see that the estimates obtained for s s continue to be
valid for s* -s < si.
This completes the proof of Proposition 3.1.
59
To prove Part (a) of Theorems A and B, we merely need to return to RN. Thus, le ERN
and define
N-N-2w (4) = a(s) , s = (3.4s=(-.. 2
(3.14)
Then w satisfies
Aw+wq =0, w >0 if 0< I t I<R0
W(t)=0 if I|I =IR0
where
R = (N-2)SOl/(N-2 ) , CN = (N(N-2))} -2 M .
As an intermediate step we prove the following lemma about the solution v (,y) of the problem
v''+N - v'+ v i + VP = 0 p > 0P
v(0)=y, v'(0)=0
and
R(y) = sup(p >0: v(,) > 0on(0,p)).
LEMMA 3.3. Set a = [N-q (N-2)}-'. Then
(a) 2a y(1pe-1)I I,y) -+ w() as y_-+ c;
(b) f- (-)R (y) -+ R o as'y - oo.
Proof. (a) By Lemma 3.1
7"y (y'-vs, y) -+a(s) as y -+o . (3.15)
But
]-vs N-2 N-2
and hence, in view of (2.5)
y (Y-vs, Y) = v (f(q-1) 1,y).
Thus, since v = 2a, (3.14) and (3.15) imply that
60
2aV (ya~- 1,y) --+ w (t) as y -+ co .
(b) Recall from (2.7) that
R (y) = (N-2)T~1"w-2)(y) . (3.16)
If we use the limit
Yv-lT (y) -+So as Y-+00,
which was established in Theorem 3.2, in (3.16), the desired limit follows.
Finally, we return to the solution (u, A) of Problem P. By (2.7),
T(y) = N -2/1-q)(Y) , N1 = (N-2)N-2
and hence, in view of Theorem 3.2,
1-VX2(p-q)() -+* N1/So as Y-+coo. (3.17)
By (2.2),
c (y) det u(0) = lI(p-q)( 7y)y, (3.18)
and hence, using (3.17) to eliminate A, we obtain the asymptotic expression
Y~(+v2c(y) -+ (N1 /So)"' as y -+ oo. (3.19)
Note that this means that
c(y)-+ oo as y -+ 0.
To obtain an asymptotic expression involving only c and A, we combine (3.17) and (3.18) to
eliminate y with the following result:
C 1-q(Y)X(Y) -+ (N1/SO)1-1- as y -oo . (3.20)
To return to solutions defined on the unit ball, we transform once more, setting
w (t) = (S0 /N i )$(x ) , x = E/R o .
Then 4 satisfies
A$+ 4=0, $>0 if 0< IxI <1
4(x)=0 if IxI =1
$(x)-cN~x|2 -- 0 asx -+0,
where
N= (NIS)*-1.
61
Thus
C 1-q(Y)A.(Y) -+ as y--+ oo ,
which completes the proof of Part (a) of Theorem A.
Next, to transform v (p,y) to u defined in B1 , we use (2.1), (2.2), and (2.4) to define
(y) = y( I) /R (y)
and
u (, c) = R1/(k~2 )v ((q~ 1) II,Y)
Thus, by Lemma 3.3(a),
,?aR-1/(k- 2 )u (,c) = ayv(YQ-~ 1 ,) -> (S0 /N1 )4(x).
However, by Lemma 3.3(b) and (3.19),
C~1 aR-1/(k- 2)(Y) -+ So/N1
and so
cu(, c)--4(x) as c - o . (3.21)
To complete the proof of Part (a) of Theorem B, we observe that by Lemma 3.3
Q(Y)-x = ~Q-1) - 1 -+ 0 as y --+ oo . (3.22)R0 R (Y)
By arguments similar to the ones we have used to estimate cu (x,c), one can show that there Exists
a constant M, which does not depend on c, such that
suP IcVu(x,c)I <M .
Hence, by the Mean Value Theorem
I cu(1,c)-cu(x,c)1 <M I (y)-x I
and it follows from (3.22) that
cu(1,c)-cu(x,c) -+0 as c-o- .
Thus, we may replace Q by x in (3.21), and conclude that
cu(x,c)-+4(x) as c-+oo,
which finishes off the proof.
62
4. The Case q z k-2
It was shown in [Atkinson and Peletier 1986b] that
(i) ifq = k-2:
T(y) ~y3-klogy as y -+ oo ; (4.1)
(ii) if q > k-2:
T(y) ~ y+s-2k as y -oo. (4.2)
We shall begin by proving the following theorem, which sharpens these asymptotic estimates.
THEOREM 4.1. Let y (t, y) be the solution of Problem 111 in which f is given by (2.6). Then
(a) ifq = k-2:
S-3-c3T (y) -+ kik-5 s -+o;
logy a y -k kS
(b) if k-2 < q < 2k-3:
9--T (y) -+ A (k,q),
where
k...3 2- ( - 1)r(k + 1)Akkq) 2k -3-q k -2 k -2
k-2 q+1 ___
k -2
and k1 = (k-1)'1(k-2).
About the graph of y (t, y), we shall establish the following result.
THEOREM 4.2. Let y (t, y) be the solution of Problem III in which f is given by (2.6). Then if
k-2 q< 2k-3,
yy(t,y) = k1 [t-T(y)]+o(1) as y-+ o ,
uniformly on [T (y), T (y)+t] for any t> 0.
We shall first prove Theorems 4.1 and 4.2 and then show that they yield Parts (b) and (c) ofTheorems A and B.
The proof of Theorem 4.1 is based on a monotonicity property of the Pohozaev functional
H(t) = ty2_y+ 1- t1kyf(y). (4.3)kfli
If y (t) is a solution of Equation (2.8), then
63
H(t ) =- k t k-ky'{(2k-3)f (y) - yf'(y)) .k-i
If, in addition, y (t) -+ y as t -4 , then y'(t) = O(t1~k) as t - oo, and
H(t) -+ 0 as t -oc.
Finally, at t = T(Y),
H(T) = Ty'2(T),
where we have written T = T(y). Thus, if we integrate H'over (T, oo), we obtain
Ty' 2(T) = Js -k{(2k-3)f (y (s)) - y (s)f '(y (s)))y'(s)dsk-1 1
= Jskg(y(s))ds,T
where
U
g (u) = J((2k-3)f(v)-f'(v))dv.0
If f is given by (2.6), we obtain for g:
g(u)= - u4+1q+1
and thus we finally arrive at the identity
Ty'2(T) -= Js~y4+I(s)ds . (4.4)q+1 T
In the next two lemmas we shall obtain sharp estimates for both sides of (1). Equating
these estimates then yields the desired estimate for T(y). Theorem 4.2 will follow as a corollary
of the next lemma.
LEMMA 4.3. Let q k -2. Then
lim yy'(t,y) =k
uniformly on intervals [T (y),T 1(y)], where T 1 (y) = O(f ), in which 5-2k+3 < a < 2, as y-+ .
Proof. We first prove an upper bound for y'(t,y). In view of the concavity of y, it is sufficient to
consider t = T(y).
Integration of (2.8) yields
64
y'(T) = Jsk(y (s)+yP(s)ds.T
ByLemma2.1,y <zandhence
y'(T) < Js~k{z(s)-+ ?P(s))ds = _ + szsds,(4.5)T Iy 3 T
where we have used the differential equation (2.12) for z.
To estimate the integral in (4.5), we use Lemma A2 of the Appendix which states that
S-kzq(s)ds = K (y)B(T,a,b) ,T
where
K (Y) =k f To-k , T(y) = (T /To)k~2
k -2
and
a = g-k+l b = k -1k-2' k-2
Observe that
K (y)=y+2-2 ,(y)-+0 as y -4oo
in view of (4.1) and (4.2).
If q > k-i, a > 0 and B (0,a,b) exists. In that case
S= oe+2-2k) =(') as y->+coo
because q < 2k-3. If q = k-1, a =0 and
I = 0 (ylklogy) as y- 0 0
If k-2 <q <k-1,
I = 0 (f +q-2+3X-k+2)) as y -* oc.
If q = k-2, a = -1/(k-2) and
I = 0 (1/(ylogy)) as y- oo.
Therefore, in all four cases,
Y fs~*z(s)ds -+ as y -+ . (4.6)T(Y)
Thus, if we multiply (4.5) by y and let y tend to infinity, we obtain using (4.6) and Lemma
Al,
65
lim sup yy'(T(y),y) k 1 . (4.7)
To obtain a lower bound, we note that for anyt > T(y), and for any to > t,
y (to)-y (t) iy'(t) > y(o- t > -- y (to)-z (t)} . (4.8)
to-t to
Suppose k-2 < q < 2k-3. Then, by Lemma 2.2,
)y'"(t)> 1 yz(to)_ k-1 y-2t +q[ ll _ . r(4t).)14 2k+q to q-k+2 to t to
We now set to = to and choose 6oso that
to(Y) r,21ty)- 0 and f-2+q _+ 0 as Y -+ 0,0(.1.To(y) to(y) (4.10)
i.e., 5-2k+q < a < 2. This means, according to (4.2), that to(y) > T(y) for y large enough.
Iftry satisfies (4.10), it follows from Lemma Al that
.ireYz(t o(Y), Y =kY y) 1"(.1hm = i .(4.11)
y-... to~y)Finally, if t 5 T1(y), and T1(y) = o(to(y)), we deduce from (4.9), (4.10), and (4.11) that
lim inf yy'(t, y) k 1 (4.12)
uniformly on (T(y), T1(Y)).
Next, suppose q = k-2. Then, by Lemma 2.2,
Yy'(t) > 1 ^0z(to) _ k-1 - log{l + (To/to)2 -_'Pz(t) ._t_
1+y1~ ['to k-2 to t to
As before, we set to = f, and we choose 3-k <a < 2. Proceeding as in the previous case, wearrive again at (4.12).
This completes our proof of Lemma 4.3.
Next we turn to the right hand side of (4.4). For convenience we write
1(y) = Jskyq+l(s)ds .T (Y)
LEMMA 4.4. (a) Suppose q = k-2. Then
-1(Y) -+ kk- 3 as y--*oo.logy
(b) Suppose k -2 < q < 2k -3. Then
66
F(A -1)F( k-1)-k_ k-2 k-2 .
k -2 q+1
k-2
Proof. (a) Let a > 0, and assume that y is so large that aTo > T(y). For such y we break up the
integral 1(y) into integrals over (T, aTo) and (aTo,oo). We write the corresponding decomposi-
tion of the integral 1(y) as
1(y) = 1i(y,a) +1 2(y,C) .
Observe that by Lemma 2.1 and Lemma A2
12 (y,a) f s-kzk-(s)dsaTo,
1- Y,1To kB a, 0, k-1k -2 k-2
Remembering that T0 (y) ~y2 , we see that
lim -1 1 (y,a) = 0 . (4.13)-- logy
To estimate 11, note that by Lemma 4.3, for any E> 0
y (t) < (k 1 +e)(t-T)y
provided y is large enough. On the other hand, by the concavity of y,
y(t)>y(aT0 )tT , T <t <aTo.aT 0 -T
By Lemma 2.3,
y ((X) > z (a)(, -Lyl~k)
and by Lemma Al,
z(czTo) > cay .
Hence, since T0(y) = kit y2 [1+o(1)] as y-+ oo, and ca = a[l+o(1)] as a -+*0,
y (t)> (k 1-)(t-T)f'1 , T < t < aTo ,
for y large and a small enough.
It follows that
67
aT
11(y,a) < (k +E)k-1Y-k J s-k(sT)k-ldsT
aT0 rlT
= (k 1+E)k-lyk J u-k(u_)k-ldu
Recall that in view of (4.1)
aTo(y)/T(y) = yk~/logy as Y -+ oo.
Hence
J u--(u)k- du = (k-1)logy[l+o(1)] as y -+oo.
Thus
lim sup-I1 (y,a) < (k 1 +E)k-l(k1), (4.14)- logy
and similarly
lim i nf I1(Y~a) > (k i-E)k~I (k -1) . (4.15)Y logy
Because e can be made arbitrarily small by choosing a small, (4.13), (4.14), and (4.15) imply that
lim I( ) = kl-1(k-1) = k- 3 . (4.16)'-... logy
(b) As before, we split I in an integral I1 over (T, aTo) and an integral 12 over (aT0 ,oo).
By Lemmas 2.1 and Al,
aT
11( , tt) < 1s~kzq*l(s)dsT
aT
< k1 Y-7-~1 r s~-*** dsT
Y-q-1 q-k+2 a-k+2(4.17)q -k +2
and hence
w r -3-so i(Ya)eCa-k+2
where C is some positive constant.
68
By Lemma 2.3, for any a> 0,
c (y,ca)z(t, y) <y (t, y) < z(t, y) for aTo < t < oo,
where c (y,a) -+ 1 as y -+ 0. Thus, we may replace y by z in 12. This yields, with Lemma A2,
s-kz (s)ds = f I T [-Ba, k k- . (4.18)aT 0
k-2 k -k2 k -2J
Finally, because we may choose a arbitrarily small, we may conclude from (4.17) and
(4.18) that
lim -3-1(Y) - k- B [0, -k+2 k-1Yn- k-2 k-2 k-2
Since B (0,a,b) = F(a)F(b)/F(a+b), we have proved the desired estimate.
Proof of Theorem 4'1. (a) If we multiply (4.4) by y-'/logy and divide by yy'2(T), we obtain by
Lemmas 4.3 and 4.4
T (Y) -+ ki2k k~3 =kk?-slogy
(b) We now multiply (4.4) by fi-3-q and divide by #y' 2(T). Using Lemmas 4.3 and 4.4, we
arrive at the desired limit.
Proof of Theorem 4.2. Plainly,
yy(t,y) = Iyy'(s)ds.T(y)
Thus, by Lemma 4.3 on any interval [T(y),T(y)+t]
yy@(t,y)=k1[t--T(y)]+o(1) as y-+oo.
Parts (b) and (c) of Theorem A and Part (b) of Theorem B follow from Theorems 4.1 and
4.2 when we replace y and T by c and X using the identities (2.2) and (2.7).
Acknowledgment
We are indebted to J. B. McLeod for a helpful discussion about Section 3.
69
References
M Abrainowitz and 1. A. Stegun 1965. Handbook of Mathematical Functions, Dover, New York.
F. V. Atkinson and L. A Peletier 1986a. "Ground states of -Au = f(u) and the Emden-Fowlerequation," Arch. Rational Mech. Anal. 93, 103-127.
F. V. Atkinson and L. A. Peletier 1986b. "Emden-Fowler equations involving criticalexponents," Nonlinear Anal., TMA 10, 755-776.
F. V. Atkinson and L. A. Peletier 1988. "On nonexistence of ground states," Quart. J. Math., toappear.
H. Brezis 1986a. "Some variational problems with lack of compactness," Proc. Symp. in PureMath. 45, 165-201.
H. Brezis 1986b. Private communication.
H. Brezis and L. Nirenberg 1983. "Positive solutions of nonlinear elliptic equations involvingcritical Sobolev exponents," Comm. Pure Appl. Math. 36, 437-477.
B. Gidas, W.-M. Ni, and L. Nirenberg 1979. "Symmetry and related properties via the maximumprinciple," Comm. Math. Phys. 68, 209-243.
J. B. McLeod 1986. Private communication.
W.-M. Ni 1982. "A nonlinear Dirichlet problem on the unit ball and its applications," IndianaUniv. Math. J. 31, 801-807.
S. I. Pohozaev 1965. "Eigenfunctions of the equation Au + ,f(u) = 0," Dokl. Akad. Nauk.SSSR 165, 36-39 (in Russian) and Sov. Math. 6, 1408-1411.
Appendix
In this Appendix we establish some properties of the function
-1/(k -2)
z(t, y) = Yt tk-2 + k1l(l 1(Al)
introduced in Lemma 2.1 and used throughout the paper. As we noted, it is the solution of the
equation
z'+ t~kY3 2kf (y)z22-3 = 0 (A2)
which has the property
limz(t,y) = Y. (A3)I-
The following notation will be convenient:
k1 = (k-1)1/(k- 2 ) , k2 =k-k -2
anda n d
E ll 1/(k - 2 )
To0 k-1 Y
LEMMA Al. Boundsfor z and z':
0<yz(t,y)<k1 t for t>0; (A4)
z'(r,Y) < k 1 for t > 0 ; (A5)
Iyz(t,y)-k1 t I <k2 t(t/To)k- 2 ; (A6)
z (t, y) c a(t /aT) for 05 t < aT0 , (A7)
z(t,y) c ayfor t->aT0 , (A8)
where ca = 0( 1+ak-2-1/(k-2) and a > 0.
The inequalities (A4)-(A8) all follow readily from (Al). In the next lemma we use the incom-plete beta function
B (,,a,b) = Jxa-1(l+x)--bdx
in which a e R and b is a positive number. Recall that if a > 0 [Abramowitz and Stegun 1965],
71
B(o,a,b)= .a br(a+b)
LEMMA A2. Supposem > 1 and r e R. Thenfor t >0
fs-mZr(s)ds =K(y)B Ikr--m+l m-1k -2 k -2
where
K(y) = l r?.l-m(7 0T-rn(Y)k -2
and
S=(t/Tof-2
Proof. If we insert the expression for z into the integral, and introduce the new variable
x = (s/To)k-2 , we can write it as
yr Tu~" xa-l(1+x)-a-bdx,
(tm/T)( )
where a = (r-m+l)/(k-2) and b = (m -1)/(k-2).
72
NON-NEGATIVE SOLUTIONS FOR A CLASS OF RADIALLY SYMMETRICNON-POSITONE PROBLEMS
Alfonso Castro*Department of Mathematics
North Texas State UniversityDenton, TX 76203-5116
R. ShivajiDepartment of MathematicsMississippi State University
Mississippi State, MS 39762
Abstract
This paper shows that the superlinear Dirichet problem (1.1)-(1.2) has a non-negative solution for small positive values of X when f(O) < 0. The casef (0)> 0 was considered previously using upper-lower solutions. The proofspresented here are based on "shooting" arguments.
1. Introduction
Here we consider the existence of radially symmetric non-negative solutions for the boun-
dary value problem
-Au(x) = Af() SI X II <-1, x E RN(1)
u(x) = 0 1XII = 1, (1.2)
where aX> 0, and f : [0,oo) -+ R is a locally Lipschitzian nondecreasing function. As is well
documented, the study of (1.1)-(1.2) is equivalent to the problem
-u "- n-u'= Af(u) re [0,1] (1.3)r
u'(0) = 0 (1.4)
u(l) = 0, (1.5)
where n = N-1. We will assume that
lim (f(u)/u) = oo, i.e., f is superlinear. (1.6)
* Part of this research was done while this author was visiting Argonne National Laboratory.
73
f(O)'<0.
For some k e (0,1) A = lim ( d )N(F(kd) -- 2 df(d)) = co, (1.8)d-o. f(d) 2N
where F(x) = Jf(r)dr.0
If f (0) > 0 and Xk> 0 is small, it can be seen (see Shivaji [1983]) that (1.1)-(1.2) has two
solutions: one near zero, the other bifurcating from infinity. If f (0) > 0, comparison arguments
(upper-lower solutions) show that for A > 0 small enough, the solutions bifurcating from infinity
persist. However, such arguments fail when f(0) < 0. This is why we have been motivated to
undertake this study. Our main result is given in Theorem 1.1.
THEOREM 1.1. Under the above assumptions, there exists X > 0 such that if 0 < A < A0, then
(J.1)-(1.2) has a non-negative solution.
Castro and Shivaji [1988] have made an extensive study of the one-dimensional problem
(N = 1). Our proof of Theorem 1 is based on the shooting method. That is, to prove that prob-
lem (1.3)-(1.5) has a solution, we consider the problem (1.3), (1.4) subject to u(0) = d. By
analyzing this problem depending on the parameter d, we show that for an adequate value of d, u
satisfies also (1.5). To prove Lemma 3.2, we use an identity of Pohozaev type (see Section 2)
used by Castro and Kurepa [1987a and 1987b] to study oscillatory solutions of other radially
symmetric problems. For other applications and extensions of this type of indentity, see Ni and
Serrin [1986] and Pucci and Serrin [1986].
2. Preliminaries and Notations
First of all we extend f to (-oo,oo) by defining f(x) = f(0) for x <0. By (1.6) we see that
lim F(d) = c. Hence (see (1.7)) there exist positive real numbers 0 < 0 such thatd-
0 = f([) = F(0). (2.1)
Since (see (1.8)) A = cc, we see that there exists y > (0/k) such that
2NF(kd) - (N-2)df(d) 0 for d y. (2.2)
For each real number d, the initial value problem (1.3), (1.4), u(0) = d has a unique solution
u(t, d, A). This solution depends continuously on (d,A) in the sense that if ((d.,A))} -+ (d, A), then
{u( ,d.,,)) converges uniformly to u(,dA) on [0,1]. To see this, we observe that for each (d, A)
the map
74
(l .7)
u(s) -+ d +A. 1 t~" j r"(-f(u (r)))dr (2.3)0 0
defines a contraction on C([0,e],R) for e small enough.
Given deR, Ae R, we define
E(t,d, X) (u'(t,d,)))2 + XF(u(t,d, A)), (2.4)2
N -2H(t,d, X) = tE(t,d,) +N2 u (,d,k)u'(t,d,X). (2.5)
Multiplying (1.3) by rNu' and integrating over [?,t], and then multiplying (1.3) by rN-lu and
integrating over [,t], we obtain
tN-1H(t,d,A) = 4H(f d,A) + r"A[NF(u(r,d,X) - N-2 f(u(r,d, X))u(r,d,X)]dr. (2.6)4 2
This identity is a form of "Pohozaev identity." For more details see Castro and Kurepa [1987a]
and Pucci and Serrin [1986].
From (2.2) we see that for d kT we have
NF(d) - N2k f(d)d 2 NF(d) - N-2 f()( ) 0. (2.7)k k
Multiplying (2.7) by (- 2k)d -(2.+N-2yf(N-2), we obtain that for d 'y
F(d) cd "(2k,- (2.8)
and
f (d) c(d)(2N-N+2y(N-2) = cd' , (2.9)
where c is a constant independent of d. We observe that
N+21 :q < N-2 . (2.10)
Thus for d y we have (see (2.10))
(d If(d ))N /2F(dk ) >-c(df(d )))N 2(df(d )
Scd 1**NI2(f(d))'-(N2)
cd(1+(N/2))+q(1-(N/2))
-+ oo as d -+ oo, (2.11)
where c is a constant independent of d.
75
For d 2y, let to:= to(d,k) be such that d z u(to,d,) >-kd for all te [0,to) and
u(to,dA) = kd. Multiplying (1.3) by r", we have for tE [O,tol,d > y
u '(t,d,A) = -'t" Jrf(u(r,d,k))dr < -ktf(d )/N. (2.12)0
Integrating on [Ot o], we have kd z d - to Af(d)/N. Thus
to c(d/Xf(d)) 1", (2.13)
where c is again a constant independent of (d,A). Now, combining (2.6), (2.11), and (2.13), we
have
t5H(t0) = A. J r"[NF(u(r,d, A.)) - N-2 fu(r,d,A)u(r,d,A.I dr0
to
SAr" r"cf(u(r,d,A.))u(r,d,A)dr0
cz tLf(kd)(kd)/N
>_ c (d /kf(d))(N2 F(kd ), (2.14)
where we have used that because f is increasing f(d)d z F (d) for all d _0.
3. Main Lemmas and Proof of Theorem 1.1
LEMMA 3.1. IfAe (O,A.1 := (N((y- 1)/f(y)), then u(t,y,A) >3 for all t member'[0,1].
Proof. Given a,> 0, let t1 (A) = t 1:= sup(t 5 1; u(r,y, ) >$ for all r e (0,t 1 )). Since
u'(t,yA)_= -At " "s"(f(u(s,yA.)))ds, we see that u is a decreasing function on [0,t1]. Thus if0
X E (0,A1), t E [0,t1 ], we have
I u'(t,y,A)I 5 < Y -a$. (3.1)N
Thus u(t ,y,,) > y- (y-)t 1. In particular, if t1j < 1, then u(t ,y,A.)> $, contradicting the
definition of t 1. Hence t1 = 1, which implies that u(t,yA) 3 for all t e [0,1 ], and the lemma is
proven.
76
LEMMA 3.2. There exists X > 0 such that if XE (0,12), then E(t,d, X) > 0for all t e [0,1 ],
d E [7,o).
Proof. By (2.13) we have
to c(d/Xf(d))", (3.2)
where u(to,d,k) = kd. Hence (see (2.14))
toH(to) c(d/2f(d))NA2 XF(kd). (3.3)
Thus by identity (2.6) we have for t to
t"H(t) c(d/Xf(d))N/1 2 XF(kd) + . r"[ NF(u(r)) - N-2 f(u(r))u(r)]drto
> cX 1 '2 (d/f(d))N/2 F(kd) + AB
- as + d2- , (3.4)
where we have used that (d/f(d))N/2 F(kd) -+ oo as d -+ cc (see (2.11)). By (3.4) we see that if
d y and X is sufficiently small, then H(t) > 0 for all t E [0,1]. In particular,
u2(t, d,X) + (u'(t, d,))2 > 0, which proves the lemma.
LEMMA 3.3. Given any X there exists d > such that u(t,d,X) < 0 for some t e [0,1].
Proof. Let p > 0 be such that if co satisfies
w''+ n-o'+ po= 0, w(0)= 1, w'(0) = 0, (3.5)r
then the first zero of o is (1/4). By (1.6) there exists do = do(k) >?0 such that if x do, then
1> (3.6)x X
By (3.4) we see that there exists d 1 > do such that if d > d 1 , then
E(t,d,X) > AF(do) + 2do . (3.7)
Now let d > d 1 . Since (do)+-(do)'+ p(do) = 0 (see (3.5)), and u''+ n-ur r
+ X( ~u) u = 0, by (3.6) and the Sturm comparison theorem we see that for some t E [0,1/4]u
77
u(ti,d,X) = do.
Since d1 > a, without loss of generality we can assume that u(- ,d,X) is strictly decreasing on
[0,t1]. In particular
u '(t,d,A) -2d0 . (3.9)
Now let t2 > t1 be such that 0<- u(t,d,k) do for all t e [t1 ,t2] (the existence of such a t2 fol-
lows from (3.9)). Since for t r [ 1 ,t 2 ] AF(u(t)) < XF(do), from (3.7) we have
(u'Q,d, A)) 2 >_4do for all t E [t 1 ,t2 ] . (3.10)
From the continuity of u1',(3.9), and (3.10), we see that
u'(t,d,) <_-2do for all t e [t1 ,t2 ]. (3.11)
Thus u( - ,d, X) is a decreasing function and
u(t1 +(1/2)-do-do <-0 .
Since t1 + (1/2) < 1, we see that there exists t E [0,1] with u(t,d,X) = 0, which proves the
lemma.
Proof of Theorem 1.1. Let
0 = min{L1, X2}. (3.12)
Let ? E (0,A), and
a( ) = a = sup{d E [y,o); u(t,d, A) z0 all t e [0,1]). (3.13)
By Lemma 3.1 we know thata>- y. By Lemma 3.3 we know that a < oo. We claim that
u(1,a,) = 0 (3.14a)
u(t,aX) > 0 (3.14b)
for all t E [0,1). In fact, if u(i,, X) = 0 for some ItE [0,1), then by Lemma 3.2 we have
(u'(E,ak))2 > 0. Since u(0,a,) = 0, without loss of generality we may assume that
u',, )< 0. (3.15)
Hence for some t2 E (, 1) we have u(t2,a,X) < 0. Since u is continuous in the variable (t,,dA),
we have that for some d < a, u(t2 ,d,) < 0, but this contradicts the definition of a. This proves
(3.14b). By the continuity of u we have then that u(1, a, )0. Suppose u(1,,L) > 0. This and
(3.14b) imply that for some it > 0
u(t,a, ) ? a for all t E [0,1]. (3.16)
Hence by the continuity of u on (t,d), there exists d > 0 such that if a < d < a + y, then
78
(3.8)
u(t,d, X) 2 . (3.17)
Since this contradicts the definition of , (3.14) is proven. Thus u(t,a,%) is a non-negative solu-
tion to (1.1)-(1.2), and Theorem I is proven.
4. Remarks
Unlike the case N = 1 (see Castro and Shivaji [1988, Theorem 1.1]), for N z 2 the problem
(1.1)-(1.2) does not have non-negative solutions with interior zeros. Indeed, from the definition
of E(t,d, X) (see (2.4)), we see that if 0 < tI < t2 S 1 are two zeros of a non-negative solution
u(t,d, A), then E(t;,d,?) = (u '(t,d,A)) 2 = 0. However, since (dE/dt) = -n(u')2/t, we see that
u =0 for [t1 ,t2 ]. But u(t,dA,)=0 is not a solution to (1.1), (1.2) because f(0)<0. Thus
u(t,d,A) cannot have interior zeros.
References
A. Castro and A. Kurepa 1987a. "Energy analysis of a nonlinear singular differential equationand applications," Rev. Colombiana Mat. 21, 155-166.
A. Castro and A. Kurepa 1987b. "Infinitely many solutions to a superlinear Dirichlet problem ina ball," Proc. Amer. Math. Soc. 101, no. 1, 57-64.
A. Castro and R. Shivaji 1988. "Non-negative solutions for a class of nonpositone problem,"Proc. Royal Soc. Edinburgh 108A, 291-302.
W. Mo Ni and J. Serrin 1986. " Non-existence theorems for quasi linear partial differential equa-tions," Rend. Circolo Mat. Palermo, Suppl. 5, 171-85.
P. Pucci and J. Serrin 1986. "A general variational identity," Indiana Univ. Math. J. 35, no. 3,281-303.
R. Shivaji 1983. "Uniqueness results for a class of positone problems," Nonlinear AnalysisT.M.A. 2, 223-230.
79/60
QUENCHING FOR SEMILINEAR SINGULAR PARABOLIC PROBLEMS
C. Y. Chan*Department of Mathematics
University of Southwestern LouisianaLafayette, LA 70504-1010
Hans G. KaperMathematics and Computer Science Division
Argonne National LaboratoryArgonne, IL 60439
Abstract
Let the real-valued function f be monotone nondecreasing and continuouslydifferentiable on [0, c) for some finite c (c > 0), such that f (0) > 0 andlim _ f (u) = w. This manuscript is concerned with positive solutions u of thesemilinear singular parabolic differential equation u, = u. + (b/x)u, + f (u),b < 1, on bounded intervals (0, a). The solutions satisfy the initial conditionu(x, 0) = 0 and the boundary conditions u(0, t) = 0 and u(a, t) = 0. Let II1- IIdenote the sup-norm over the interval [0, a]. It is shown that a solution uquenches (i.e., there exists a T < 00 such that lim,.-T, s<T 1 1u1( -, t) II = 00) if
II u (-, t) II tends to c from below as t approaches T Furthermore, there exists acritical length a* such that u may exist for all t > 0 if a < a*, but IIu(- , t)IItends to c in finite time if a > a*. An upper bound for the quenching time T isobtained, and a numerical method is given to compute a*. An example is givento illustrate the results.
1. Introduction
The concept of quenching of the solution of a nonlinear heat equation was first introduced
by Kawarada [1975], who studied the following problem:
u, = u.a + (1 - u)-1, (x, t) E (0, 1) x (0, T); (1.1a)
u(x, 0) = 0, x E (0,I1); u(0, t) = 0, u(1, t) = 0, t E (0, T). (1.lb)
The solution u of (1.1) quenches if there exists a T <0o such that
lim 1-)T, 1<T II u,("- , t) II = 0o (1.2)
Here, II " II denotes the sup-norm over the interval [0, 1]. The value of T is referred to as the
quenching time.
*Visiting Scientist, Mathematics and Computer Science Division, Argonne National Laboratory, summer 1987.
This work was supported in part by the State of Louisiana under Grant LEQSF(86-89)-RA-A-11.
81
If the solution u of (1.1) quenches at some finite time T, then
lim T, 7<T II u (- , t)II = 1. (1.3)
Kawarada claimed that (1.3) implied quenching of the solution of (1.1). If this claim were
correct, it would follow that the conditions (1.2) and (1.3) are equivalent and either can be taken
to define quenching. However, Kawarada assumed without justification that a function 1>, con-
structed in the course of the proof, satisfied the heat equation on curves s ('t) for t arbitrarily
close to the quenching time T. Hence, Kawarada's claim needs to be reexamined.
Furthermore, it would be desirable 1.o extend Kawarada's claim, if true, to nonlinear heat
equations with a general forcing term f. Since Kawarada made use of the explicit expression
f (u) = (1 - u)-1, the extension of his proof is not obvious.
In his investigation of (1.1), Walter [1976] adopted (1.3) as the definition of quenching.
The same definition was adopted by Acker and Walter [1976, 1978], who studied the solutions of
the general equations
u, = u,.+f (u), (x, t) E (0,I) x (0, T), (1.4)
and
u, = u, + f (u, ux), (x, t) E (0, 1) x (0, T),
subject to the initial and boundary conditions (1.Ib). The results of Acker and Walter imply that
there is a critical length J* such that a solution u exists globally (i.e., for all t > 0) if 1 < 1*, but u
quenches if 1> 1*.
An upper bound for 1* for the problem (1.1) can be inferred from Kawarada's article [1975],
namely, J* 5 24 = 2.8284. Walter [1976] showed that the critical length 1* for (1.1) was in the
range 1.5303 : 1* <'/ = 1.5708. The exact value of 1* was identified by Acker and Walter
[1976] as 2M N, where M is the maximum value of Dawson's integral, F (x) = e- 2t e1 2 dt, on
the interval (0, ao); cf. Abramowitz and Stegun [1964, Section 7.1]. The numerical value of 1* is
1.53030416 (to eight decimal places).
Results on the behavior of the solution of (1.4) and (1.1b) at I = I* were given by Levine
and Montgomery [1980]. We refer to the survey article by Levine [1985] for further results and
references.
The purpose of the present article is to study the semilinear singular problem
u, = u. + (b/x)u +f(u), (x, t) E (0, a) x (0, T); (1.5a)
U (x, 0) = 0, x E (0, a); u (0, t) = 0, u,,,(a, t) = 0, t E (0, T). (1.5b)
Here f is a real-valued function that is continuously differentiable on the interval [0, c) for some
c > 0, f (0) > 0, and lim ~,f (u) = oo. The constant b satisfies the inequality b <I1. If b = 0,
82
then (1.5) reduces to (1.1) with 1 = 2a, because the solution of (1.1) is symmetric about x = 'l.
We remark that, if f (0) = 0, then u = 0 is the only solution of (1.5).
Throughout this article we use the abbreviation i2 = (0, a) x (0, T). We always assume that
T is maximal. We use the symbol L to denote the differential expression Lu = u. + (b/x)ux - u,.
Thus, (1.5a) may be abbreviated as Lu +f (u) = 0 on f.
The transformation u (x, t) = v (z, t), where z = (1/4)x2 , reduces the degenerate elliptic-
parabolic expression zv, + '2(1 + b)v, - v, to Lu. This degenerate expression arises in probability
theory; it was studied by Brezis et al. [1971] for b > - 1. Also, the stochastic process described
by the expression 'Mv, +%'(b/z)v - v,, which was studied by Lamperti [1962] for b > - 1,reduces to Lu upon the transformation u(x, t) = v (z, t), where z = 2- 'x.
The existence of unique solutions of nonhomogeneous problems for the linear differential
expression L was studied by Alexiades [1980, 1982] and by Alexiades and Chan [1981]. In par-
ticular, it follows from Alexiades [1982] that initial-boundary value problems described by equa-
tions of the form Lu = g (x, t) on 0, where g is a given function on fl, with the initial and boun-
dary conditions (1.5b), have unique classical solutions if b < 1.
In Section 2, we prove that the (unique) solution u of (1.5) quenches if there exists a T <oo
such that Iim,T,,<7 IIu (-, t) II = c. This result proves and generalizes Kawarada's claim;however, our method of proof is different from Kawarada's. In Section 3, we establish the
existence of a critical length a*. In Section 4, we obtain an upper bound for the quenching time T
by solving a singular Sturm-Liouville problem and using results from the theory of differential
inequalities. In Section 5, we show that a* is determined by the solution of the corresponding
steady state problem. Picard's iteration scheme gives a strictly monotone sequence of functions,
which converges upwards to the minimal steady state solution. We also give an algorithm to
compute a*. In Section 6, we illustrate our results by considering the special case
f(u)=(1-u)-'.
2. Quenching
In this section we prove that (1.5) has at most one solution. This solution is necessarily
positive (i.e., nonnegative and nontrivial) and nondecreasing in each variable separately. Assurr:-
ing that the solution exists, we show that it quenches if there exists a finite time T such that the
value of u at a approaches c, the critical value of f, as t tends to T from below.
LEMMA 1. The initial-boundary value problem (1.5) has at most one solution u. This solution
has the following properties: (i) u (x, t) > 0 for all (x, t) E f u ((a} x (0, T)); (ii) u (x, t) is a
strictly increasing function of t for all x E (0, a ]; (iii) u(x, t) is a nondecreasing function of x forall t e (0, T).
83
Proof. Let u 1 and u2 be two distinct solutions of (1.5) and let y = u1 - u2. Then y satisfies the
differential equation [L +f '(l)]y = 0 in l for some Tt between u 1 and u 2, the initial condition
y (x, 0) = 0 and the boundary conditions y (0, t) = 0 and yx(a, t) = 0. Since f '(ri) is bounded
above, the uniqueness of u follows from the strong maximum principle and the parabolic version
of Hopf's lemma; cf. Kawarada [1975, Sections 3.2 and 3.3].
(i) Because f (0) > 0, we have Lu + [f (u) - f (0)] <0 in , so [L +f '(rt)]u < 0 for some 11between 0 and u. It follows from the strong maximum principle and the parabolic version of
Hopf's lemma that u > 0 in ( u ((a) x (0, T)).
(ii) Let h be a positive constant less than T. Let the function uh be defined in
QA = (0, a) x (0, T - h) by the expression uh(x, t) = u(x, t + h), and let y = uh -u. Then
[L +f '()]y = 0 in 0,, for some between u and uh. On the parabolic boundary a3,ph of fh we
have y (x, 0) > 0, y (0, t) = 0 and yx(a, t) = 0. The inequality y > 0 in 0, follows from the
strong maximum principle and the parabolic version of Hopf's lemma. Hence, u (x, t) is a strictly
increasing function of t for all x E (0, a).
(iii) Let OE = (e, a) x (0, T), where e is a positive number less than a. As in the proof of
part (i), one shows that the solution u E of the (regular) problem
LuE = -f (uE), (x, t) E (E; (2.1a)
uE(x, 0) = 0; uE(E, t) = 0, E, -) = 0; (2.b)
is positive in "E u ((a) x (0, T)). Differentiating (2.la) with respect to x, we obtain
[L +f '(uE) - b/x2]uE,, = 0, (x, t) E QE.
From (2.1b) we have uE(x, 0) = 0, E, t) 0, u Ex( t) =0. It follows from the strong max-
imum principle that uE, > 0 in . It also follows from the strong maximum principle and the
parabolic version of Hopf's lemma that uE is strictly monotone increasing as e decreases. In par-
ticular, we have 0 < uE E<u in (. Since uE is bounded, the limit lim E - o uE exists; we denote it
by Z. Thus, Z 0and0 <uE Z $uon Q.
Let a E (E, a) be fixed. Consider the (regular) problem
Lu0 = -f (u0 ) (x,t) E Q., (2.2a)
ua(x, 0) = 0; u(a, t) = u E(a, ) , (at ) = 0. (2.2b)
Green's function R*(, r; x, t) of the adjoint L* of L subject to the adjoint boundary conditions
exists; cf. Friedman [1964, Section 3.7 and Chapter 5, Problem 5] and Polozhiy [1967, Section
6.2]. In Green's identity
vLu - uL*v = (vu - uvX + (b/x)uv)X - (uv),,
we take u = uE and v (t, ) = R* (, c; x, t). Integrating over the domain (a, a) x (0, t - S),
84
where S is some small positive constant less than t, using Green's theorem, and letting S tend to 0,
we find
La
uE(x, t) = JJ R*(4, T; x, t)f (ue(t, t)) d dt + JR* (a, r; x, t)u(a, r) d (x,t) e f20.OcT 0
Since R*(4, t; x, t) > 0 for (, t) E (a, a) x (0, t) (cf. Friedman [1964, Section 3.7]), it follows
that R*4(a, r; x, t) >0. Because f and uE are nondecreasing as E decreases, it follows from the
monotone convergence theorem that
ta
Z(x, t) = fR* (4, r; x, t).f(Z (4,)) dt dt +f R*4(a, r; x, t) Z(a,T) dT (x, t) E f .
Oa 0
Thus, LZ = - f (Z) on 0Q. Since a is arbitrary, it follows that LZ = -f (Z) on Q. Also,
Z,(a, t)=0 and Z(x, 0)=0. From the inequalities 0:u e Z<-u in 0 it follows that
Z(0, t) = 0. Since u is unique, we have u = Z and, hence, u, >0. 0
THEOREM 2. Suppose that the function f is nondecreasing and satisfies the condition
C
ff(u) du = oo. (2.3)0
If
lira,-r u (a, t) = c, (2.4)
then the solution u of (1.5) quenches.
Proof. The proof is by contradiction, where we assume that (2.4) holds, but II u,("-, t) II remains
bounded as u(a, t) tends to c. (We recall from Lemma 1(iii) that Iu(- , t)II = u(a, t).)
By assumption, there exists a constant M 2c/a2 such that u,(x, t) M for all (x, t) e f,
the closure of Q. Thus,
ux, + (b/x)u = x-b(xbux)x ; M - f (u), (x, t) E f2. (2.5)
Because u is a nondecreasing function of both arguments and u (a, t) tends to c as t -+ T, there
certainly exists an open rectangle Q = (, a) x (T, T) such that f (u) 2M in Q. Thus,
x-b~(xb ux)x < - M, (x, t) E Q = (f, a) x (,T). (2.6)
The following arguments show that 4 is bounded away from 0.
If b < 0, it follows from (2.6) that (xbux)x S - M/a-b. Upon integration from a point
x e ( , a) to a, we obtain xbux(x, t) (M/a-)(a - x), so u(x, t) > M(,/a)-b(a - x). A second
integration from 4 to a yields the inequality u (a, t)- u (, t)> 'VM(/a)- (a -k)2. By increas-
ing and r if necessary, we can certainly achieve the inequality u (4, t) z (3/4)c. Since
85
u (a, t) < c, it follows that (/a)-b(l - l/a)2 5 c/(2Ma 2) < 1/4, whence a positive lower bound for
E/a can be determined. If b >0, then it follows from Lemma I(iii) and (2.6) that u 5 - M. Two
integrations yield the inequality u (a, t) - u (, t) '2/fJ(a - t)2, whence we conclude that
A/a >- %.
We return to the inequality (2.5). Because f (u) 2M on Q, we have
2X-bbu 5-f ,t) E '. (2.7)
If b < 0, it follows that 2(xbu)(xbuo x 5 -f(u)u/a-2. Upon integration from t to a we find
u(a t)
u 2(, t) (-/a)-2b f (u) du, t e (T, T).
As t approaches T, the integral grows beyond bounds, because of (2.3), so the same must be true
for the (nonnegative) quantity u,(, t). If b 0, then it follows from (2.7) that 2uxx 5 -f (u).
Since u,, >0, we have
u(a )ux2 ( , t) ? 5 f (u) du, t E (T, T).
Again we arrive at the conclusion that uK(, t) grows beyond bounds as t tends to T.
Since f is nondecreasing and u is nonnegative, we have f (u) > 0. The inequality (2.5)
therefore yields (xbu)X Mx ' on Q. It follows upon integration from some point x t t that
ux(x, t) 2x-b[4bu,, t) -(M'/(1 + b))(41+b - xl*)], b # -1,
ux(x, t ) ? (x/t)uz( , t ) - M'x In (;/x ), b = - 1.
A second integration from 0 to 4 yields
u,t) ( , :) M'42 r 1 11U (4, t) > 2 )- 1 - - , b * -1,1-b 1+b 1-b 2)
u(, t) ->2 u(4, t) -AM'42, b = -1.2 4
As t tends to T, these lower bounds become arbitrarily large, so it would follow that u (, t)
becomes arbitrarily large. But now we have a contradiction, because u (4, t) is less than u (a, t),which tends to the finite limit c. 0
3. Existence of a Critical Length
In this section we establish the existence of a critical length.
THEOREM 3. Suppose f is nondecreasing. If T = cc and u < c on Q, then u(-, :t) converges from
86
below to a solution U of the singular nonlinear two-point boundary value problem
U"'+ (b/x)U'= -f (U), x e (0, a); U(0) = 0, U(a) = 0. (3.1)
The convergence is uniform on [0, a ]. Furthermore, u < U on (0, a ] x (0, -0).
p. oof. Since the homogeneous problem corresponding to (3.1) has only the trivial solution,
Green's function G (x; y) for (2.1) exists. A direct computation gives
(1-b)~1xl-a, 0- x 5y,G(x;y)= (1 -b)-y1 -, y xa,
Let F denote the function
a
F(x, t)= JybG(x; y)u(y, t) dy, (x, t)E E(. (3.2)0
If u satisfies the initial-boundary value problem (1.5), then Green's identity yields
a
F,(x, i) =-u (x, t) +- JybG (x; y) f (u (y, t)) dy. (3.3)
0
According to Lemma 1(ii), u is stnctly increasing in t for x e (0, a 1. Since f is nondecreasing,
the integrand in (3.3) is monotone nondecreasing with respect to t. By the monotone convergence
theorem and the continuity of f,a
lin F,(x, t) = - limu(x, t)+ ybG (x; y) f (limu(y, t))dy.t -+e. t- 0 t -+..
From (3.2) and Lemma 1(ii) we infer that lim ,..... F,(x, t) 0. We claim that the limit is exactly
0. If the limit were (strictly) positive at some point x, it woulc follow that F (x, t) would increase
without bound as t tends to infinity, so u would reach c in a finite time, contradictory to the
assumption that T is infinite. The identity (3.3) implies that
a
limu(x, t) = yG(x- y) f (limu(y, t))dy.
That is, lim,_. u (x, t) = U(x). The uniform convergence follows from Dini's theorem.
Lemmas 1(i) and 1(ii) imply that U > 0 on (0, a ]. Furthermore, [L +f '(J)](U - u) = 0 in
fl; U -u > 0 at t = 0, U - u = 0 at x = 0, and U'- u, = 0 at x = a. Thus U - u >0 for
0<x a. 0
THEOREM 4. Let u a denote the solution of the problem (1.5), where a is replaced by a + a for
some constant a > 0. If f is nondecreasing, T = o and lim ,.. u(a, t) = c, then u a quenches.
Proof. Assume that ua does not quench. Let y = ua - u. Then IL +f '(n)]y = 0 in 0 for some
87
71 between u and ua. Furthermore, y(x, 0) = 0, y(O, t) = 0, and yx(a, t) 0 by Lemma 1(iii).
Therefore, ua > u in i.
Let E and to be positive numbers, so chosen that f (z) (4E/a)(2/a + 31 b I /a) + a2 for
z E [c - e, c) and u (a, to) c - e. Let S denote the domain (a, a + a) x (to, oo).By assumption, ua exists for all t > 0; in particular, it must be the case that ua < c in S. It
follows from Lemma 1(ii) and (iii) that u c - e on the parabolic boundary aS of S. Consider
the function z, defined by the expression z (x, t) = c - E + (x - a)(a + a - xXt - to) on S.
Clearly, z = c - c on aS. Furthermore,
Lz +f (z)>-2(t--to)+(b/x)[2(a -x)+a](t-to)-(x-a)(a+a-x)
+ (4eIa)(2/a+ 31b I/a)+ a2.
The expression in the right member is nonnegative on (a, a + a) x (to, to + 4E/a 2). It follows
from the strong maximum principle that u a > z in this domain. In particular, the inequality must
hold at the point (x, t) = (a + a/2, to + 4e/a2 ), where z (a + a/2, to + 4e/a 2 ) = c. But now we
have a contradicition, since it would follow that ua 2 c at this point. 0
Theorem 3 implies that there exists a critical length a* such that u exists on [0, a ] for all
t > 0 if a < a*. The critical length a* is determined as the supremum of all positive values a for
which a solution U of (3.1) exists; if U(a*) exists, then u(a*, t) exists also. According to
Theorem 4, u quenches if a > a*.
4. Quenching Time
In this section we obtain an upper bound for the quenching time T. We use a comparison
result from the theory of differential inequalities.
Consider the singular Sturm-Liouville problem
- x~(xbw)'=2w, x E (0, a); (4.la)
w(0) = 0, w'(a) = 0. (4.1b)
The general solution of (4.1Ia) is
w (x) = x"[c IJ,(x)+ c2Y"(x)], (4.2)
where c1 and c2 are arbitrary constants, X > 0, and J~ and Y., are Bessel functions of the first and
second kind, respectively, with v = /(1 - b); cf. Abramowitz and Stegun [1964, Section 9.11.
The eigenvalues 72 are found from the equation J"_i (Aa) = 0. Let j.,1 denote the first positive
zero of J~_1. Then the smallest positive eigenvalue is X2 = (j,./a)2 ; the corresponding eigen-
function is xv x
88
Any function w that satisfies the differential inequality
Lw _> - f (w), (x, t) E i2; (4.3)
and the initial and boundary conditions w (x, 0) = 0, w (0, t) = 0, wx(a, t) = 0, is a lower bound
for u, by virtue of the strong maximum principle and the parabolic version of Hopf's lemma. We
will seek a lower bound of the form w (x,t) = xVJ%()g (t). According to (4.3), we have
g'(t) Jv-1 25f(x"Jv(l v-ix/a)g (t))g(t axJvrr,-xag()(4)
g (t) + x"yj-i/~ (t)
The expression in the left member, which is independent of x, is bounded by the infimum of the
expression in the right member. If b _0, this infimum exists. Let G (g (t)) satisfy the inequality
G (g(t)) < inf f(vvj-xagt):x¬E [0, a ]. (4.5)x Jv(jv-ix/a)
Then we can determine g as the solution of the initial value problem
g'(t)+ (jv-i /a)2g (t) = G(g (t)), t >O; g (0) =0,
where the initial condition comes from u (x, 0) = 0. Let x0 denote the value of x for which the
infimum in (4.5) occurs. Then the time t 1 determined by the condition
xOv/v(jv-1x0/a)g (t 1) = c (4.6)
is an upper bound for the quenching time.
If b > 0, the solution of (1.5) is strictly greater than the solution for b 5 0. Thus, an upper
bound for the quenching time for b > 0 is given by that for b = 0.
5. Determination of Critical Length
We now give a procedure to compute the critical length a*.
Let jv-i be the first positive zero of the Bessel function of the first kind of order v - 1.
Choose a < j ,i and let UO = 0 on [0, a]. We construct a sequence (U,E N by defining U as
the solution of the boundary value problem
U,"+(b/x)U,'+f (U.i 1) = 0, x e(0, a); U,(0)=0,U(a)=0.
or, in terms of Green's function G,
a
U,,(x) = f G (x, t) f (U,_I-( )) d , n = 1, 2, -..-. (5.1)
0
The sequence is well defined.
89
THEOREM 5. The sequence {U} ne N converges monotonically upwards to the minimal solution 1I
of the h-vndary value problem (3.1). This minimal solution satisfies the inequality U < c on
[0, a].
Proof. Since f (0) > 0 and G ("-, ) > 0 in f2, it follows that U 1 > 0 on (0, a ]. From Lemma
1(ii) and Theorem 3, it follows that U > 0 on (0, a]. Since f is nondecreasing, we have
(U - U 1 )''+ (b/x)(U - U 1 )'<_0. Furthermore, (U - U1 )(0) = 0 and (U -U 1 )'(a) = 0. The
positivity of Green's function then yields the inequality U > U 1 on (0, a ]. Similarly,
U > U2 > U 1 on (0, a ], and, by induction,
S< U < Ui+1 <U < con (0, a], n =1, 2, -" -. (5.2)
Hence, there exists a function V(x) such that lim_.. U = V. The integrand in (5.1) is nonde-
creasing with respect to n and integrable, so it follows from the monotone convergence theorem
that V satisfies the equation
V(x)= J bG(x;V x)f(V())d .0
Hence, V is a solution of (3.1). It follows from (5.2) that V is minimal. 0
A determination of a* for any given b < I is to start with a = jv_ -S, a > 0, where 8 is a
positive constant. Then U can be computed by means of (3.4), with U0 = 0. To compute U.,
we may divide the interval [0, a ] into N equal subintervals. We then use the IMSL library sub-
routine ICSCCU (the cubic spline interpolation in single precision) to interpolate the function
U_1 at the N + I subdivision points. The subroutine ICSEVU (for the evaluation of a cubicspline to single precision) is used to evaluate U,_ 1 in the integrand, in order that the subroutine
DCADRE (to perform, to single precision, numerical integration of a function using cautious
adaptive Romberg interpolation) can be used to integrate the right-hand side of (3.4). In this way
we obtain U,(x) at the subdivision points. If U,(a) < c and
max {U,(6) - U,_1 (.): i = 2, 3, - - -, N) <0.5x10-d for a desired accuracy of d decimal
places, then u exists globally. If U,(a) c, then u quenches. In that case, we decrease a to
a = j,-1 - 28 and repeat the procedure, until we find that u exists globally for a = jj - pS for
some positive integer P. We then use the method of bisection to determine the value a** between
J v-r- 08 and j,_- - (a - 1)8 such that u exists globally for a 5 a**, but u quenches for a > a**.Since a* - a** can be made arbitrarily small, it follows that a** can be taken numerically to be
a*.
90
6. Example
We illustrate the results of the previous sections for f (u) = (1 - u)~1.
By Theorem 2, u quenches as u (a, t, tends to 1. The inequality (4.4) is
>a 2 _____________ __
gj)+ Jv-i 1
g (t) a xv/v( v-lx/a)g (t)[1 - xvJv(jv-tx /a)g (t)]
Consider the case b 0. Let xo denote the value of x where the infimum in (4.5) occurs, and let
m denote the quantity xovJv(jv-ixo/a). Then
g'(t) + (jv-i/a)2 g(t) = [m(1-mg(t))]~1 , 0 <g(t) S (2m)-, (6.1)
Here, the variables g and t can be separated. Let to denote the time when g (to) = (2m)~1.
Integrating the first of the equations (5.1) from 0 to to, we obtain2 -/ . 2 -A
to = a 4 _Jv-i tan-i {vi r4 [Jv-]
j V-1 a a a
2 . 2
--- 4 In I- (h- .)62
2 j v-1 2a(62
Next, integrating the second of the equations (6.1) from to, we obtain
g (t) = (2m)~'exp([4 - (jv-, /a)2 ](t - to)). (6.3)
According to (4.6), an upper bound t,' of the quenching time is given by mg (t 1 ) = 1. Using 6.3),
we find txp([4 - (jv. 1/a)2](tI - to)) = 2, from which we obtain
In 2t = to + . 2(6.4)
4 -(%v_1/a)
The quenching time for b = 0 is an upper bound for that for b > 0.
We deduce from (6.2) that 4 - (j v_1i/a) 2 > 0. Hence, u quenches when a > V 2j 1. In partic-
ular, when b = 0, v = /2 and J_%(z) = [2/(,tz)]J'cos z, so j_% = %A. Thus, u quenches if
a > (1/4)n. This result agrees with the conclusion of Walter [1976] that I* 5 'v for (1.1). The
solution quenches for a = 'i; (6.2) and (6.4) give the following estimate for the quenching time:1li 4
t = -( +ln ) = 0.31837 (to five decimal places).
Using the algorithm of Section 5, we have determined a* to five decimal places for various
values of b. We started the algorithm with a = 2jv-1 -8, instead of a = jv--8, since we
already know that u quenches if a > 'jv-1. The results are given in Table 1.
We note that 2a* = 1.5303 (to four decimal places) if b = 0, in agreement with the result of
Acker and Walter [1976] for the initial boundary value problem (1.1).
91
Acknowledgment
We thank Professor Man Kam Kwong for helpful discussions.
References
M. Abramowitz and I. A. Stegun, eds. 1964. Handbook of Mathematical Functions, NBS Appl.Math. Series, Vol. 55.
A. Acker and W. Walter 1976. "The quenching problem for nonlinear parabolic differentialequations," Lecture Notes in Mathematics, Vol. 564, Springer-Verlag, New York, pp. 1-12.
A. Acker and W. Walter 1978. "On the global existence of solutions of parabolic differentialequations with a singular nonlinear term," Nonlinear Anal.: Theory, Methods & Applic. 2,499-505.
V. Alexiades 1980. "A singular parabolic initial-boundary value problem in a noncylindricaldomain" SIAM J. Math. Anal. 11, 348-357.
V. Alexiades 1982. "Generalized axially symmetric heat potentials and singular parabolic initialboundary value problems" Arch. Rat. Mech. Anal. 79, 325-350.
V. Alexiades and C. Y. Chan 1981. "A singular Fourier problem with nonlinear radiation in anoncylindrical domain," Nonlinear Anal.: Theory, Methods & Applic. 5, 835-844.
H. Brezis, W. Rosenthkrantz, B. Singer, and P. D. Lax 1971. "On a degenerate elliptic-parabolicequation occurring in the theory of probability," Comm. Pure Appl. Math. 24, 395-416.
A. Friedman 1964. Partial Differential Equations of Parabolic Type, Prentice-Hall, EnglewoodCliffs, N. J.
H. Kawarada 1975. "On solutions of initial-boundary problem for u, = u. + (1 - u)ft," Publ.RIMS, Kyoto Univ. 10, 729-736.
J. Lamperti 1962. "A new class of probability theorems," J. Math. Mech. 11, 749-772.
H. A. Levine 1985. "The phenomenon of quenching: A survey," in: Trends in the Theory andPractice of Non-Linear Analysis, V. Lakshmikantham, ed., Elsevier Science Publ., NewYork, pp. 275-286.
H. A. Levine and J. T. Montgomery 1980. "The quenching of solutions of some nonlinear para-bolic equations," SIAM J. Math. Anal. 11, 842-847.
G. N. Polozhiy 1967. Equations of Mathematical Physics, Hayden Book Comp, Inc., New York.
M. H. Potter and H. F. Weinberger 1967. Maximum Principles in Differential Equations,Prentice-Hall, Englewood Cliffs, N. J.
W. Walter 1976. "Parabolic differential equations with a singular nonlinear term," Funkcial.Ekvac. 19, 271-277.
92
EXISTENCE RESULTS OF STEADY STATES OF SEMILINEARREACTION-DIFFUSION EQUATIONS AND THEIR APPLICATIONS
C. Y. Chan* and Man Kam Kwong
Mathematics and Computer Science DivisionArgonne National Laboratory
9700 South Cass AvenueArgorne, IL 60439
Abstract
In this paper we study conditions that qurantee the existence of steady-state solu-tions of a semilinear parabolic equation in a bounded interval in one spacedimension. The nonlinear reaction term can blow up at some finite value of thedependent variable. Applications are given in the study of boundary value prob-lem of second-order ordinary differential equations, finite-time quenching andblow-up of nonlinear heat equations.
1. Introduction
Let n = (a,b) x (0,oo). Let us consider the one-dimensional parabolic equation
u,=uX+f(x,u,ux) infl , (1.1)
subject to the initial-boundary conditions on its parabolic boundary Ba:
u(x,0)=g(x)for a5x5b, (1.2)
u(a,t)=h(t), u(b,t)= k(t) fort >0, (1.3)
where f: [a,b] x (c,d) x (-o,oo)--+(-o,co) is continuous; a, b, c, and d are constants with
c ? - o, d S oo; and g, h, and k are continuous functions; g (a) = h (0), and g (b) = k(0). We are
interested in the asymptotic behavior of the classical solutions. In particular, we would like to
know whether a classical solution will approach a steady-state as t -+ . We remark that in gen-
eral, however, a solution of (1.1), (1.2), and (1.3) may not exist globally for all t > 0. If there is a
finite time to at which lim u (x,t) = o, we have a finite-time blow-up. On the other hand, if
there is a to at which lim u (x,t) = d <0o, causing f to blow up to infinity, we have the
phenomenon of quenching.
* Partially supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S.
Department of Energy, under Contract W-31-109-Eng-38 while a Visiting Scientist at Argonne National Laboratoryduring the summer of 1987, and in part by he Board of Regents of the State of Louisiana under Grant LEQSF(86-89)-RD-A-11. Permanent address: University of Southwestern Louisiana, Lafayette, LA 70504-1010.
93
In Section 2, we quote the well-known Nagumo lemma (cf. [Walter, 1970]) for (1.1). This
is used to prove existence of a steady-state solution for the problem (1.1), (1.2), and (1.3) under
fairly weak smoothness conditions on fin Section 3. These results are then applied in the rest of
the paper to the study of three interesting problems. In Section 4, we extend some results on
quenching phenomena by Acker and Walter [1976 and 1978]. In Section 5, existence of the
minimal and maximal solutions of boundary value problems of second-order ordinary differential
equations is established by embedding them as steady-state solutions of a parabolic equation.
This gives an alternative to the monotone iterative methods used, for example, by Ladde, Laksh-
mikantham, and Vatsala [1985]. In Section 6, we give sufficient conditions on the initial data that
give rise to the phenomena of finite-time blow-up and the decay of the solution, respectively.
2. Nagumo's Lemma
In the rest of the paper, by a solution of (1.1), we mean a classical C2 solution.
The classical monograph by Walter [1970] has an extensive and excellent account of
Nagumo's lemma on parabolic differential inequalities. We quote here a version that suffices for
our purpose.
A function f(x,u,v) is said to satisfy a local one-sided Lipschitz condition if, given positive
constants c 1 and c2, there exists a positive constant L such that
f(x,u,v) -f(x,w,v) < L (u - w) (2.1)
whenever u > w, Iu I S c1, Iw I Sc 1 , and Iv1i s5c2 .
THEOREM 1. Suppose that the function f satisfies a local one-sided Lipschitz condition. If w (x, t)
is a C -function in t, and a C 2 -function inx such that
w, 2( )w=+f(x,w,ww) in sl, (2.2)
w(x,0) (S)u(x,0) foraSxsb,
w(a,t) ( )u(a,t), w(b,t) ( )u(b,t) fort>0,
then w (x,t) (S ) u (x, t) in f, where u (x,t) is the solution of the initial-boundary value problem
(1.1), (1.2), and (1.3).
It can be shown that Nagumo's lemma still holds if the local one-sided Lipschitz condition
(2.1) is weakened to a condition similar to one used by Kamke in the study of uniqueness of ini-tial value problems of ordinary differential equations (cf. [Harman 1973, p. 311).
As applications of Theorem I we give two examples, whose results will be used later on.We now assume that f satisfies the hypotheses of Theorem 1.
94
EXAMPLE 1. The steady-state equation associated with (1.1) is
U'x)+f(x,U(x),U'(x)) = 0. (2.3)
If it has a solution such that for t > 0, U (a) h (t), and U(b) k (t), and if
U(x) u (x, 0) = g (x), then by Theorem 1, we have U (x) u (x,t) in Q.
EXAMPLE 2. Let us consider the case of zero boundary conditions. If there exists an initial
condition g (t) z 0 such that its corresponding solution u1 (x,t) tends to zero for all x as t tends to
infinity, and if for some to, the solution u < g 1(x), then by Theorem 1,
u(x,t) u1(x,t -t0 ),
and hence lir u (x,t) = 0.
Similarly, if we would like to show lim u (x,t) = oo, we need only show that for some g (x),
the corresponding solution u 1 satisfies lim u 1(x, t) = c, and that for some to, u (x,to) 2 g 1 (x).
3. Steady-State Solutions
In this section, we give conditions that guarantee existence of steady-state solutions for
(1.1), (1.2), and (1.3). The first lemma is an extension of Theorem 1(a) of Acker and Walter
[1978].
LEMMA 2. Let f be such that Nagumo's lemma holds. If h (t) and k (t) are nondecreasing, and
g"'(x) +f(x,g(x),g'(x)) 0, (3.1)
then for each fixed x, u (x,t) is nondecreasing with respect to t.
Proof. Using Theorem I on u and g, we obtain u (x,t) 2 g (x) in s. Again by using Theorem 1
on u(x,t+e) and u(x,t), for any given E> 0, we have u(x,t+E) u(x,t). Since e is arbitrary, the
lemma follows.
Next, we show existence of a steady-state solution.
THEOREM 3. Suppose that h (t) and k (t) are nondecreasing, (3.1) holds,
sup u(x,t) < d , (3.2)
sup iu(x,t)I <0, (3.3)
where d is the constant appearing in the domain of definition of f, and f is locally Lipschitz
95
continuous in its second and third variables, then lim u(x,t) = U (x) exists uniformly, where U ist-00
a solution of the steady-state problem (2.3), with
U (a) = lim h (t) , U (b) = iim k (t) . (3.4)
Proof. By Lemma 2 and (3.2), lim u(x,t) exists for each x. We need to show that it is a steady-
state solution. For each fixed x,
! u1(x,)dt = lim u (x,t) - g (x) <o.o '-+**
Hence,
be.
f J u,(x,t)dtdx < .aO
From Lemma 2, u,(x,t) z 0. By Fubini's theorem,
I~b
f u,(x,t )dxdt < cn . (3.5)
Oa
Let
b
K(t) = ju,(x,t)dx .a
Then, K(t) is nonnegative, but it need not approach 0 as t tends to on. By (3.5), there exists a
sequence tM tending to a such that
K(t.) tends to0 . (3.6)
Let xo be any fixed point in the interval (a,b). >From (3.3), the sequence (u(xo,t.)) is bounded,
and hence, by passing to a subsequence if necessary, we may assume without loss of generality
that its limit, denoted by y, exists.
Let V(x) be the solution of the initial value problem,
V'(x) +f(x,V(x),V'(x)) = 0 , (3.7)
V(x0 ) = lim u(xo,t.)= U(xo) , V(xo) = Y . (3.8)
The proof of the theorem will be complete if we can show
lim u(x,t) = V(x),
since then V(x) * U (x). To do so, let us transform (3.7) and (3.8) into a system of first-order
integral equations:
96
- U(xo) +V '(s) d 39V(x)_= + [ T ] , (3.9)Y -f(s,V (s))
where VT(x) = (V(x),V'(x)) is the transpose of V. Similarly, (1.1) becomes
ut(x)= kxot.) )+ -u,(s~t)( ds ,(3.10)ux u(xo, t) - f(s~un (s))+ u,(st)
where u~ (x) = (u (x,t), u(x,t)). Since f is locally Lipschitz continuous, there exists a constant
c4 such that-T - T
I f(s,V T(s)) - f(s,unT (s)1 <_ (c4-1) I V(s) - E,(s) I.
Let us subtract (3.10) from (3.9), and use the norm, IX I = 1X 1 I + IX 2 1, where XT = (X1 ,X 2).
We have
IVf(x) - 5(x)1 <- V(xo) - E(xo)I + c4 j IV(s) - u(s) ids +K(t) . (3.11)x
For any given e > 0, it follows from (3.6) and (3.8) that by choosing n sufficiently large,
I V(x0) - u(xo) I +K(t) < E .
Using Gronwall's inequality on (3.11), we have
I V(x) - U(x. I SEexp[c4(b - a) ,
which implies that as n tends to a*, E(x) converges to V(x) uniformly with respect to x e [a,b 1.This completes the proof of the theorem.
By applying Theorem 3 to -u (x, t), we obtain the following result.
THEOREM 4. Under the hypothesis (3.3), if h (t) and k (t) are nonincreasing, and f is locallyLipschitz continuous in its second and third variables,
g"(x) +f(x,g (x),g'(x)) 5 0, (3.12)
infu(x,t)>ca
where c is the constant appearing in the domain of definition of f, then for each fixed x, u (x,t) is
nonincreasing with respect to t, and lim u (x,t) = U (x) exists uniformly.f-
We remark that although the proof of Theorem 3 also shows that lim u1(x,t.) = U(x) uni-
formly, it is not clear whether lim u1(x,t) = U'(x) uniformly. Theorems 3 and 4 are still true ifI-
97
one of the boundary conditions is replaced by Neumann's condition, because such a problem can
be extended, with respect to either x = a or x = b, into the form given by (1.1), (1.2), and (1.3).
Below, we give a sufficient condition for (3.3) to hold. This is related to Nagumo's condi-
tion in the study of boundary value problems for second-order ordinary differential equations.
THEOREM 5. If f satisfies a local one-sided Lipschitz condition, the boundary values h (t) and
k (t) are both constants, (3.1) holds, u is bounded, and there exist continuous functions r (u), and
q (u,2) > 0 such that for all I u,I > a, which is a positive constant,
f(x,u,u2) S r(u)q(u 2) , (3.13)
dR 0=j / (3.14)ul 7(R )
then (3.3) holds.
Proof. By Lemma 2, u, z 0. Thus,
u, +f(x,u,u,) 20 . (3.15)
For an arbitrarily fixed t, say to, let x1 be any point in (a,b). If Iu(xi,to)I 5a, then the
theorem is proved. Therefore, suppose that either u(x ,to) > a or u (x,,to) < -a. We would
like to show that there exists a constant c 5 > a which , independent of x 1 such that
l u1(x i,to)| < C5.
Let us consider the case u(x 1 ,to) > a first. At least in a neighborhood to the right of the
point x1 , u is strictly increasing. Thus, we have two subcases: u(x,to) >0 for x 5x < b, or
there exists a point x 2 (chosen closest to x1 ) in (x 1 ,b) such that u,(x2,to) = 0.
Case: u(x,to) > 0 for x 1 < b. Let
c6 = max(a, -g'(a)+1, g'(b)+1).
There must exist a point x3 in (x1 ,b) such that
g(b)-g(x3 )
b-x3
By Lemma 2, u (x,t) z g (x). Since u(b,t) = g (b), it follows that
u(b,to)- u(x3 to)
b -x 3 <c6 -
By the Mean Value Theorem, there exists a point x2 in (x3,b) such that
98
u(b,to) - u(x3 ,to)ux(x2,t) = b-X 3
If ux(x 1 ,t) c6, then the theorem is proved; if u(xi,t)> c6 , then by applying the Intermediate
Value Theorem to ux, there exists a point x0 in (x1 ,x2 ) such that ux(xo,to) = c6. On the interval
[x1 ,xo], u is strictly increasing; we may use u as the independent variable. Let R = Iu1 I 2. From
(3.15),
dR-+2f(x,u,ux)0 foru(x1 ,to) u 5 u(xo,to).du
By (3.13), -2r (u)q(R). Thus,du
u,2(x,10) u(x0 S10 )
J <- 2 f r (u)du . (3.16)
C6 2 q (R) Ux,0
Since u is bounded, the right-hand side is bounded. By (3.14), u 2(x,,t) must be bounded, and
hence (3.3) holds.
Case 2: ux(x 2 ,to)= 0. As in the previous case, we may apply the Intermediate Value
Theorem to u, to conclude that there exists a point xo in (x,,x2 ) such that u(xo,to) = c6. The
rest of theproof is the same as in Case 1.
Now let us consider r the situation ux(x l,to) <-a. In this case, u (x,to) is decreasing at least
in a neighborhood to the left of x 1. Again, we have two subcases: ux(x,to) < 0 fora < x 5 x1 , or
there exists a point x2 (chosen closest to x1) in (a,x1 ) such that ux(x2,to) = 0. In either case, a
similar proof as in ux(x 1 , t)> a shows that there exists a point x0 in (a,x1 ) such that
ux(xo,to) = -c 6 and u is strictly decreasing on the interval [xo,xi]. By using u as the indepen-
dent variable and R as the dependent variable, it follows, as before, from (3.13), (3.14), and (3.15)
that ux(x 1,t) is bounded.
Similarly, we have the following theorem.
THEOREM 6. Under the hypotheses of Theorem 5, except that (3.1) and (3.13) are replaced
respectively by (3.12) and
f(x,u,ux) -r(u)q(ux 2 ) , (3.17)
then (3.3) holds.
We remark that the hypothesis on the boundedness of u in Theorems 5 and 6 is used to
deduce the boundedness of the righthand side of (3.16). If we impose instead the assumption
99
j r(u)du <00,
then the boundedness of u follows from the conclusion that u is bounded.
As a consequence of Theorem 3, we have the following result, which is used in Sections 4
and 6 in discussing the phenomena of quenching and finite-time blow-up, respectively.
THEOREM 7. Under the hypotheses of Lemma 2 and (3.3), if f is locally Lipschitz continuous in
its second and third variables, and the problem (2.3) and (3.4) has no solution, then either there
exists afinit : time T such that
lim sup u(x,t) = d <oo,
or
lim sup u(x,t) = 00.1-+n x
4. Quenching
The terminology quenching was first introduced by Kawarada [1975] when he studied the
problem
u, = uxx+ 1 infl2,
with g, h, and k being identically zero. The solution u is said to quench if there exists a finite time
T such that
limsup(lu,(x,t)I: a xSb)= o. (4.1)t- T
He proved that there existed a length b - a beyond which
lim sup u(x,t) = 1 . (4.2)-+T- x
He claimed that (4.2) implied (4.1). Obviously, (4.1) is a sufficient condition for (4.2). If his
claim were true, the two conditions would be equivalent. Thus in studying quenching
phenomena, Walter [1976] used the necessary condition (4.2). Similarly, Acker and Walter [1976
and 1978] determined the maximum length of the interval beyond which
lim sup u(x,t) = d < cc, (4.3)
where lim f = co when they investigated such phenomena for the more general equations
u, = u+f(u), (4.4)
100
u, = u +f(u,u) , (4.respectively. Further references on this topic can be found in the survey by Levine [1985].
We note that Kawarada's proof that (4.2) implied (4.1) was not entirely correct. Recently,
we [Chan and Kwong 19891 used a completely different proof to establish the fact that (4.3)
implied (4.1) for (4.4) and (4.5), respectively, subject to (1.2) and (1.3). We have thus covered
Kawarada's claim as a special case.
Below, we extend the result of Acker and Walter [1978] for (4.5) to (1.1) under weaker
hypotheses.
THEOREM 8. Under the hypotheses of Theorem 5, if f(x,u,u ) is continuously differentiable, f,
is nondecreasing,
f(x, 0,0) > 0, (4.6)
and lim f(x,u,ux) = oo uniformly with respect to x and u, then there exists a critical length
below which the problem (1.1) with g (x) a.0 (and h (t) a 0 . k (t)) has a solution for t > 0, and
beyond which (4.3) holds.
Proof. Let I * denote the supremum of all values I = b - a such that the steady-state problem
(2.3) subject to
U(a)=O=U(b)
has a solution. It follows from Theorem 3 and Example 1 that for I <0', u exists for t > 0. To
show that (4.3) occurs for I > I , let u(x,t ; b) denote the solution of (1.1) with zero initial and
boundary conditions for all t > 0. >From (4.6) and (1.1),
u, > uzx +f3(x~tll,(i)u +f2(x, Bi, I) .
where f; denotes the partial derivative of f with respect to the idt variable, and Tb and t lie
between u and 0, and u, and 0, respectively. Since f2 is bounded above, it follows from the
strong maximum principle that u > 0 in Q. For any constants a > 0, it follows from our
hypothesis, fr 0, that the function
y(x,t) .u(x +6,t ;b + a)-u(x,t ;b) for0 < :5 a,
satisfies
y, Zy7+f3( ,1,)yz +f2( ,r, )y in (f,
where 4, rt, and ( lie, respectively, between x + S and x, u(x + S,t ; b + a) and u(x,t ; b), and
u,(x + 8,t ; b +a) and u(x,t ; b). We have made use of the Mean Value Theorem under the
assumption that u(x+8,t ; b+a) exists for t >0. Since f(x,0)=0, '(a,t)> 0, )(b,t) 0, we
have
101
(4.5)
u(x+bS,t;b+c)>u(x,t;b) in f2 for0<S5a
by the strong maximum principle. Let us choose positive numbers to and E such that 0 < e < 1,
f(x,z,z)2 a2 +8e/a 2 for 0<d-E5z <d and -4/azx 4/a, (4.8)
'4(x(to),to ; b) = d - E,
where x(to) is the point at which u(x,to ; b) attains its maximum. Let us consider the domain
D E (x(to), x(to)+a) x (to,oo). By (4.7) and Lemma 2, u(x,t ; b+ a) d - e on the parabolic
boundary 3D. On the other hand, the function,
z(x,t) = d - e + [x - x(to)] [x (to) +a - x](t - to),
attains the value d - e on 3D, and by (4.8),
z. +f(x,z,zx) >_z, in (x(to), x(to)+ a) x (to,to+4ea2 )uaD.
By Theorem 1, z iu(x,t ; b +a) inD_. Since
z(x(to)+ a/2, to + 4e/a2 ) = d ,
it follows that u (x,t ; b + a) attains d in a finite time. This contradiction proves the theorem.
We note that the above proof does not require the existence of a curve $(t) such that
u (xt; b) is monotone increasing in x on [a,$(t)] and monotone decreasing in x on [$t),b] (cf.
Theorem 1(b) of Acker and Walter). We also note that Acker and Walter required the assumption
that there exists a constant L (B) where B is any constant less than d such that whenever 05 z B
and Ip I 1, the following inequalities hold:
f(z,p) S L (B)p2 , (4.9)
f=(z,p) 5L (B)Ip I , (4.10)
-pf,(z,p) SL(B)p 2 . (4.11)
We do not require assumptions (4.10) and (4.11), and instead of (4.9) we need the weaker
hypotheses (3.13) and (3.14).
5. Minimal and Maximal Solutions
Let us consider nonnegative solutions of the problem
y'+ f(x,y,y') = 0 , (5.1)
y(a)=0=y(b). (5.2)
In general, there can be more than one solution, and they need not be ordered in any way. Thus,
102
(4.7)
there may or may not be a minimal or a maximal solution. A common situation in which such a
solution exists occurs when f satisfies certain monotonicity conditions, which allows the use of
monotone methods to set up successive approximation schemes. Theorems 3 and 4 enable us to
establish, respectively, existence of a minimal and a maximal solution without any monotonicity
requirement.
THEOREM 9. In addition to the hypothesis of Theorem 5, we assume that (4.6) holds and that f is
locally Lipschitz continuous in its second and third variables. If the problem (5.1) and (5.2) has
a nonnegative solution, then it has the nonnegative minimal solution.
Proof. Let us consider the associated problem (1.1) with zero data on its parabolic boundary. By
Example 1, u (x,t) S y (x) for any nonnegative solution y of the problem (5.1) and (5.2). From
(4.6), u 0 in 0 by Theorem 1. By Theorem 3, lim u(x,t) exists uniformly and is a solution of
the steady-state problem. Hence,
lim u(x,t) Sy(x),-+e
and lim u(x,t) is the minimal solution.-+M
We remark that if f(x, 0,0) = 0, then the minimal solution is the trivial solution. Below, we
give criteria for existence of the maximal solution.
THEOREM 10. Under the hypotheses of Theorem 5, if f is locally Lipschitz continuous in its
second and third variables, and if all nontrivia wlutions y of the problem (5.1) and (5.2) are
bounded by a constant M > 0, and there exists a solution Y (x) of (5.1) with inf Y (x) z M, thenx
the problem (5.1) and (5.2) has the maximal solution.
Proof. Let h (t) Y(a), k (t)a Y (b), and g (x) * Y(x). By Theorem 1, the solution u of the prob-
lem (1.1), (1.2), and (1.3) is an upper bound of y. By Theorem 5, u, is bounded. By Theorem 4,
u converges to U, and hence the problem (5.1) and (5.2) has the maximal solution.
As an application of the above theorem, we have the following result.
COROLLARY 11. If f is locally Lipschitz continuous in its second and third variables, and if there
exist positive continuous functions r (y) and q (y'2) such that for all I y' a, where a is a posi-
tive constant,
103
f(x,y,y') 5 r (y)q(y 2),
Jr(y)dy<J d =?0 0 q(R)
then the problem (5.1) and (5.2) has the maximal solution.
Proof. Let y be any solution of (5.1). Using an argument similar to the proof of Theorem 5, we
arrive at the inequalities
y (x1 ) y(x2)
j (Y 2 Jf r(u)du < o.«2 q(R) y(x1)
Thus Iy'(x) I <c7 for some constant c7 .
Now, if y satisfies (5.2), then !yY(x) 5 c7 (b - a). On the other hand, if y is a solution of
(5.1) subject to the initial conditions, y (a) = 2c7(b - a), y'(a) = 0, then y (x) z c7 (b -a). The
hypotheses of Theorem 10 are satisfied, and hence the problem (5.1) and (5.2) has the maximal
solution.
Our approach also applies to more general boundary conditions and to more general bound-
edness requirements. A solution ym (yM) of the problem (5.1) subject to
y(a)=c8 ,y(b)=c9 , (5.3)
where cS and c9 are arbitrary constants, is said to be the minimal (maximal) solution relative to a
function y 1 (Y2) if
ym(x) zy 1 (x), ym(x) 5 y (x) , (YM(x) sy2 (x), YM(x) -y (x)) (5.4)
for any other solution y 2zy 1 (y Sy2).
The following result can be obtained by using Theorems 1, 5, and 3 for the minimal solu-
tion, and Theorems 1, 6, and 4 for the maximal solution since we may choose h(t) and k(t) as
nondecreasing (nonincreasing) functions such that h (0) = y 1 (a) (h (0) = y2 (a)), k (0) = y (b)
(k(0) = y2 (b)), lim h (t) = c 8, and lim k(t) = c9.
THEOREM 12. Let f be locally Lipschitz continuous in its second and third variables, and (5.1)
have two solutions y 1(x) y2 (x) such that
y1 (a)5c s y2 (a), y1 (b)5c9 y2 (b).
Under the hypotheses of Lemma 2 and Theorem 5, the minimal solution relative to y 1 (x) for the
problem (5.1) and (5.3) exists. On the other hand, under the hypotheses of Theorems 6 and 4, the
maximal solution relative to y2 for the problem (5.1) and (5.3) exists. If the minimal and the
maximal solutions coincide, then the problem (5.1) and (5.3) has a unique solution y such that
104
y1 (x) y(x) y 2 (x).
6. Asymptotic Decay and Blow-Up
Let us consider the problem (1.1) subject to
u(x, 0) = g (x) >-0, u(a,t) = 0 = u(b,t), (6.1)
where f is continuous, locally Lipschitz continuous in its second and third variables, and
f(x,u,ux) z0 , f(x, 0,0) = 0 . (6.2)
A classical prototype is given by
f(x,u)=u', s> 1 . (6.3)
It is well known (cf. [Weissler 1984], and references quoted there) that if g is large enough, then u
may blow up in a finite time, whereas if g is small enough, u is bounded for all t. We would like
to show that for a certain class of f, including (6.3), if g is smaller than the unique nontrivial, non-
negative, and bounded solution U of the corresponding steady-state problem
U''(x)+f(x,U(x),U'(x)) = 0 , U(a) = 0 = U(b) , (6.4)
then u decays to zero, and if g is larger than U, then blow-up occurs. In particular, our results
apply to equations in which the nonlinear term may have a non-constant coefficcient. For the
simpler class given by (6.3), similar results are known; see, for instance, [Ni, Sacks, and
Tavantzis 1984].
From (6.2), u .0 is a solution of the problem (6.4). We assume that besides this, there is
exactly one nontrivial nonnegative solution. This is true in the case of (6.3). It was shown by
Coffman [1972] that the same is true for some classes of f, including x"u',v > 0, s > 1, and
a > 0. More recently, Ni [1983], and Ni and Nussbaum [1985] studied Coffmans's problem and
extended his results.
LEMMA 13. Assume that the problem (6.4) has a unique nontrivial, nonnegative, and bounded
solution U. Let
f(x,au,ux) S a f(x,u,u.) for 0 < a < 1 , (6.5)
(3.17) and (3.14) hold. If for some to > 0, u (x,to) 5 aU (x), then the solution of the problem
(1.1) and (6.1) decays to 0 as t tends to infinity.
Proof. By Theorem 1, u 0 in 0. From Example 2 of Section 2, it is sufficient to show that the
solution ui(x,t) of the problem (1.1) and (6.1) with g (x) = aU(x) decays to zero as t tends to
infinity. >From (6.5), g satisfies (3.12). It follows from Theorems 6 and 4 that lim u (x,t)I--
105
exists. By Theorem 4, u 1 (x, t) is nonincreasing with respect to t. Thus lir u1 (x,t) = 0.
Similarly, the following lemma can be proved.
LEMMA 14. Under the hypotheses of Lemma 13 with (3.17) replaced by (3.13), if for some
to > 0, u (x,to) U(x)/a, then the nontrivial solution of the problem (1.1) and (6.1) becomes
unbounded as t tends to infinity.
Below, we give some comparison results.
LEMMA 15. Assume the problem (6.4) has a unique bounded solution U which is positive for
a < x < b. Let
f(x,u,u,) / u be nondecreasing in u . (6.6)
If
g(x) U(x), g *U, (6.7)
then there exists a constant c 10 < I such that
u(x,t) S c1 oU(x) for sufficiently large t. (6.8)
On the other hand, if
g(x) Z U(x), g(x)*U(x), (6.9)
then there exists a constant c 11 > 1 such that
u(x,t) z c 11 U(x) for sufficiently large t .
Proof. Let us prove the first part since the proof of the second part is similar. By Theorem 1,
g (x) U(x) gives u(x,t) U(x) fort > 0. Let y(x,t) be the solution of the linear problem
%, = Wu + f(x,U, U')y/ U in s2 , (6.10)
y(x, 0) = g (x), y(a,t) = 0 = i(b,t) . (6.11)
From (6.6), yi, iW +f (x,u,ur)y/u. By Theorem 1,
u(x,t) y(x,t) innA . (6.12)
Let us solve the problem (6.10) and (6.11) by the method of separation of variables. Let
{$,: n = 1,2,3,...) be the set of normalized eigenfunctions of the Sturm-Liouville problem
$"(x)+f(x,U(x),U(x))$(x)/ U(x) = -4(x), $(a) = 0 = $(b), (6.13)
with k,: n = 1,2,3,...) as its corresponding eigenvalues. Obviously, zero is an eigenvalue of the
problem (6.13) with U (x) as its corresponding eigenfunction. Now, U is positive for a < x < b.
106
b
By the classical Sturm-Liouville theory, U / II U II, where II UI112 = J U2(x)dx, is the eigenfunc-a
tion $1 corresponding to the smallest eigenvalue X1 = 0. Using the eigenfunction expansion
%V(x,t) = a~e~ k'$(x),X3=I
where g (x) = a,$n(x) , and a. = J g (x)$ (x)dx , we obtainR=1 a
b
lim y1(x,t) = a 1$1(x) = [gI(x)U(x)dx jU(x) ) U 2 <U(x).
From (6.12), we have (6.8).
By combining Lemmas 13 to 15, and noting that (6.6) implies (6.5), the following result can
be deduced.
THEOREM 16. Let f be continuous, locally Lipschitz continuous in its second and third variables,
and satisfy (3.14), (3.17), (6.2), and (6.6). Also, assume that the problem (6.4) has a uniquebounded solution U which is positive for a < x < b. If (6.7) holds, then lim u (x,t) = 0. On the
other hand, if(6.9) holds, then lim u(x,t) = .
Let us give some more results on unbounded solutions.
THEOREM 17. Under the hypotheses of Theorem 16, for u to be unbounded, either there is a finite
blow-up time, or u exists for t > 0, and
lim u(x,t) = oo for a < x < b ; (6.14)
furthermore, the divergence in (6.14) is unform in any proper closed subinterval of (a,b).
Proof. By Lemma 15, we may assume u(x, 0) = c ,1 U(x), where c 11 > 1. Let us assume that the
blow-up time is not finite.
Case 1:
b
lim J u(x,t)U(x)dx = 00 .I -f"" a
Thus, there exists a sequence (t.: n = 1,2,3,...) such that
107
b
b 1 ' E u(x,t U (x)dx
tends to infinity. By using t 1 as the initial time, a proof similar to that of Lemma 14 gives
lim u(x,t) b1 U(x).
The relation (6.14) and the uniform divergence in any proper closed subinterval then follow.
Case 2:
b
lim J u(x,t)U(x)dx < 0 .I -- +_ a
Suppose there is only one point x3 E (a,b) such that lim u(x3 ,t) = c. Then, for
x e (a , x3 ), lirm u (x,t) <c00. By Theorems 5 and 3, U is a solution of1-"
U'(x)+f(x,U(x),U'(x))=0 for a <x <x3 .
From (6.2), f(x,U,U') 0, and hence U' 0, which implies that U is concave downwards for
a < x < x3 . This contradicts lim U(x) = c.X-.X 3
Let us suppose that there are two points x4 and x5 between a and b with x4 < x5 such that
lim u(x4 ,t) = G = lir u(x5 ,t).t --- 1 -+oo
For any given number N > 0, there exists a time t 1 such that u(x4 ,t1 ) N, and u(x5 ,t1 ) N. By
Lemma 2, u is nondecreasing with respect to t. Let us consider the problem V, = VX for
x4 <x <x5 and t1 < t, V(x,t1) = u(x,t 1), V(x 4 ,t) = N = V(x5,t) for t > t1. By Theorem 1,
V(x,t) S u(x,t) for t1j < t <coc. On the other hand, lim V(x,t) = N for x4 < x < x5 . Thus,-*m
lim u(x,t) = co for x4 5 x 5 x 5. Hence,
lim u(x,t)U(x)dx= c.
Thus, Case 2 is really void.
Let us give a result on the finite blow-up time.
THEOREM 18. Under the hypotheses of Theorem 17, if f(x,u,uL) ' Eu' for some positive con-
stants e and s with s > 1, then a finite blow-up time must occur.
Proof. It follows from Theorem 17 that if a finite blow-up time does not occur, then for any
given large constant N > 0, it is possible to find a time to such that on any proper closed subiner-
val [a+,b_] of (a,b), u(x,t) z N for t > to. By Theorem 1, a solution of the problem,
108
v, = v + Ev Pin [a+ ,b-.] x [to,**) ,
v(x,to) = N = v(a,t) = v(b.,t) ,
is a lower bound of u. It is well known (cf. [Weissler, 1984]) that for N sufficiently large, v blows
up in a finite time. Hence, the theorem is proved.
References
A. A. Acker and W. Walter 1976. "The quenching problem for nonlinear parabolic differentialequations," Lecture Notes in Mathematics 564, Springer-Verlag, New Yoik, pp. 1-12.
A. Acker and W. Walter 1978. "On the global existence of solutions of parabolic differentialequations with a singular nonlinear term," Nonlinear Anal. 2, 499-505.
C. Y. Chan and Man Kam Kwong 1988. "Quenching phenomena for singular nonlinear para-bolic equations," Nonlinear Anal. 12, 1377-1383.
C. V. Coffman 1972. "Uniqueness of the ground state solution fc. \u - u + u 3 = 0 and a varia-tional characterization of other solutions," Arch. Rational Mech. Anal. 46, 81-95.
P. Hartman 1973. Ordinary Differential Equations, Baltimore.
H. Kawarada 1975. "On solutions of initial boundary problem for u, = u, + 11(1-u)," Pbl.RIMS, Kyoto Univ. 10, 729-736.
G. S. Ladde, V. Lakshmikantham, and A. S. Vatsala 1985. Monotone Iterative Techniques forNonlinear Differential Equations, Pitman Advanced Publishing Program, Boston.
H. A. Levine 1985. "The phenomenon of quenching: A survey," in Trends in the Theory andPractice of Non-linear Analysis, ed. V. Lakshmikantham, North Holland, New York, pp.275-286.
W. M. Ni 1983. "Uniqueness of solutions of nonlinear Dirichlet problems," J. Diff. Eq. 50,289-304.
W. M. Ni and R. Nussbaum 1985. "Uniqueness and nonuniqueness for positive radial solutionsof Au +f(u,r) = 0," Comm. Pure Appl. Math. 38, 67-108.
W. M. Ni, P. E. Sacks, and J. Tavantzis 1984. "On the asymptotic behavior of solutions of cer-tain quasilinear parabolic equations," J. Diff. Eq. 54, 97-120.
109
W. Walter 1970. Differential and Integral Inequalities, Springer-Verlag, Berlin.
W. Walter 1976. "Parabolic differential equations with a singular nonlinear term," Funkcial.Ekvac. 19, 271-277.
F. B. Weissler 1984. "Single point blow-up for a semilinear initial value problem," J. Diff. Eq.55, 204-224.
110
UNIQUENESS RESULTS FOR SOME NONLINEARINITIAL AND BOUNDARY VALUE PROBLEMS
Hans G. Kaper and Man Kam KwongMathematics and Computer Science Division
Argonne National LaboratoryArgonne, IL 60439
Dedicated to Professor James Seiun on the occasion of his 60th birthday
Abstract
In the first part of this article a uniqueness theorem is presented for nonlinear ini-tial value, problems of the type
dx /dt = $(x) +p (t), t > 0; x (0) = 0; (*)
where p is a real-valued continuous function that may change sign near the ori-gin.
The theorem is subsequently applied to establish a result related to the unique-ness of nonnegative nontrivial radial solutions of nonlinear boundary value prob-lems of the type
V-(A (I Vu I)Vu +f (u) = 0, x E R"; lim x_,.. u(x) = 0; (**)
where A is a positive continuous function such that the product pA (p) is mono-
tone. The class of equations (**) includes the mean curvature equation.
1. Statement of Results
In the first part of this article we are concerned with nonlinear initial value problems of the
type
- = $(x(t)) +p(t), t > 0; x(0) = 0. (1.1)dt(11
Here, is a differentiable function, which is defined on some (bounded or unbounded) interval
(0, a), vanishing with a vertical tangent at the origin, monotonically increasing and concave (i.e.,
$'is decreasing) on (0, a); p is a real-valued continuous function which may change sign near the
origin. Let P denote the integral of p,
P(t) =Jp (s) ds, t ? 0. (1.2)0
We prove the following theorem.
111
THEOREM 1. If P (:) 0 for t _0 and there exists a sequence (tj) e N of points t > 0, which
converges to 0, such that P(t1) > 0 for all j, then (1.1) has at most one nonnegative (nontrivial)
solution.
Theorem 1 can be generalized trivially to initial value problems of the type
dx- = g (t)4(x(t))+p (t), t > 0; x(0) = 0; (1.3)
where g is a (strictly) positive continuous function of its argument. A transformation of the
independent variable to g (s) ds from t reduces (1.3) to an equation of the type (1.1). Thus we
obtain a generalization of our earlier result [Kaper and Kwong 1988] for the special case
4(x)= xa 0< 1.
The uniqueness criterion given in Theorem 1 appears to be different from any of the unique-
ness criteria given in the literature; cf. Hartman [1964, Section 1.6]. Special cases of equation
(1.1) have been studied by several authors. Murakami [1966] considered the case $(x) = x 1/2,
p (t) E 1, but, as was shown by Bownds and Diaz [1973], his result can be obtained as a corollary
of a theorem of Bownds [1970]. The article by Bemfeld, Driver, and Lakshmikantham [1975/76]
gives a general uniqueness theory for equations of the type (1.1), but under the assumption that
1/$ e L (0, a) and p (t) > 0 for all t. It is not clear whether the technique of Bernfeld, Driver, and
Lakshmikantham [1975/76] can be modified to allow for functions p that change sign near 0.
As we showed in [Kaper and Kwong 1988], the condition P(t) 0 in Theorem 1 cannot be
relaxed, unless other conditions are strengthened. Perhaps, if the positive part of P somehow
dominates the negative part near zero, uniqueness can still be shown, but such cases are not
covered by the present theorem.
In the second part of this article we apply the theorem to establish a result related to the
uniqueness of nonnegative nontrivial radial solutions of the quasiinear elliptic partial differential
equation
V-(A (I VutI)Vu +f (u) = 0, x ERR"; limrn,..u(x) = 0. (1.4)
Here, f is a real-valued function on [0, oo), which is Lipschitz continuous on (0, oo), with
lim U + o f (u) = 0 and f (0) = 0; A is a continuous positive function on [0, a*), such that the pro-
duct pA (p) is a strictly incr sing function of p on [0, oo). Among the equations covered is the
mean curvature equatic-n, for which A (p) = (1 +p 2)-1 2 . We discuss this case in some detail in
Section 4.
Radial solutions of (1.5) are functions u of the variable r = Ix I that satisfy the boundary
value problem
112
(A( Iu'I)u)'+ n-1 A ( Iu'1)u'+f (u) =0, r >0; u'(0) =0; lim, .. u(r) =0. (1.5)r
Let F denote the integral
F(u)=Jf(v)dv, u>0, (1.6)0
and let P denote the constant
1=inf(u>0:F(u)>0). (1.7)
We prove the following theorem.
THEOREM 2. Let u 1 and u 2 be two distinct nonnegative nontrivial solutions of (1.5). If 3>0,
then u 1(r) = u 2(r)for at least one value of r e (0, oc).
Theorem 2 can be generalized to equations of the type
V-(A ( I Vu I ) u) + a (I x I)f (u) = 0, x E R"; lim .. u (x) = 0; (1.8)
where a is (strictly) positive. In this case one defines r to be a weighted radial variable,
dr = a (Ix I) d Ix 1. The factor (n-1)/r in the second term of (1.5) is replazd by g (r), where
g (r) = (d/dr)(ln(lx"-4a( Ix ))) (1.9)
Notice that g is monotonically decreasing on (0, o0).
Theorem 2 generalizes our earlier result [Kaper and Kwong 1987, Theorem 1] for the con-
stant coefficient case A (p) m 1.
A corollary of the theorem is the monotone separation lemma. Since every solution of (1.5)
is monotonically decreasing, it can be inverted on its support. The monotone separation lemma,
which was first formulated for the constant coefficient case by Peletier and Serrin [1986], states
formally that the difference between the inverse functions of two different solutions of (1.5)
increases monotonically beyond the last point of intersection of their solution graphs.
COROLLARY. Let the conditions of Theorem 2 be satisfied. Let u 1 and u 2 be two distinct non-
negative nontrivial solutions of (1.5), and let r 1 and r2 be their respective inverses. If
u I(r) = u2 (r) = rfor some r er(0, a), then r 1 (u) - r2(u) is a monotone nonincreasing function
ofuon [0, ].
Theorem 2 and its corollary are at the heart of a uniqueness proof of Franchi, Lanconelli,
and Serrin [1986] for nonnegative nontrivial solutions of (1.5). These authors prove uniqueness
by showing that, if the graphs of two solutions of (1.5) intersect at some point, then the solutions
must be identical. If the intersection occurs at a point where the function value is less than a, no
further conditions on fare needed. If the function value is greater than 1, an additional condition
must be imposed on the behavior of f beyond 1, requiring that the region between the positive
113
real axis and the graph of f be star-shaped with respect to the point (0, $).
2. Proof of Theorem 1
The proof of Theorem 1 is preceded by three lemmas. We assume throughout that the con-
ditions of Theorem I are satisfied.
LEMMA 1. If ilk 4 L (0, a)for any a > 0, then (1.1) has at most one solution.
Proof. The lemma is a consequence of the concavity oft. The right-hand side of the differential
equation (1.1) is a continuous function f (t, x), which satisfies the inequality
If (t, x1)-f (t, x2)I = I4(x 1)-O(x 2 )I S (Ix1 -x 2 1) for any pair of points (t, x1 ), (t, x2),
with t > 0. If i/$ 4 L (0, a) for any a > 0, the only solution of the initial value problem
dy /dt = $(y (t)), t > 0; y (0) = 0; is the trivial one. The assertion of the lemma follows from
Kamke's general uniqueness theorem; cf. Hartman [1964, Section 111.61. 0
In the proofs of Lemmas 2 and 3 we assume that 1/$ e L (0, a) for some a > 0. Then the initial
value problem
= $y(t)), t > 0; y (0) = 0; (2.1)dt
has at least one solution, besides the trivial one. Let yM denote the maximal solution; yM is given
implicitly by the equation
yM()
5- = t, t 20. (2.2)
We assume that yM is defined on the interval [0, T). The next lemma shows that yM is a lower
bound for any solution of (1.1). (Here and throughout the remainder of this article, inequalities
(equalities) of the type x > y (x = y) involving two nonnegative functions x and y are understood
to hold pointwise everywhere on the common domain of definition.)
LEMMA 2. Every solution of (1.1) satisfies the inequality x , yu, where yM is the maximal solu-
tion of (2.1).
Proof. Every solution x of (1.1) satisfies the integral equation x = K (x), where K is given by
I
K(x)(t) = P(t)+J(x(s)) ds, t 0; (2.3)0
and vice versa. K is monotone, in the sense that K (x) z K (y) if x z y 0. We use K to define a
114
sequence of functions (x.), e N :
x 41 = K(x), n eN; x 1 = P. (2.4)
Clearly, x 1 z 0. Furthermore, x2 x 1 and, by induction, x, zx for all n E N. Hence, the
sequence (x) E N is monotonically increasing. Since it is also bounded from above, it converges
as n -4 oo to an element x... The limit satisfies the integral equation x = K (x) and is therefore a
solution of (1.1).
Let y be another solution of (1.1). Then y = K(y) z x1 . Because K (y) = y and K is mono-
tone, it follows that y zx forall n ErN. Hence, y x., sox., is the minimal solution of (1.1). It
suffices therefore to establish the lemma for x.
Consider the initial value problem (2.1) and the equivalent integral equation y = L (y),
where
L (y)(t) = f(y(s))ds, t >0. (2.5)0
Like K, L is monotone. We use L to define another sequence of functions (y)n EN , which will
converge to a solution of (2.1). The choice of the initial function y is critical.
Let E be any of the points tj of the sequence mentioned in the theorem. Let YM,E be the
maximal solution of the differential equation
-=$(y (t)), t > E; y (E) = 0; (2.6)dt
and let this function be trivially extended backwards to t = 0. Because P (E) > 0 and yME (E) = 0,
there certainly exists a c > e, such that P > yME on [E, r]. Taking such a r, we define the initial
element y 1 by
y 1(t) = yME (t), t E [0, 1]; y 1(t) = 0, t E [T, TI]; (2.7)
and subsequent elements by
y,+1 = L (y,), n e N. (2.8)
The sequence (y),, EN converges as n -+ 00 to a limit function y.., which is a solution of (2.1).
A direct computation shows that y coincides with yM,E on [0, or] for all n E N, so the same
is true for y... Notice, however, that yM,E is the unique maximal solution of the initial value prob-
lem (2.1) beyond e. Hence, y.. and yM,E are one and the same function.
Clearly, x1 y z0. Because K and L satisfy the ordering relation K (x) L (y) whenever
x y 20, it follows that x~ >_y4 for all n E N. In the limit n -4o we have, therefore, x., >_y..
We complete the proof by taking E successively equal to each of the points in the sequence
(tN)jE . This sequence converges to 0, and yM,E converges pointwise to yM, the maximal solu-
tion of (2.1), as E tends to0 through the points of the sequence. 0
115
The next lemma establishes an upper bound for any solution of (1.1).
LEMMA 3. Let xm be the minimal solution of (1.1). Any solution x of (1.1) satisfies the inequality
x S x,,, + yM, where yM is the maximal solution of (2.1).
Proof. Since x = K(x) and xM = K(xM), where K is the integral expression (2.3), we have
r r
x(t) - XM(t) = J ($(x(s)) - $(xM(s))) ds 5 Jf$(x(s) - xM(s)) ds, (2.9)0 0
so the difference w = x - xM satisfies the integral inequality w 5 L (w), where L is the integral
expression defined in (2.5). It follows that w is dominated by the maximal solution of the integral
equation w = L (w), i.e., by the maximal solution of the initial value problem (2.1), which is yM.
0
We are now ready to prove Theorem 1.
Proof of Theorem 1. In view of Lemma 1, it suffices to prove the theorem under the assumption
that i/$ e L (0, a) for some a > 0.
Suppose (1.1) has more than one solution. Let u and v be two distinct solutions with com-
mon domain [0, T); we assume, without loss of generality, that u > v. The difference w = u - v
satisfies the initial value problem
dt= $(u (t)) - $(v (t)), t > 0; w (0) = 0. (2.10)dw
According to the mean value theorem, $(u (t)) - $(v (t)) = $'((t))w for some t(t) E (v (t), u(t)).
The derivative $- is decreasing, so estimating t(t) from below by v(t) and v (t), in turn, by yM(t)
(cf. Lemma 2), we obtain the following differential inequality for w:
dw- ' YM(t))W (t). (2.11)dt
Since yM satisfies (2.1), we have the identity
d"-(yM(t)) = -ln 4(YM). (2.12)dt
Hence,
dIn w:5 dIn Myu). (2.13)d d
Upon integration, it follows that
w(t2)S YM 2 w(tl) ) (2.14)(yMft 1))
116
for any two points t1 , t2 with 0 < t l < t2 < T. We estimate w(t 1) from above by yM(tl) (cf.
Lemma 3) and observe that $y(t1)) = 0-(rO)y(t 1) for some Ti e (0, yM(t1)). Thus,
W(Y2)M $(YM(t2)). (2.15)
Since $ has a vertical tangent at the origin, the coefficient 1/$-(9) tends to 0 as t1 -4 0. There-
fore, we can make the right-hand side arbitrarily small by choosing t1 sufficiently close to 0. It
follows that w(t 2) = 0. Since t2 is arbitrary, we conclude that w = 0, i.e., u = v. This result con-
tradicts the assumption that u and v are two distinct solutions of (1.1). 0
3. Proof of Theorem 2
The proof of Theorem 2 is preceded by two lemmas. We assume that the conditions of
Theorem 2 are satisfied.
Let [r 1, r2] be any compact subinterval of [0, oo), and let u satisfy the differential equation
(A(Iu'I)u)'+ A (Iu'I)i'+f(u)=0 (3.1)r
on [r1, r2]. Upon multiplication by u', (3.1) yields the identity
(A(Iu'I)Iu'I)'1Iu'1 + n-A(Iu'i)I1u'12 +f (u)u'=0. (3.2)
Because pA (p) is monotone and strictly increasing on [0, oo), the equation pA (p) = v can besolved for p on the range of pA (p). The solution defines a function p (-), which is also monotone
and strictly increasing, with p (0) = 0. Hence, if we define the function v in terms of u by
v =A (I u'l) Iu'l, (3.3)
then we can rewrite (3.2) as
p (v)v'+ p(v)v + f (u)u' = 0. (3.4)r
We integrate (3.4) over [r 1, r2 ]. Recall that F is the integral of f. If P is the integral of p,
V
P(v)=Jp(s)ds, v 0, (3.5)0
then we have the following identity:
[P(v)+F(u)]'= -(n - 1)J p(v(r))v(r) . (3.6)r
We use this identity in the proof of the following lemmas, which give general properties of solu-
tions of the boundary value problem (1.5).
117
LEMMA 4. Let u be a nonnegative nontrivial solution of (1.5). Then
(i) lim, _.. u'(r) =0;
(ii) u satisfies the identity
"r dsP(v(r))+ F (u(r)) = (n - 1)Jp(v(s))v(s) -- , r 0; (3.7)r
Proof. Let r 0 be fixed. We take any R E (r, oo) and apply (3.6) to u on the compact interval
[r, R],
R
P (v (R))+ F(u(R))] - P (v(r))+ F(u (r))] =-(n - 1)jp (v (s))v(s) s. (3.8)r S
Since v and p (v) are positive, the expression in the right member tends to some negative constant,
possibly -o, as R -+ c. The same must then be true for the expression in the left member. Since
r is kept fixed, it must therefore be the case that the quantity P (v (R)) + F (u (R)) tends to some
constant as R -+ c*. This constant may be -o, but it is certainly less than oo. Because
lim R -.. u(R) = 0 and lim~. , 0 F(u) = 0, the quantity F(u(R)) tends to0 as R -+ o, so it must
then be the case that P(v (R)) tends to some constant as R -+ o. We recall that P is the integral
of a nonnegative function, so the constant cannot be negative, and conclude that P (v (R)) tends to
some finite positive constant. But then p (v (R)) must tend to 0 as R -. Co. Because I u'I = p (v),
this proves (i). The identity (3.7) follows from (3.8) as R -+ co. 0
Because u'(0) = 0, v and therefore P vanish at the origin. So evaluating (3.7) at r = 0, we
find that F(u(0)) > 0. Hence, any nonnegative nontrivial solution of (1.5) must necessarily
satisfy the condition u(0)> $a.
LEMMA 5. If u is a nonnegative nontrivial solution of (1.5), then either
u > 0 on [0, oo), u'<0 on (0, oo), (3.9)
or there exists an R e (0, oo) such that
u > 0 on [0, R), u'< 0 on (0, R), u = 0 on [R, oo). (3.10)
Proof. Let R be the right-most point of the support of u:
R = inf(r e (0, oo) : u (s) = 0 for all s E (r, oo)). (3.11)
Because u is nontrivial, we have R > 0; R may be finite or infinite. Let a be defined by
a = inf(r e [0, R): u'(s) < 0 for all s e (r, R)) (3.12)
Because u is nonnegative, we have u(a) > 0; furthermore, u'(a) = 0.
118
Suppose that a > 0 and that u has a local maximum at a. Then there exists a point
b E [0, a), such that u'(b) = 0 and u'> 0 on (b, a). Because u(R) = 0, there must be a point
c e (a, R ] where u (c) = u (b). Evaluating (3.7) at b and at c and subtracting the resulting equa-
tions, we obtain the identity
- P(v(c)) = (n - l)jp(v (s))v(s) -. (3.13)b S
But here we have a contradiction: the expressions on either side of the equation have opposite
signs. It follows that either a = 0or, if a > 0, then u(r) z u(a) for all r Er[0, a ].
Suppose that a >0 and u(r) u(a) for all r E [0, a]. Then u is constant on an interval
(a - q, a) of positive length, or u has an inflection point at a. In either case, u -(a) = u--(a) = 0.
Since u satisfies the differential equation in (1.5), it must be the case that f (u(a)) = 0 as well.
Let ua = u (a) and let the constant function w be defined on [a, co) by the expression u(r) = ua.
Both u and w are solutions of the initial value problem
(A(ly'l)y')'+ n A(Iy'I)y'+f(y)=0, r>a; y(a)=ua; y-(a)=0. (3.14)
Because ua > 0, f is Lipschitz continuous at a; hence, the solution of (3.14) is unique. It follows
that u(r) =wx(r) = ua for all r 2!a. But lim ,., u(r) = 0, so it must be the case that ua = 0, i.e.,
u (a) = 0. This conclusion contradicts the earlier statement that u (a)> 0. The configuration
a > 0 and u(r) z u (a) for all r e=[0, a ] is therefore impossible. It follows that a = 0. O
Lemma 5 implies that the definition (3.3) of the variable v can be recast in the form
v = -A (I u'i)u'. (3.15)
The pair (u, v) satisfies the system of equations
u'= -p(v), v' .= - - -v +f (u). (3.16)r
Conversely, any solution (u, v) of the system (3.16) defines a solution u of (1.5) through the rela-
tion (3.15).
Usually, u and v are taken as the coordinates in a phase plane analysis of (1.5). However,
here an alternative choice is preferable. According to Lemma 5, any solution u of (1.5) is
(strictly) decreasing on its entire support, so it can be inverted there. We denote the inverse func-
tion, which is defined on [0, u(0)], by r(-). Let the function P be defined by
v(r M))
P(u) = .pI(s) ds, u e [0, u(0)]. (3.17)0
Clearly, P is nonnegative. In fact, P (0) = 0, because of Lemma 4, and P (u (0)) = 0, because
u'(0) = 0. Furthermore, P is (strictly) positive in the open interval (0, u (0)), because of Lemma
5. We also observe that v (r) = $(P (u (r))) for all r 0.
119
We take u and P (u) as the coordinates of a phase plane analysis of (1.5). One readily
verifies that P -(u) = - v -(r (u)), so P satisfies the initial value problem
-n- = -14(P(u)) - f (u), u > 0; P(0) = 0. (3.18)du r (u)
Thus, as r increases from 0 to co, the solution of (1.5) generates a curve in the first quadrant of the
phase plane, starting at the point (u (0), 0) and terminating at the origin. At any point along the
curve the tangent is given by (3.18).
One recognizes (3.18) as an initial value problem of the type considered in Theorem 1. The
derivative of $ is lip, which is always positive, infinite at the origin, and a decreasing function of
its argument. The graph of $ is therefore concave and has a vertical tangent at the origin, so $satisfies the conditions imposed on (1.1).
If f is such that a > 0, then it is certainly possible to find a sequence of points (tj)j eN of the
type described in Theorem 1, because F is negative over the entire interval (0, 3). Hence, if the
initial value problem (3.18) has a solution which is positive in some interval (0, u) with u > 0,
this solution is unique. This conclusion is used in the proof of the theorem.
Proof of Theorem 2. Suppose us1 and U2 am two distinct nonnegative nontrivial solutions of
(1.5). We use the notation introduced in the preceding paragraphs, distinguishing the variables
related to ui1 and u2 by the subscripts 1 and 2, respectively.
The functions P 1 and P; defined by the solutions u 1 and us2, respectively, as in (3.17),
satisfy the initial value problems
dPi _n -idan- $(P1V (u)) -f (u), u > 0; P 1 (0) = 0; (3.19)du r 1(u )
and
dP2 _n -idu - rnu)$(P2(u)) -f (u), u >0; P2 (0) = 0; (3.20)du r2(u)
respectively.
Suppose that u1(r) > u 2(r) everywhere. Then r1 (u) r2(u) for all u E [0, u2 (0)]. Hence,
P 1 satisfies the same initial value problem as P2 , but with a differential inequality,
__ n-I $(P I(u)) - f (u), u > 0; P(0) = 0. (3.21)du r2 (u)
In general, the theory of differential inequalities enables us to compare the maximal solutions of
(3.20) and (3.21). However, under the conditions of the theorem, (3.20) and (3.21) have at most
one nonnegative solution, so there are no other solutions besides the maximal solutions, and we
conclude that
120
P 1 (u) 5 P2(u), U E [0, u2(0)].
if u1 (0) > u2(0), then P1 (u2 (0)) > 0, while P2 (u2(0)) = 0. As this conclusion is incompatible
with (3.22), we must assume that u 1(0) = u2 (0).
Because P is the integral of the positive function p, it follows from (3.22) that
v1 (r 1 (u)) 5 v2 (r2 (u)). (3.23)
Next, we apply the identity (3.7) to u 1 and u2 and subtract the resulting equations,
Jp(v1 (s))v,(s) = Jp(v2 (s))v 2 (s) -- , (3.24)0 S o S
or, after a transformation of variables,
u,(0) u2(O) d1P (v 1(r 1(u))) du -=IP (v 2(r2(u))) du (3.25)
0 r 1(u) 0 r2(u)
We recall that u 1(0) = u2(0) and r1 (u) > r2 (u), and conclude that (3.23) is compatible with
(3.25) if and only if v 1 (r 1 (u)) = v2(r2 (u)) for all u e[0, u2(0)]. This equality, in turn, implies
that u 1 and u2 coincide everywhere. But here we have arrived at a contradiction, since we had
assumed that u 1 and u2 were distinct. Hence, if u 1 and u2 are distinct, their graphs must inter-
sect at some point r e (0, 00). 0
The monotone separation lemma (corollary) is an immediate consequence of (3.22); cf. the
proof of the corresponding result for the constant coefficient case in [Kaper and Kwong 1987].
4. Mean Curvature Equation
The arguments of the previous section apply, in particular, to the mean curvature equation.
This equation, studied by Concus and Finn [1979] in their investigation of the shape of a pendent
liquid drop and discussed extensively in a recent article by Atkinson, Peletier, and Serrin [1987],
is
v.Vuo- V +f (u)= , x E R"; lim ,....u (x) = 0. (4.1)
(1 + I~u 112
It is a special case of (1.4), where A (p) = (1 +p2 )-1 2 . Clearly, A satisfies the conditions of
Theorem 2.
The equation pA (p) = v defines the inverse function p (-),
p(v) = v(1 - v2 -1 2, (4.2)
the integral of which is
121
(3.22)
P(v) = 1 -(1 -v 2 )112
The inverse of P is $,
$(P) = (P(2 - P))"2. (4.4)
Radial solutions of (4.1) satisfy the boundary value problem
u' +n -1 u' +()= ,r>;
(1 +(u)2)112 r (1 +(u)2)l +fi )0,r0
u'(0) = 0, lim, .u(r) = 0. (4.5)
A phase plane analysis of (4.5) based on the coordinates u and P leads to the following evolution
equation for P:
dP _ n-1P(2-P)-f(u), u>0; P(0)=0. (4.6)du- r(u)
Following the methods of Section 3, one shows that this nonlinear initial value problem has at
most one solution if a > 0. As shown by Franchi, Lanconelli, and Serrin [1986], it then follows
that (4.5) has at most one nonnegative nontrivial solution.
Acknowledgments
We thank Professors J. Serrin and E. Lanconelli for stimulating discussions during the
Microprogram on Nonlinear Diffusion Equations and Their Equilibrium States, held at the
Mathematical Sciences Research Institute in Berkeley (August/September 1986), and for com-
municating their unpublished results to us.
References
F. V. Atkinson, L. A. Peletier, and J. Serrin 1987. "Ground states for the prescribed mean curva-ture equation: The supercritical case," to appear in: Nonlinear Diffusion Equations andTheir Equilibrium States, Wei-Ming Ni, L. A. Peletier, and J. Serrin (eds.), MSRI Conf.Proc., Springer-Verlag, New York.
S. R. Bemfeld, R. D. Driver, and V. Lakshmikantham 1975/76. "Uniqueness for ordinarydifferential equations," Math. Sys. Theory 9, 359-367.
J. Bownds 1979. "A uniqueness theorem for y -= f (x, y) using a certain factorization," J. Diff.Eq. 7, 227-231.
122
(4.3)
J. M. Bownds and J. B. Diaz 1973. "On restricted uniqueness for systems of ordinary differentialequations," Proc. Amer. Math. Soc. 37, 100-104.
P. Concus and R. Finn 1979. "The shape of a pendent liquid drop," Phil. Trans. Royal Soc.London 292, 307-340.
B. Franchi, E. Lanconelli, and J. Serrin 1986. "Existence and uniqueness of ground state solu-tions of quasilinear elliptic equations," Presentation at the Microprogram NonlinearDiffusion Equations and Their Equilibrium States, MSRI, Berkeley.
P. Hartman 1964. Ordinary Differential Equations, John Wiley & Sons, Inc., New York.
H. G. Kaper and Man Kam Kwong 1987. "Uniqueness of non-negative solutions of semilinearelliptic equations," to appear in: Nonlinear D5usion Equations and Their EquilibriumStates, Wei-Ming Ni, L. A. Peletier, and J. Serrin (eds.), MSRI Conf. Proc., Springer-Verlag,New York. Also: Proc. 1986-87 Focused Research Program on "Spectral Theory and Boun-dary Value Problems," ANL-87-26, vol. 4, Hans G. Kaper, Man Kam Kv"ng, and AntonZettl (eds.), Argonne National Laboratory, Argonne, Illinois.
H. G. Kaper and Man Kam Kwong 1988. "Uniqueness for a class of non-linear initial valueproblems," J. Math. Anal. and Applic. 130, 467-473. Also: Proc. 1986-87 Focused ResearchProgram on "Spectral Theory and Boundary Value Problems," ANL-87-26, vol. 4, Hans G.Kaper, Man Kam Kwong, and Anton Zett (eds.), Argonne National Laboratory, Argonne,Illinois.
H. Murakami 1966. "On nonlinear ordinary and evolution equations," Funkcial. Ekvac. 9, 151-162.
L. A. Peletier and J. Serrin 1986. "Uniqueness of non-negative solutions of semiinear equationsin R"," J. Diff. Eq. 61, 380-397.
123
UNIQUENESS OF NON-NEGATIVE SOLUTIONSOF A CLASS OF SENILINEAR ELLIPTIC EQUATIONS
Hans G. Kaper and Man Kam KwongMathematics and Computer Science Division
Argonne National LaboratoryArgonne, IL 60439
Abstract
This article is concerned with boundary value problems of the type
u''+g(r)u'+f (u) = 0, r>0; u(0)=0, lim, ,.u(r)= 0; (BVP)
where f (0) = 0. Such problems arise in the study of semilinear ellipticdifferential equations in R". It is shown that (BVP) has at most one non-negativenon-trivial solution under appropriate conditions on f and g. The conditions areweaker than those given by Peletier and Serrin [J. D . Eq. 61 (1986), 380-397],who considered the special case g(r) = (n - 1)/r, n = 2, 3, - - -.
1. Introduction
In this article we are concerned with boundary value problems of the type
u"+g(r)u'+f(u)=0, r>0; u'(0)=0; limr..u(r)=0. (1.1)
Such problems arise in the study of semilinear elliptic differential equations in R". For example,
radial solutions of the boundary value problem
Au+f(u)=0, xER"; lim,.. u(x)=0; (1.2)
satisfy (1.1), where r = Ix I and g (r) = (n - 1)/r. Radial solutions of the more general problem
Au+a(Ix)f(u)=0, xER"; lim,...u(x)=0; (1.3)
satisfy (1.1), where r is a weighted radial variable, dr = aa(Ix l)dIxI, and g(r) =
(d/dr)ln [Ix I"n' a(Ix I)).
If f (0) = 0, as we assume throughout this investigation, then (1.1) has at least one solution,
viz., the trivial solution u . 0. In addition, there may be other solutions. Our interest focuses on
the uniqueness of non-negative non-trivial solutions of (1.1). We use the following notation:
U
F(u)= f (s)ds, (1.4)0
G (r) = exp[ g (s) ds , (1.5)
125
where a is some fixed positive number. Notice that G '(r) = g (r)G (r) for all r. Furthermore,
a=inf(u>0:f(u)>0), (1.6)
s(=inf(u>0:F(u)>0). (1.7)
These constants satisfy the ordering relation 05 a 5 $.
The boundary value problem (1.1) with g (r) = (n - )/r and the corresponding problem (1.2)
have been the subject of numerous investigations; cf. Gidas, Ni, and Nirenberg [1979] (radial
nature of the solutions); Berestycki and Lions [1978], and Berestycki, Lions, and Peletier [1981]
(existence of positive solutions); Peletier and Serrin [1983], and McLeod and Serrin [1981]
(uniqueness of positive solutions); Peletier and Serrin [1986] (uniqueness of non-negative non-
trivial solutions); and the references cited therein. Our investigation was, in fact, motivated by
the results of McLeod and Serrin [1981] and Peletier and Serrin [1986].
Peletier and Serrin [1983] showed that (1.2) has at most one positive radial solution if fsatisfies the following conditions (we follow their notation):
(H1) f is locally Lipschitz continuous on (0, oo);
(H2) lim ._.of (u)/u = - m for some m > 0;
(H3) F(u) > 0 for some u > 0; and
(S) f (u)/(u - 3) is a monotone non-increasing function of u on the set (u > (3: f (u) > 0).
The conditions (H2) and (H3) concern the behavior off for small values of the argument, (S) its
behavior for large values of the argument; (S) is equivalent with the condition that the region D
= ((u, v) E R2 : u > (3, 0< v <f (u)) is star-shaped with respect to the point (0, 0).
Peletier and Serrin [1986] also showed that the condition (H2) is stronger than necessary.
They proved that (1.2) has at most one non-negative non-trivial radial solution if (HI) through
(H3) are replaced by the following conditions (we follow the notation of Peletier and Serrin[1986]):
(Al) f is locally Lipschitz continuous on (0, oc);
(A2) F(u) > 0 for some u > 0;
(A3) lim,_ of(u)=0andf(0)=0; and
(H*) $3>Oifn =2,a>Oifn =3,4, ---.
If n is considered as a continuous variable, then the condition 3> 0 in (H*) is sufficient for all n
in the range 3/25 n 5 2, while the condition a > 0 applies to all n > 2.
We note that the original condition (H2) implies that f(u) -+0 as u -40, and also that
f (0) = 0. As these properties do not necessarily follow from the condition (H*), the condition(A3) must be added for consistency.
126
The new condition (H*) is clearly weaker than the original condition (H2). However, it will
follow from the present investigation that it is still stronger than necessary and that the single ine-
quality 3> 0 is sufficient for all values n = 2, 3, - - - (or n z 3/2, if n is continuous).
In this article we shall prove that (1.1) has at most one non-negative non-trivial solution if fand g satisfy the following conditions:
(F1) f is locally Lipschitz continuous on (0, cc), lim0e of(u) = 0, and f (0) = 0;
(F2) $ > 0;
(F3) f (u)/(u -13) is a monotone non-increasing function of u on the set (u > 3: f (u) > 0);
(G 1) g is (strictly) positive and monotone, with g 'S0, on (0, cc);
(G2) GG 'is monotone, with (GG ')'z 0, on (0, u );
(G3)Jd s =c,or elseG(s)
1/2
=f as r -+o. (1.8), G (su)G(s ) , G (s)
In the case of the boundary value problem (1.2), where g (r) = (n - 1)/r, the conditions (G 1),
(G2), and (G3) are satisfied for all n = 2, 3, - - - (or n >3/2, if n is continuous).
The plan of this article is as follows. In Section 2 we establish several general properties of
solutions of the boundary value problem (1.1). Some of these are straightforward generalizations
of the corresponding properties derived by Peletier and Serrin [1986, but in particular those
given in Lemma 4 are new. In Section 3 we give a new proof of the fact that the graphs of two
distinct solutions of the boundary value problem (1.1) must intersect (Theorem 1). The proof is
based on the introduction of a new dependent variable, which is the solution of a nonlinear initial
value problem. A uniqueness result for this type of initial value problem was shown by Kaper
and Kwong [1988]; it makes a direct comparison of the two solution graphs possible. The mono-
tone separation lemma [Kaper and Kwong 1988, Lemma 9] is an immediate consequence of
Theorem 1 of this section. In Section 4 we show that the graphs of two non-negative non-trivial
solutions of the boundary value problem (1.1) cannot intersect, unless the solutions are identical
(Theorems 2 and 3). Thus we arrive at the desired uniqueness result, which is stated in Section 5
(Theorem 4).
127
2. General Properties of Solutions
Let [r1, r2] be any compact subinterval of [0, oo), and let u satisfy the differential equation
u''+ g (r)u'+ f (u) = 0 (2.1)
on [r 1, r2]. Multiplying (2.1) by u' and G2u', respectively, and integrating the resulting equa-
tions over [r1, r2 ], we obtain the following identities:
u '(r)2 + F (u(r)) = -J g(s)u'-(s)2 ds , (2.2)
r=r, r
r =r2 2
G (r)2 1u'(r)2 + F (u(r)) = 2JG (s)G '(s)F(u(s)) ds . (2.3)
r=r 't
We use these identities in the proofs of the following lemmas, which give some general properties
of solutions of the boundary value problem
u"+g(r)u'+f(u)=0, r >0; u'(0) =0; limr -,. u(r) =0. (2.4)
In the proofs of Lemmas 1 and 2 we assume that f and g satisfy the conditions (Fl) and (G1). In
the proof of Lemma 3 we require, in addition, that f satisfies the condition (F2). Lemma 4 holds
if f satisfies the conditions (FI) and (F2), and g satisfies the conditions (G1) and (G3).
LEMMA 1. Let u be a non-negative non-trivial solution of (2.4). Then
(i) lim ,, .u(r) = 0;
(ii) u satisfies the identity
u'(r)2 + F(u(r)) = J g (s)u '(s)2 ds, r 0. (2.5)(25
Proof. Let r 0 be fixed. We take any R E (r, o) and apply (2.2) to u on the compact interval
[r, R ],
R
u '(R)2 + F (u (R)) - 2u'(r)2 + F (u (r)) = - fg (s)u'(s)2 ds . (2.6)
Since g is positive, the expression in the right member tends to some negative constant, possibly
-o, as R -+ 0. The same must then be true for the expression in the left member. Since r is kept
fixed, it must therefore be the case that the quantity '%u '(R)2 + F (u (R)) tends to some constant as
R -+ 00. This constant may be -o, but it is certainly less than oo. Because lim R .. u(R) = 0 and
lim.u . 0 F(u) = 0, the quantity F(u(r)) tends to0 as R -+oo, so it must then be the case that
u'(R)2 tends to some constant as R -+ c. Clearly, this constant cannot be negative, so u'(R)2,
128
and therefore u'(R) tends to some non-negative finite constant as R -+ oo. In fact, the only value
compatible with the limiting condition lim.R u (R) = 0 is zero, so lim R u '(R) = 0. This
proves (i). The identity (2.5) follows from (2.6) as R -+ oo.
Evaluating the identity (2.5) at r = 0, we find that F (u (0)) > 0. Hence, taking into account
the definition (1.5) of the constant $, we conclude that any non-negative non-trivial solution u of
(2.4) must necessarily satisfy the inequality u (0) > P.
LEMMA 2. If u is a non-negative non-trivial solution of the boundary value problem (2.4), then
either
u > 0 on [0, oo), u'<0 on (0, oo) , (2.7)
or there exists an R e (0, oo) such that
u > 0 on [0, R), u'< 0 on (0, R), u = 0 on [R, oo) . (2.8)
Proof. Let u be a non-negative non-trivial solution of (2.4). Let R be the right-most point of the
support of u:
R = inffr e [0,oo) : u(s) = 0 for all s Er(r,oo)} . (2.9)
Clearly, R > 0, because u is non-trivial; R may be finite or infinite.
Let a be defined by
a = inf{r e [0, R): u'(s)<O for all s E (r, R)). (2.10)
Because u is non-negative, we have u (a) > 0; furthermore, u '(a) = 0.
Suppose that a > 0 and u has a local maximum at a. Then there exists a point b e [0, a),
such that u'(b) = 0 and u'> 0 on (b, a). Because u(R) = 0, there must be a point c e(a, R Iwhere u (c) = u (b). Evaluating (2.5) at b and at c and subtracting the resulting equations, we
obtain the identity
- u'-(c)2 =Jg(s)u'(s)2 ds . (2.11)b
But here we have a contradicition: the expression on the left-hand side is obviously negative,
while the expression on the right-hand side is positive. We must therefore conclude that either
a = 0 or, if a > 0, then u (r) u (a) for all r e [0, a ]. We claim that the latter configuration is
impossible.
Suppose that a > 0 and u(r) u(a) for all r E [0, a ]. Then u'50 on (a - r, a) for some
rl > 0. Also, u '(a) = 0, as we have seen above. If u' = 0 on (a - i, a), i.e., u is constant there,
then the C2-continuity of u at a implies that u"(a) = 0. If u' < 0 on (a - Ti, a), then u has an
inflection point at a, so u"(a) = 0 as well. In either case, u'(a) = u"(a) = 0. Since u satisfies
the differential equation and g is continuous at a, it must be the case that f (u (a)) = 0. We shall
129
show that these properties together imply that u(a) = 0.
Let ua = u (a), and let the constant function i be defined on [a, oo) by the expression
~(r) = ua, r E [a, oo) . (2.12)
Both u and are solutions of the initial value problem
y +g(r)y'+f(y)=0, r>a; y(a)=ua, y'(a)=0. (2.13)
Because we have assumed a > 0, it follows from the condition (Fl) that f is Lipschitz continuous
on [a, oo); hence, the solution of (2.13) is unique, and we conclude that u(r) = W (r) = ua for all
r z a. But lim r -.. u(r) = 0, so it must be the case that ua = 0, i.e., u(a) = 0.
This conclusion contradicts the earlier statement that u(a) > 0, which followed from the
definition (2.10). Hence, the configuration a > 0 and u(r) u(a) for all r E [0, a 1 is impossible,
as claimed. It must therefore be the case that a = 0. 0
LEMMA 3. Let u be a non-negative non-trivial solution of the boundary value problem (2.4).
Then there exists a number K _0, such that
(i) lim, r.~ G (r)2 [u(r)2 +F (u(r))] = K;
(ii) u satisfies the identity
G (r)2 [ u '(r)2 + F(u(r)) = K - 2J G(s)G '(s)F (u(s)) ds, r 20. (2.14)
Proof. Let r 0 be fixed. We take any R e (r, oo) and apply (2.3) to is on the compact interval
[r, R ],
G(R) 2 u(R)2 +F(u(R))] -G(r) 2 [iu '(r)2 +F(u(r))
R
= 2 JG(s)G '(s)F(u(s))ds . (2.15)r
The first term in the left member is always positive; cf. (2.5). Consequently, (2.15) implies that
R
2JG (s)G '(s)( - F (u(s))) ds < G (r)2 u'(r)2 + F(u(r)) . (2.16)
Since u(s) -+ 0 as s -* c, u(s) is less than or equal to $ for all sufficiently large s, so F (u(s)) is
zero or negative beyond a certain point. (Here, we use the condition (F2).) From that point on,
the integrand in (2.16) is zero or positive. These arguments show that the integral in the left
member of (2.16) is a monotone non-decreasing function of R for R sufficiently large. Because
130
the expression in the right member is positive and fixed, the integral is also bounded above by
some (positive) constant. It must therefore be the case that the integral converges to some finite
limit as R - oo. Going back to (2.15), where we now know that the expression in the right
member converges as R -+ oo, we conclude that the same must be true for the expression in the
left member. Since the second term is fixed, it must be the case that the first term converges as
R -+ oc. Property (i) states that K is its limiting value. The identity (2.14) follows from (2.15) as
R -+ oo. 0
LEMMA 4. Let u be a non-negative non-trivial solution of (2.4). Let K be the number defined in
Lemma 3(i). if = o, then K = 0; otherwise,G (s )
lim u (r)i / d = 2K (2.17)r -+- , G(s)
Proof. First, we consider the case where J = . Suppose that K #0. Then necessarilyG(s)
K > 0, so taking any K' E (0, K), we have
1,2 K' __
-u '(r)2 >G 2 -F(u(r)) K'2 G (r)2 G (r)2
for all sufficiently large r. Hence,
- u(r) > .7 (2.18)G(r)
Since the right-hand side is not integrable at infinity, whereas the left-hand side is, we have a
contradiction and conclude that K = 0, as claimed.
rdsNext, we consider the case where the integral J Gs)converges. Let E be arbitrarily smallG (s)
positive. It follows from Lemma 3(i) that
G(r) 2 [u'(r)2+F(u(r))] > K -e
for all sufficiently large r. Hence, - u '(r) > 2(K - E)/G (r); so, upon integration, we find the
inequality
u(r) > 2(- E)J ds(2.19)Sa G(s)
for all sufficiently large r. On the other hand, we also have the inequality
131
G (r)2 [4u'(r)2 + F (u(r))] <K+ E
for all sufficiently large r, so - u '(r) < (2(K + E)/G (r) 2 - 2F (u(r))) . Since u(r) - 0 as
r -+ oo, u (r) is less than or equal to $ beyond a certain value of r, and from that point on F (u (r))
is zero or negative. Using the elementary inequality (a2 + b2)1/ 2 a + b for a, b 0, we find
that - u '(r) < '2(K + e) /G (r) + ( - 2F (u (r))1 2 for all sufficiently large r. Hence, upon integra-
tion,
u(r) < 42(K + E) +J(-2F(u(s)))1/2 ds . (2.20), G(s) r
We estimate the last integral by means of Holder's inequality:
Go 001/2 1/2
S(- 2F (u (s))) " ds G (s)G '(s)( - 2F(u (s))) ds] 12 G s1/2 (2.21)r r G(s (s)
Lemma 3 implies that the first integral in the right member becomes arbitrarily small as r tends toinfinity, so by taking r sufficiently large we can certainly achieve the inequality
1/2
(- 2F(u(s)))1 ds E J'(2.22)r G(s)G'(s)
We recall from the condition (G3), that the asymptotic relation (1.8) is satisfied in the present
case. Hence, the expression in the right member of (2.22) is of the same order of magnitude as E
times the integral sas r -+ oo. By increasing r further if necessary, we can therefore cer-r G (s)
tainly achieve the inequality
J(-2F(u(s)))12dscJ . (2.23)G(s)
Combining (2.20) and (2.23), we find that
u(r) < [2(K+ E)J E G(2.24), (s )
for all sufficiently large r. The relation (2.17) follows from (2.19) and (2.24) as E tends to zero. 0
3. Distinct Solutions Must Intersect
Throughout this section we assume that u and v are two distinct non-negative non-trivial
solutions of (1.1). Our purpose here is to show that the graphs of u and v have at least one point
in common.
132
According to Lemma 2, u and v are strictlyy) decreasing on their respective supports, so they
can be inverted there. We denote the inverse functions by r and s, respectively. Thus, r is defined
on the interval [0, u (0)], s is defined on the interval [0, v (0)], and
u(r (t)) = t, t e [0, u(0)]; v (s (t)) = t, t E [0, v (O)] . (3.1)
Let the functions R and S be defined by
R (t) = u'(r (t))2 , t E[0, u (0)] ; S(t) = v(s (t))2 , t E [0, v(0)] . (3.2)
Clearly, R and S are non-negative on their respective domains of definitions. In fact,
R (0) = S(0) = 0, because of Lemma 1(i), and R(u(0)) = S(v(0)) = 0, because of the boundary
condition that is satisfied by u and v at 0. Furthermore, R and S are (strictly) positive in the open
interval (0, u (0)) and (0, v (0)), respectively, because of Lemma 2. We also observe that
u '(r) = - 4 R((u(r)) and v '(r) = - S(v (r)) for all r > 0.
One readily verifies that R '(t) = 2u "(r(t)) and S'(t) = 2v '(s(t)), so R and S satisfy the ini-
tial value problems
R '(t) = 2{g (r (t))4(t)~-f (t)), t > 0; R(0) = 0; (3.3)
and
S'(t) = 2(g (s (t))y -f (t)), t > 0; S(0) = 0; (3.4)
respectively. We prove the following theorem.
THEOREM 1. Let f and g satisfy the conditions (Fl), (F2), (GJ), and (G2). If u and v are two dis-
tinct non-negative non-trivial solutions of the boundary value problem (1.1), then u (r) = v (r) for
at least one value r E (0, oo).
Proof. Suppose that u(r) > v (r), say, for all r e (0, oo). Then r (t) > s (t) for all t E (0, v (0)).
Because g is non-increasing, it follows that g (r (t)) 5 g (s (t)) for all t e (0, v (0)). Hence, R
satisfies the same initial value problem as S, but with a differential inequality:
R '(t) ; 2{g (s (t))"FR (t) - f (t)) , t > 0; R (0) = 0 . (3.5)
The theory of differential inequalities enables us to compare the maximal solutions of the
initial value problems (3.3) and (3.4). For our purpose, such a comparison is not meaningful,
unless there are no other solutions besides the maximal solutions. The conditions of the theorem
indeed imply that (3.3) and (3.4) have at most one non-negative solution.
The initial value problems (3.3) ano (3.4) are of the type
x'=p(t)x1 2 +q(t), t>0; x(0)=0; (3.6)
where p (t) = 2g (r (t)) and p (t) = 2g (s (t)), respectively, and q (t) = - 2f (t). Uniqueness of
133
non-negative non-trivial solutions of this type of initial value problem has been established byKaper and Kwong [1988] under the following conditions: (i) the coefficients p and q are integr-
able near 0; (ii) p and the integral q1 , defined by q (t) = q (s) ds, t 0, are non-negative near
0; and (iii) for every t > 0, there is a points E(0, t), where q 1 (t) > 0.
One readily verifies that the conditions (i) and (ii) are satisfied by the coefficients p and q
above. The condition (iii) is certainly satisfied if $> 0 and f does not vanish identically near 0.
If f vanishes identically near 0, f = 0 on [0, ] for some y > 0, say, then the differential equation
(3.6) reduces to x'= p(t)x1a on [0, y]. The solution that satisfies the initial condition x(0) = 0 is
either trivial or strictly positive on (0, y]. In the former case, we consider the same initial value
problem for t y. Since y < $, the condition (iii) is satisfied to the right of y, so there is at most
one non-negative non-trivial solution of (3.6) for t >y. When combined with the trivial solution
on [0, y], this solution becomes the unique non-negative non-trivial solution for t 0. In the
latter case, when the solution is already strictly positive at y, we can use the Lipschitz continuity
of f at y to show that the solution has a unique continuation beyond y. Thus, we conclude that theinitial value problem (3.6) has at most one non-negative non-trivial solution.
A simple comparison argument then shows that
R (t): S(t) , t e [0, v(0)] . (3.7)
If u (0) > v (0), then R (v (0)) > 0, while S (v (0)) = 0. This would clearly contradict (3.7), so at
this point we must conclude that u (0) = v (0).
The inequality (3.7) implies that I u'(r(t)) I SIv '(s (t)) I. Furthermore, 0 < g (r (t))
g(s(t)), so
g (r (t)) I u'(r (t))I ISg (s (t)) I v '(s(t))I , t e [0, u(0)]. (3.8)
Next, we apply (2.5) to u and v at r = 0 and subtract the resulting equations. We find
g (r)u '(r)2 dr = g (s)v '(s)2 ds , (3.9)0 0
or, after a transformation of variables,
u o) y o)
g (r (t)) I u '(r (t))I dt = g (s (t)) I v '(s (t)) I dt . (3.10)0 0
We recall that u(0) = v(0) and conclude that the inequality (3.8) is compatible with the identity
(3.10) if and only if u '(r(t)) = v '(s(t)) for all t e [0, u(0)]. This equality, in turn, implies that u
and v coincide everywhere. But here we have arrived at a contradiction since we had assumed
that u and v were distinct. Hence, if u and v are distinct, their graphs must intersect at some point
r E (0, oo). 0
134
The monotone separation lemma of Peletier and Serrin [1986, Lemma 9] is an immediate
consequence of Theorem 1. We formulate it as a corollary.
COROLLARY. Let the conditions of Theorem 1 be satisfied. If u and v are two distinct non-
negative non-trivial solutions of the boundary value problem (1.1) and u (r) = v (r) = trfor some
r e (0, oo), then r(t) - s (t) is a monotone non-increasing function oft on [0, t].
Proof Suppose u (r) > v (r) beyond the point of intersection of the two graphs. According to
(3.7), we have R (t) S (t) for all t e [0, t]. That is, - u '(r (t)) - v '(s (t)) for all t up to the ordi-
nate of the point of intersection. Since u '(r (t))r '(t) = I and v '(s (t))s '(t) = 1, the last inequality
is equivalent with r '(t) <-s '(t) for all t e [0,T], which shows that the difference r - s is monotone
on [0, '], as claimed. 0
4. Intersecting Solutions Are Identical
In the previous section we showed that the graphs of two non-negative non-trivial solutions
of (1.1) either coincide everywhere or intersect in at least one point r e (0, oo). In this section we
prove that the latter alternative is excluded by the conditions (F1) through (F3) and (G1) through
(G3).
THEOREM 2. Let f and g satisfy the conditions (F), (F3), and (G1). Let u and v be two non-
negative non-trivial solutions of the boundary value problem (1.1). If u (r) = v (r) > a for some
r 0, then u(r) = v (r)for all r -0.
Proof. The theorem follows from condition (F3) and is shown in the same way as the Sturm
comparison theorem.
Suppose that u(a)= v(a) = r for some a z0 and that u(r)> v(r) for 05 r < a. If ti>,
we obtain the following inequality from the condition (F3):
(v (r) -a) f (u) -(u(r) -a) f (v)<- 0, 0<- r:5 a .
Since u and v satisfy (1.1), this inequality can be rewritten,
[G(r) [-(v(r) -I)u'(r)+(u(r)~i)v'(r)] 5 0.
Hence, upon integration over 10, a ],
(- u'(a)+ v '(a))(t- $) 50 .
Our assumptions imply that - u '(a) > - v'(a), so the expression in the left member is positive.
Clearly, we have a contradiction, unless u '(a) = v '(a). But if both u (r) = v (r) and u '(r) = v '(r)
135
at r = a, then u(r) = v (r) for all r; cf. the last part of theproof of Lemma 2. 0
THEOREM 3. Let f and g satisfy the conditions (F), (GI), (G2), and (G3). Let u and v be two
non-negative non-trivial solutions of the boundary value problem (1.1). If u (r) = v (r) $ for
some r 0, then u (r) = v (r)for all r _0.
Proof. We prove the theorem in two steps. In the first step we rule out the possibility that the
graphs of two distinct solutions u and v have more than one point in common, once they are at or
below the horizontal line t = P. In the second step we show that they cannot even have a single
point in common. The symbols r, s, R, and S below have the same meaning as in Section 3; cf.
(3.1) and (3.2).
Step 1. Suppose that there are two points a and b, where u(a) = v (a) @ and
u (b) = v (b) $f. Without loss of generality, we may assume that a < b and u (r)> v (r) on
(a, b). Then R (u(a)) < S(v (a)) and R (u (b)) > S(v (b)).
By continuity, there exists a pair of points c, d, with a < c < d < b, such that
u(d) = v(c) = t, with u(b) <t < u(a), and R(tc) = S(t). If more than one such pair exists, we
take the one for which T is largest. Thus, R (t) < S(t) for all t e (t, u (a)).
We apply the identity (2.3) to u on [a, d] and to v on [a, c ] and subtract the resulting equa-
tions,
[G(d)2 - G (c)2][ u '(d)2 + F(u(d)) - {G(a)2 [R(u(a)) - S(v (a))]
= 2 G (r (t ))G (r (t ))_ G (s (t ))G '(s (t)) F (t) 1d2 . (4.) 1)
F (t)) dtS.((4)1)
The first term in the left member is the product of two positive terms, so it is positive. The
second term in the left member is negative, by assumption. Hence, the expression in the left
member of (4.1) is certainly positive. Since r(t) > s(t) on (a, b) and the product GG ' is non-
decreasing, we have G (r (t))G '(r (t)) G (s (t))G '(s (t)) for all t e (, u (a)). Furthermore,
R (t) < S(t) there, so the expression inside the parentheses under the integral sign is positive. On
the other hand, F(t) is zero or negative on the entire range of integration, so the integral is nega-
tive. But now we have a contradicition. The possibility of two points of intersection is thus ruled
out.
Step 2. Suppose that u(a) = v(a) S $ at some point r = a. Without loss of generality we
may assume that u (r) > v (r) for all r > a.
According to Lemma 3(i), the following limits exist:
136
K = limr G..G(r) 2 u'(r)2 + F (u(r)) , (4.2)
L = lim, r . G (r)2 2 (r)2 + F(v(r)) . (4.3)
Applying Lemma 3(ii) to u and v on [a, oo) and subtracting the resulting equations, we obtain
G (a)2 R (u (a)) - S (v (a))
= - G (r (t))G'(r (t)) _ G (s(t))G'(s()) F (t)dtS R (W 1/ S (t ) ir
The expression in the left member is negative. Under the integral sign, the expression inside the
parentheses is positive, while F(t) is zero or negative, so the integral is certainly negative.
Hence, if K > L, the expression in the right member of (4.4) is positive, and we have a contradic-
tion.
It remains to investigate the case K < L. If K < L, we take e < (L - K)/8 and choose r
sufficiently large that
G (r) 2 [u'(r)2+F(u(r))] < K + e , (4.5)
and
G (r)2 [ v(r)2+F(v(r))] >L-cE. (4.6)2
It follows from (4.5) that
u [(r) < /2(K +E)+e) fGs)(4.7)l ) , G (s)
cf. the derivation of (2.24) in the proof of Lemma 4. By reducing E if necessary, we can certainly
achieve that 42(K + E) + e < 42(L - e). Thus,
ud(r) < 2(L - E ) . (4.8)rG(s)
On the other hand, it follows from (4.6) that
v (r) > 2(L-e)Jf ds (4.9), G(s)
cf. the derivation of (2.19) in the proof of Lemma 4. The inequalities (4.8) and (4.9) together
imply that u (r) < v(r) for r sufficiently large. But this conclusion contradicts the earlier
137
assumption that u (r) > v (r) for all r > a. Thus, the possibility that the graphs of u and v intersect
is ruled out. 0
5. Conclusion and Discussion
It is clear that the conclusions of Sections 3 and 4 are compatible only if the solutions u and
v of the boundary value problem (1.1) coincide. We have therefore shown the following unique-
ness result.
THEOREM 4. If f and g satisfy the conditions (F)) through (F3) and (G) through (G3), then the
boundary value problem (1.1) has at most one non-negative non-trivial solution.
This uniqueness result generalizes and strengthens the earlier result of Peletier and Serrin
[1986]. It generalizes their result, because the boundary value problem (1.1) is a (non-trivial)
generalization of the problem considered in [McLeod and Serrin 1981]. It strengthens their result,
because the condition (F2) on the nonlinear term f is weaker than the corresponding condition
(H*) in [Peletier and Serrin 1986].
The crucial step in the proof presented here is the transformation of variables (3.2). The
new variables R and S satisfy the nonlinear initial value problems (3.3) and (3.4); and if (F2)
holds, then R and S are uniquely determined. Thus, a direct comparison of their graphs is possi-
ble.
The results obtained here can be generalized to the more general boundary value problem
[A(Iu'I)u') '+ g(r)A (Iu'I)u'+f(u) = , r >O;
u'(0)=0; lim r -.. u(r)=0; (5.1)
where the real-valued continuous function A is such that the product pA (p) is strictly increasing
on [0, oo). Problems of this type arise, for example, in the study of the semilinear elliptic equa-
tion
V-A (I u'I)Vu +f (u) = 0, xeR"; lim ., .u(x)=0; (5.2)
cf. the article by Franchi, Lanconelli, and Serrin [1986]. Details will be presented elsewhere.
Acknowledgments
We thank the organizers, Profs. Wei-Ming Ni, L. A. Peletier, and J. Serrin, for their kind
invitation to participate in the Microprogram on Nonlinear Diffusion Equations and Their Equili-
brium States at the MSRI. We express our particular appreciation to Profs. Peletier and Serrin for
their interest in this work and for several stimulating discussions during the Microprogram.
138
References
H. Berestycki and P.-L. Lions 1978. "Existence d'ondes solitaires dans des probldmes non-lindaires du type Klein-Gordon," C. R. Acad. Sci. Paris, Sdrie A, 287, 503-506.
H. Berestycki, P. L. Lions, and L. Peletier 1981. "An ODE approach to the existence of positivesolutions for semilinear problems in RN," Indiana Univ. Math. J. 30, 141-157.
B. Franchi, E. Lanconelli, and J. Serrin 1986. "Existence and uniqueness of ground state solu-tions of quasilinear elliptic equations," preprint.
B. Gidas, Wei-Ming Ni, and L. Nirenberg 1979. "Symmetry and related properties via the max-imum principle," Commun. Math. Phys. 68, 209-243.
H. G. Kaper and Man Kam Kwong 1988. "Uniqueness for a class of non-linear initial valueproblems," J. Math. Anal. Applic. 130,467-473. Also: Proc. 1986-87 Focused Research Pro-gram on "Spectral Theory and Boundary Value Problems," ANL-87-26, vol. 4, Hans G.Kaper, Man Kam Kwong, and Anton Zettl (eds.), Argonne National Laboratory, Argonne,Illinois.
K. McLeod and J. Serrin 1981. "Uniqueness of the ground state solution for Au +f (u) = 0,"Proc. Nat. Acad. Sci. USA 78, 6592-6595.
L. A. Peletier and J. Serrin 1983. "Uniqueness of positive solutions of semilinear equations inR"," Arch. Rat. Mech. Anal. 81, 181-197.
L. A. Peletier and J. Serrin 1986. "Uniqueness of non-negative solutions of semilinear equationsin R"," J. Diff. Eq. 61, 380-397.
139/4
A NON-OSCILLATION THEOREM FOR THE EMDEN-FOWLER EQUATION;GROUND STATES FOR SEMILINEAR ELLIPTIC EQUATIONS
WITH CRITICAL EXPONENTS
Hans G. Kaper and Man Kam KwongMathematics and Computer Science Division
Argonne National LaboratoryArgonne, IL 60439
Abstract
Let i be a ball centered at the origin in RN (N > 2), a its boundary, andf : [0, co) -4 [0, oo) a given function. This article is concerned with radial solu-tioit of the boundary value problem
Au+f(IxI)u =0, u>0 in ; u =0 on A. (1)
Radial solutions of (1) are functions u of the variable r = Ix I that satisfy theordinary differential equation
u"+ N-1 u'+f(r)u = 0. (2)r
It is shown that, if p = p*, where p* is the critical Sobolev exponent
(p5 = (N+2)/(N-2)), then (2) is non-oscillatory at the origin if there exists aa > 0 such that f (r)(log(1/r))0 is a nondecreasing function of r near the origin.
The result is a corollary of a similar result for the Emden-Fowler equation
x''+ t-2-Yg(t)x'+ 2 Y = 0 , (3)
where g is a given positive-valued function and y a positive constant. The rela-tion between the solutions of (2) and (3) is established via the transformationsx(t) = u(r), g (t) = f (r), where t = ((N-2)/r)N-2 . It is shown that (3) is non-oscillatory at infinity if there exists a a> 0 such that g (t)(logt)G is a nonincreas-ing function oft for all sufficiently large t. The proof of this result constitutes thecore of the article. The result is of independent interest in the theory of theEmden-Fowler equation and improves earlier oscillation results of Nehari andChiou.
1. Introduction and Statement of Results
Let it be a bounded domain in RN (N > 2), Baf its (smooth) boundary. It is w. I known that
the problem
Au +uP = O , u > 0 in it; u=0 on ail; (1)
has a solution for any star-shaped domain it if p is less than the critical (Sobolev) exponentp* = (N+2)/(N-2), but no solution for any star-shaped domain fl if p > p*; cf. Pohozaev [1965].
Recently, several authors have considered problems like (1), where the nonlinear term uP is
141
perturbed by some other term, as in
Au+uP +au =0, u>0 in(2; u =0 on ail; (2)
where 0 < q < p. The dichotomy at p = p. noted above can then be resolved by means of the
additional parameters A and q; cf. Brezis and Nirenberg [1983], Budd [1985a, 1985b], Brezis
[1986], Atkinson and Peletier [1986a, 1986b, 1987], and Budd and Norbury [1987].
In this article we are concerned with another type of perturbation of the nonlinear term in
(1). We consider the case where is a ball in RN and the problem (1) is modified to
Au+f(IxI)u =0, u >0 inA; u =0 on an. (3)
Here f is a given positive-valued function, which depends on the radial variable r = Ix I only.
It follows from the results of Gidas, Ni, and Nirenberg [1979] that any solution u of (3) is
radially symmetric, i.e., u depends on the radial variable r = Ix I only. Consequently, u satisfies
the ordinary differential equation
u''+ N-1 u'+f (r)u = 0, (4)r
where 'denotes differentiation with respect to r. In this article we are interested in finding condi-
tions on f which guarantee that the equation (4) is non-oscillatory at the origin if p = p'.
We recall that a solution u of (4) is oscillatory at the origin if, for every r 1 > 0, there exists
an r2 E (0, r 1) where u (r2) = 0. A nontrivial solution of (4) that is not oscillatory at the origin is
called non-oscillatory at the origin, and the equation (4) is called non-oscillatory at the origin if
every nontrivial solution is non-oscillatory at the origin.
Clearly, if (4) is non-oscillatory at the origin, it is meaningful to ask for solutions u of (4)
that are positive on some interval [0, R) of positive length. Such solutions, if they exist, give rise
to ground states for the semilinear elliptic problem (3). Thus, the objective of this investigation
bears directly upon the investigation of the existence of ground states for problems described by
(3).
The change of variables
t = ((N-2)/r)N- 2 , x(t) = u(r) , (5)
transforms the differential equation (4) into an equation of the Emden-Fowler type [Emden 1907,
Chap. XII; Fowler 1931]. If u satisfies (4), where p = p", then x satisfies
x'+ t-2-Yg(t)x 1+* = 0, (6)
where g (t) = f (r) and y = -(N-2). Notice that 0 < y < co.2
The transformation (5) maps the origin to the point at infinity. Thus we are led to study the
oscillatory behavior of nontrivial solutions of (6) for large t. In particular, our interest focuses on
142
establishing conditions on the function g which guarantee that the equation (6) is non-oscillatory
at infinity. (Henceforth, we omit the quantifier at infinity when we discuss solutions of (6).)
A study of the oscillatory or non-oscillatory behavior of solutions of the Emden-Fowler
equation (6) is of independent interest as well. Such studies have been undertaken before by
researchers in oscillation theory, notably Nehari [1969], Coffman and Wong [1970, 1973], and
Chiou [1971, 1972].
Nehari [1969] proved that (6) is non-oscillatory if there exists a a > 2+ 1/y such that
g (t)(logt)0 is a nonincreasirg function of t for all sufficiently large t. This result was subse-
quently improved by Chiou [1971], who showed that the conclusion still holds if a satisfies the
weaker inequality a > -(3+1/y). A further improvement was announced by Chiou [1972], who2
claimed that it was sufficient that a satisfy the inequality a > 0. But, as was pointed out by
Nehari [1974], Chiou's proof contained an error, after correction, the result was that (6) is nonos-1
cillatory if a > (- + y)/((1 + y)). The corrected result still amounted to an improvement of2
Chiou's earlier result [Chiou 1971].
Finally, we mention a result of Erbe and Muldowney [1982], which is refinement of
Chiou's. These authors proved that (6) is non-oscillatory if g ()(logt) 0 is nonincreasing and
g(t)(logt)" is bounded for some pair (a, TI) E D, where Dy = ((a, y) E R2 :
a > 0, T + a/(1 + 2y)> 1/y). This result reduces to Chiou's if a = 1.
In this article we shall prove the following theorem.
THEOREM 1. The differential equation
x''+ t-2-"g(t)x1 "Y = 0, (7)
where g is a positive-valued function and y a positive constant, is non-oscillatory at infinity if
there exists a a > 0 such that g (t)(logt) 0 is a nonincreasing function of t for all sufficiently large
t.
Since the conclusion of the theorem is not true if g is constant, the result is likely to be best
possible.
As a consequence of Theorem I we have the following result for the partial differential
equation (3).
COROLLARY 1. Let C be a ball centered at the origin in RN. Let f be a positive-valued function
of the radial variable only. Letip be equal to the critical exponent p * (p * = (N + 2)/(N - 2)). If
there exists a neighborhood 0 of the origin and a a > 0 such that r -+ f (r)(log(1/r)) is a non-
decreasing function in 0, then every nontrivial radial solution of the semilinear elliptic equation
143
Au+f(IxI)u =O, xeA, (8)
is non-oscillatory at the origin.
Because of the result of [Chiou 1972], as corrected by Nehari [1974], it suffices to prove the
theorem for 0 <a (- + y)/((1 +'y)). Our proof, which takes up the entire Section 2, is based2
on the use of Lyapunov functions. At several places we apply a generalized Sturm Comparison
Theorem from the theory of linear ordinary differential equations. This theorem, together with its
proof, is given in the Appendix.
2. Proof of Theorem 1
The proof of Theorem I is by contradiction, where it is assumed that the equation (7) admits
a nontrivial oscillatory solution. We explore i detail the consequences of this assumption and
eventually derive a contradiction. Thus, the existence of nontrivial oscillatory solutions of (7) is
ruled out, and the theorem follows.
2.1 Preliminaries
Before embating on the proof, we establish some notational conventions and state three
equivalent forms of the differential equation (7).
To explicitly bring out the logarithmic factor in the function g, we introduce the function a
by the definition
a (t) = g (t)(logt)0 . (9)
We assume throughout that the function a is nonincreasing. Without loss of generality we may
assume that 0 < a (t) < 1 for all t.
With a slight abuse of notation, we will often omit the generic argument of a function. For
example, we may write x when x(t) is meant.
When considering the value of a function at a specific argument value, where the latter is
characterized by an index to the generic argument, we will often omit the generic argument and
attach its index immediately to the function symbol. For example, we may use am as a shorthand
notation for a (tm).
In terms of a, the original equation (7) reads
X'+ (t) 1+2=0. (10)t2 +l/Y(logt)G
Changing variables,
s = logt, y (s) = t- 2x(t) , (l1)
we see that (10) is equivalent with
144
(8)
Y'--_4y +as yl+2/=0. (12)
Here, a (s)= a (t(s)).
We obtain a third form of the equation upon introduction of the function z,
z(s) = s 3y(s) . (13)
The function z satisfies the differential equation
z''+Oz'+ [a(s)z - ( + _a )]z = 0 , (14)s 4 s
where the constant a is defined in terms of y and a by the relation
a = 1(1 - 2 ) . (15)
1Since it suffices to consider values of a in the interval (0, (- + y)/('(1 + y)), we may assume that2
a satisfies the condition a > 0.
The transformations (11) and (13) preserve oscillation. Hence, if x is an oscillatory solution
of (10), as we suppose throughout the proof of the theorem, then there is an infinite sequence of
points so, s1 , - - - with so < s 1 < - - -, such that y(s) = 0 and z(s;) = 0 for i = 0, 1, - - -. In the
following subsections we explore in detail the consequences of this supposition.
2.2 Lyapunov Functions
In this subsection we introduce two Lyapunov functions, one associated with the differential
equation (12), the other with the differential equation (14), and establish some of their elementary
properties.
The first functional W is defined by the expression
W(s) = (y(s)) I(Y)2 _ly2 + a (s)a2+ 2y (16)2 8 (2+ 2/y)s0
Since y satisfies (12), we have
_'s___ aa(s) _ a'(s) 2+y(17)
2+2/y s +1 s
Since a is nonincreasing, the expression inside the brackets is always positive, so 1 is monotone
nonincreasing. This is the first assertion of the following lemma.
LEMMA 1. The functional yi is monotone nonincreasing, the limit im i(s) exists, and
y'(s)) > _y4(oo) 0.
145
Proof At a zero s; of y, the value of ilr is V(si) = 2(y';)2 , which is positive, so l(s) z (si) > U2
for all s <si. As x is oscillatory, s; can be chosen arbitrarily large. Hence, 4(s) is a nonincreas-
ing function of s which is bounded below by 0, so yj(s) must tend to a limit as s -+ oo. Clearly,
the limit y(oo) satisfies the inequality 1J(oo) _0. 0
For future reference, we also give the expressions for yu and i4' in terms of z:
V(s) = s ( -(z)2 + Ozz'+ a (s) z 1- (1 - )right]z2 , (18)2s 2+2/ 8 s
'()=- sYO aas) -a'() 2+v7(- +y s a (s)z . (19)2+ 2/y s
The boundedness of the function az21Y is an immediate consequence of Lemma 1. As this bound-
edness will play an essential role in the following analysis, we state it in a separate lemma.
LEMMA 2. The function az21 is bounded.
Proof. It follows from Lemma 1 that there exist constants M and sM such that 'i(s) 5 M for all
S ? SM. Then also
Ss a (s) / z 1y z2 5 M.2 11+ 1/y 4
Let A = az1Y. Then
A (s)[A (s) - 1(1 + I/Y)] 5 (2+ 2/y)MaT (2+ 2/y)M4 S7 sN
44The upper bound becomes arbitrarily small as s increases, so A (s) tends to 0 or - (1 + / Y). In
either case, it is bounded. 0
The second functional $ has the unusual feature of being defined nonlocally. Let s; and
si +1 be two successive zeros of z. Let sm(i) be the point in (s;, s; + 1) where z has its extremum.Then $ is defined by the expression
$(s)=4)(z(s)) 1 1 (z) 2 + a(s) z2+2Y _ + 2 +y9)zj2 aT(s.(i)), s e (s;, s;1] . (20)2 2 +2/y 2 4 s
This function is discontinuous at each zero of z; in fact, because a is nonincreasing, $ has a down-
ward jump at each zero of z. This downward jump is an essential feature in the proof of the
theorem.
Since z satsfies (14), we have
146
$'(S)= - (z')2 + 2 a'(s) 2+2/ + z2 aY(sm()) , s e (si, s;+1 ]". (21)
The derivative may be positive or negative, so $ may not be monotone. A major thrust of the
proof of the theorem is to assess the relative importance of each of the three terms inside the
brackets.
Notice that $ is positive at a zero of z. The following lemma shows that 4)(s) converges to 0
as s -+ c.
LEMMA 3. lim,._,4)(s) = 0.
Proof. (We write m, instead of m (i).) At s; we have 4)(si) = -(z;) 2a, which is positive. From2
(18) we see that yi(s;) = {sT(z;')2, so
$(s) = ay s y(s;) 5 si Yc(s;).
As x is oscillatory, s; grows beyond bounds as i -+ Because W is bounded, it follows that
lim$)( - 0 .(22)
For any s e (s;, s+ 1) we have
$(s) = $(;) + $'(t)dt .S.
From (21) it follows that
$(s) S az2(s)an ,S
so
dt aaL z2
$(s) $(s;) + aaL z2(t)-gds$(si)+ 2 . (23),i t 2s;
In the same way we conclude from the identity
Si+I
$(s) = $(s;+ i)- $'(t)dt
that
Si d I z2
$(s) $(si + 1) - aaT z -2(t) a $(s;+)- 2 .(24)
s t 2s +c1
The product amz2 in the right members of (23) and (24) remains bounded as i -+ 00; cf. Lemma
147
2. Therefore, (22), (23), and (24) imply that for any e > 0 there exists an index j (E) such that, if
i j(E), then I 4(s) I < e for all s E (si, si + 1). This proves the assertion of the lemma. 0
2.3 Qualitative Behavior of z between Successive Zeros
We now turn to a detailed investigation of the qualitative behavior of z between two succes-
sive zeros.
Again, let si and si i. be two successive zeros of z. We assume, without loss of generality,
that z is positive in the interval (s;, s; + 1). Let s.() be the point in (si, s., i) where z reaches its
maximum value.
LEMMA 4. The function z is strictly increasing on (si, sm(i)) and strictly decreasing on
(sm(i), si + + )for all sufficiently large i.
Proof. (We again write m, instead of m(i).) Suppose z has a local minimum at some point sc
between s; and sm. Then z'(se) = 0 and z'(sc) > 0.
Evaluating (18) at s = se, we find
2 aczea 1 (y)
1y(se)=si"ze a -~ -(IVs s 2 + 2/y 8 (1 sc
Because V is positive (cf. Lemma 1), it follows that
ac zca 24(1 +1/y) 1 - (.
The differential equation (14) yields the identity
-z = ac z ' -4 + a_zc
Combining this identity with the preceding inequality, we find that
A, 1 4(1 + 1 /y)(W)2 + a
zeZ 4y s
The expression in the right member is strictly positive if i is sufficiently large, so it would follow
that ze ''< 0, and we have a contradiction. A local minimum between s; and sm is thus ruled out.
A local minimum between s, and si, + is ruled out by a similar argument. 0
It is easy to show that z'(si) tends to0 as i -+ oo.
LEMMA 5. lim_... z'(si) = 0.
148
Proof. Evaluating (18) at s = si, we find (z1)2 = 2s; " y(s;). The expression in the right member
tends to zero as i -+ oo, because y is bounded. 0
The following lemma gives specific information about the value of az21Y at sm(i).
LEMMA 6. lim;_,.. a(sm(i))z' (sm(i)) = 4 (1 + 1/).
Proof. (We again write m, instead of m (i).) Evaluating (20) at s = s, we find
4S) = Eamn 2+ 2tr I +1 )z2] a.$(sm) = 2+2/Y zm 2 (4 +s )Z a .m
Let Am = amz"Y. Then
1 (___+___/___________
AY Am - (1 + 1/y) (2+ 2/Y)$(sm)+ s4 s2
Both terms in the right member tend to zero as i -+ o, so the quantity Am either vanishes in theIlimit or approaches -(1 + I/y). Because
2 Am _ 1 ( )2WSmYSmZmu[2+2/Y 81 sm
and yi is positive, it must be the case that
1 (w)2
Am>'4(1 +1/Y)1l s J
But sm grows beyond bounds as i - oc, so the lower bound is certainly greater than -(1 + 1/Y) if8
i is sufficiently large. The possibility that Am vanishes in the limit is thus ruled out. 0
2.4 Auxiliary Results on the Variation of a
In this subsection we prove an auxiliary result that puts a restriction on the variation of a.
LEMMA 7. Let s, and sq be two arbitrary points in [si, si+1 with s < sq. If there exists an 71 > 0
such that a (s)z2+21(sZ) 'qfor all s e [sp , sq], then
a(sq) - 1a (st)
Proof. Consider the expression (19), which we write as
149
V(S) =--2+2/ s W a a(s) a (s )z2+ (s) .1+C~s) 2+ 2/y Is - a (s) J~~J 1 ()
Estimating the factor az2 + 2y by j and integrating both sides of the resulting inequality over the
interval (s,, Sq), we obtain
a(s,) - V(sq) >_2 (sr - sN) - :'(s s2+2/ y 'a (s) 1We may assume without loss of generality that si is greater than 1, so we can ignore the factor sNunder the integral sign. Then we can estimate the integral and obtain the following inequality:
W(s,) - V(sq) >2+2/ q(s -s,)+log2 / a (sq)]
The quantity ye(s) is a nonincreasing function of s, which converges to a (nonnegative) limit ass -+ oo; cf. Lemma 1. Hence, the expression in the left member is positive; furthermore, by tak-ing i sufficiently large, we can make it arbitrarily small. The same must then be true for each term
in the right member, so the logarithmic term must vanish as i -> oo.Q
On the interval [s;, s,,; ] z is increasing, while a is nonincreasing, so az2 +2Y may bedecreasing or increasing there. The following lemma puts a lower bound on the rate of decrease
of this function.
LEMMA 8. Let s,(i) be an arbitrary point in [si, sm (i)]. If i is sufficiently large, then
a (s)z2+2/Y(s)> 2 a (sp(i))z2+2/y(s (;)) for all s E [s(;), sM(i)].
Proof. (We write m and p, instead of m(i) and p (i).) Let sq be the smallest value of s e [s,, s.m]
where a (s)z 2 +2 y(s) = = a (s)z2+/Y(s,). Then
a(s)z2+21T(s)> 2 a (s,)z 2+2'Y(s), s E [s,, Sq] .
Applying Lemma 7 with r = az+ 2Y, we conclude that the ratio ap/aq tends to 1las i -+ oo, so
9by taking i large enough we can certainly achieve the inequality aq y61ap. The assumption that
sq < s,, leads to a contradiction, because it would follow that
1a z2+2Y = az+2/ > a z2+*2 Qa z22/Y.ThPrefreqs P 10 Ps.
Therefore, sq > 0.
150
On the interval [sm(i), si~i], the analog of Lemma 8 is trivial, because both a and z are
nonincreasing. Thus, if sq(i) is an arbitrary point in ' [sm(i), si1], then a (s)z 2+vy(s)
>a(sq())z 2 +2"(sq(i)) for all s e [sm(i), sq(i)].
LEMMA 9. Let ij be an arbitrary positive constant. If sp,()E (si, si+ 1) is such that
a (spt;))z2"y(() = 1, then
limas) = 1 .i-. a (S. w))
Proof. (We write p and m, instead of p (i) and m(i).) If s, < sm, then
a (s)z2+2Y(s) az 2 'Y '(az, T) T
for all s e [s,, sm]. (We recall that 0 < a (s) 5 1 for all s.) If s, > s,, then
a(s)z2 +2'Y(s) apz2+Y (az2,4 ) 1 +Y
for all s E [sm s,]. In either case, the assertion of the lemma follows from Lemma 7. 0
2.5 Behavior of z near' Maximum
We now consider in more detail the behavior of z near sm(i), the point in [s;, s;+1] where z
has its maximum. We recall that the value of azmY at s(i) approaches (1 + l/Y) as i -+ oc; cf.4
Lemma 6.
Let Z be the solution of the initial value problem
Z"+ZI+er - Z = 0, t > 0; Z(0)= [7 ( + 1/y) , Z'(0) =0; (25)
and let the definition of Z be extended to negative values of the argument by the identity
Z(t) = Z(-t).
As the next lemma shows, Z is the limit of a scaled version of z in the neighborhood of sm(i)
as i -+ -.
LEMMA 10.
lim_- a 2(sm(i))z(sm(i) + - ) = Z( '),
uniformly on compact intervals containing the origin.
Proof (We write m, instead of m (i).) Let the function ( be defined by the identity
151
(s) = a Z (s ), S E (s;, Si+l."
Like z, ( has a maximum at sm, where its value approaches (-(1 + 1/y))j2 as i -+ oo; cf. Lemma
6. The function C satisfies the differential equation
a 2(s) C y___ _' = 0. (26)a (sm) 4 s s2
Being the solution of an initial value problem with continuous data, C depends continuously on
the coefficients of the differential equation and the initial data. Confining ourselves to compact
intervals of the type [sm - , sm + ], where is a fixed positive constant, we let i tend to
infinity. The ratio a (s)/a (sm) approaches I uniformly. Because C' and C are bounded, the
expression inside the parentheses tends to 0. Therefore, ((s, + -) approaches Z(-), the solution
of the initial value problem (25), uniformly on [- p., p.] as i -+ o. 0
Let sI(i) be the smallest value of s e [s;, sm(,)] where a(s)zy"(s) = 1(1 + 1/y). Similarly,8
let sn(i) be the greatest value of s E [Sm(i), si+1] where a(s)z2ms(s) = -(l + 1/y). The point of the
following lemma is that the length of the interval [s (;), sn()] approaches a fixed value as i tends
to infinity.
LEMMA 11. There exists a positive constant p. such that
lim (sm(i) - Sl(i)) = ,-
and
lim(s.) - m(i)) = J.-
Proof. (We write 1, m, and n, instead of 1(i), m(i), and n(i).) From the definitions of s and s
we obtain the identities
a(sm)zl"y(sm - (sm - s,)) = 1(1 + l/y)am 8
and
a- a(sm)z2 (s. + (s, - sm)) = -(1 + 1/y) .am 8
It follows from Lemma 2 that the ratios al/am and an/am tend to I as i -+ o . Lemma 10 implies
that sm - s and s - sm tend to p, where Z(g) = ( (1 + 1/y))y2.8
152
2.6 Behavior of z away from a Maximum
In this subsection we analyze the behavior of z a bounded distance away from the pointsm(;). As in the foregoing section, we let s1(;) be the smallest value of s E [si, sm(i)] where
a (s)z Y(s) = -(1 + 1/y), and s~(;) the greatest value of s E [s, (i), s;i +] where a (s)z (s)
= 1 (1 + 1/y).
We recall that z satisfies the differential equation (14),
z'+-z'+ [az1 -] + z = 0. (27)
A Generalized Sturm Comparison Theorem will play an important role in the following analysis.
The theorem, together with its proof, is given in the Appendix.
As we saw in Lemma 5, the derivative z' at a zero s; of z tends to 0 as i -o. The follow-
ing lemma implies that the logarithmic derivative z'/z at s; remains bounded away from 0.
LEMMA 12. There exists a positive constant c such that
(z(s))2 Z cz 2 (s), s E [S;, S(;)] U [sX(i), s;+1].
Proof. The proofs for the two intervals are similar. We restrict ourselves to [s;, sl(i)J.
Let sk(i) be the smallest value of s e [s;, s(i)], where a(s)z"(s) = 0.01. We write k, 1, and
m, instead of k (i), 1(i), and m(i).
First we consider the interval [s;, s]. By choosing i large enough, we can certainly achieve
the inequality ya/s 5 0.01. Furthermore, az -(1 + als2) S -0.24. These observations lead4
us to compare the solution z of (27) on [s;, sk] with the solution w of the linear initial value prob-
lem
w"+ 0.Olw'- 0.24w = 0 , s > si; w(si) = 0 , w'(s;) = z'(si) . (28)
The solution of (28) is
w (s) = z'(si) X,
where %1 and ? are the characteristic roots of the linear equation (28). We note that X1 and ahave opposite signs. The generalized Sturm comparison theorem yields the inequality
z'(s) w (s) ___e ____s_)-___e__
z(s) w(s) e1(-S, )_e(ss) s E [s;, Sk]
Taking X1 to be the positive root, we conclude that
153
(z(s))2 kjz 2 (s) , s E [S;,SkI . (29)
Next, we consider the interval [sk, s]. We start from the identity
aX,(z')2 = ay, [z2 _ I az2+2d+-z2] +2$(s) ,
which follows from the definition (20) oft, and estimate the various terms in the right member.
The ratio ak/a, tends to 1 as i -+ 00; by taking i sufficiently large, we can certainly achieve
the inequality a 10-al. Because a is nonincreasing, it follows tha a(s) a 2 -a(s) for10 10
all s E [Sk, si]. We use these inequalities to estimate the first and second term. We estimate the
third term, which is positive, by 0.
Thus, using the abbreviation A = az 21y, we obtain the inequality
al(z)2 SA Y9- 1 A -2$(s)I.40 1+1/]
Because s is bounded away from s;, A is certainly bounded below on [Sk, s] by a positive con-
stant. Also, because s is bounded above by s, A is bounded above by a constant which is cer-9
tainly less than -(1+1/Y). Consequently, the first term in the lower bound is bounded below by40
a positive constant on [Sk, s]. By increasing i if necessary, we can also achieve that 21$0(s) I is
less than this positive lower bound, so there exists a positive constant 1 such that aX,(z) 2 11 on
[sk, s]. We combine this result with the estimate azhl S 5(1+1/Y), which holds everywhere on
[s;, s,], and use the fact that a is nonincreasing. We thus find that
(z(s))2>I,(.18 ) (z (s))2 , S E [Sk, s]. (30)1+1/Iy
The assertion of the lemma follows from (29) and (30). 0
To conclude this subsection, we show that the length of each of the intervals [s;, si)] and
[Sn(), s; +i] grows beyond bounds as i -+ oo.
LEMMA 13. lim_.. (Sl(i) - si) = 0 and limi- _ - (Si+1 - S (i)) = 00
Proof. The proofs of the two cases are similar. We restrict ourselves to the first case.
Again, let sk(i) be the smallest value of s E [s;, sI(i)] where a(s)z2'y (s) = 0.01. It suffices to
prove that the length of the interval [s;, sk(i)] grows beyond bounds as i -+ 00. (In the remainderof the proof we write k and m, instead of k (i) and m(i).)
Consider the differential equation (27) satisfied by z. The coefficient of z' is always posi-
tive. By taking i sufficiently large, we can achieve that the coefficient of z is at least equal to
154
- 0.26 on the entire interval [s;, sk]. These observations lead us to compare the solution z of (27)
on [si, sk] with the solution w of the linear initial value problem
w-O0.26 w = 0 , s > s; ; w(si) = 0 , w'(s) = z'(si) ; (31)
which is
sinh((s -s;O2)w (s) = z'(s;) .
According to the generalized Sturm comparison theorem, we have the inequality
z'(s) S w '(s) , s E [s;, s]
Applying this inequality at sk, we find
k cosh((s - s;) ~)zi
We claim that the expression in the left member grows beyond bounds as i -* .
From the expression for $ at sk we obtain the identity
aL(z')2 = a~ r -ak z+21 + 2z + 2$(s )
The last term inside the brackets is positive. The ratio am/ak tends to 1 as i -+ c, so by taking i
sufficiently large, we can certainly achieve the inequality aL -al. Furthermore, am 5 ak.2
Therefore,
aym(zk)2 2 -1Al - - Ak - 2 I(s) I ,2 4 1+1/y
where we have used the abbreviation Ak = akz . Here the expression in the right member is cer-
tainly bounded below by a positive constant if i is large enough. Because a. S 1, it must be the
case that zk'is bounded below by a positive constant. On the other hand, z;'tends to 0 as i -4 o,according to Lemma 5, so the ratio z'/z;'gows beyond bounds, as claimed.
This result implies that cosh((sk - s;)' 4 26~), and therefore sk - s tends to infinity as i -4 0.0
2.7 Monotonicity of$
We now turn to an investigation of the monotonicity properties oft. We use the same
definitions of s1(i) and s.(;) as in the foregoing section.
LEMMA 14. If i is sufficiently large, then $ is monotone decreasing on [si, sti;] and [s(), s;+1 .
155
Proof. We write 1, m, and n, instead of 1(i), m(i), and n (i).
We recall that $ has a downward jump discontinuity at si. The derivative $' is given by the
expression
$'(s) [ = - (z')2 + a(s) z2++ z 2 a.s 2+2/ s e
Ignoring the middle term, which is negative, and using the result of Lemma 12, we see that
Sa aLz2(s)$'(s) 5- cyo-- , s E [s;, s U [s, s;i+ .
s2 s
If i is sufficiently large, the expression inside the parentheses is positive, so $'(s) < 0 on the inter-
vals considered. 0
Lemmas 11, 13, and 14 show that the functional $ is monotone decreasing on [s;, s;.i 1 J,
except possibly on a subinterval of finite length near s,,(;). Thus, $ behaves almost everywhere
like a classical Lyapunov functional for the differential equation (14).
2.8 Differential Inequality for $
We now derive a differential inequality for $, which holds almost everywhere where $ is
monotone decreasing, namely on [s;, sI(i)] v [s.(;), s;if], except for two subintervals, one adja-
cent to si, the other adjacent to s;+1. The lengths of these exceptional subintervals remainbounded as i -+ co.
As a first step, we derive an estimate for the term a7(sm(i)Xz')2 , which occurs in the expres-
sion (21) for$'.
LEMMA 15. There exists a positive constant v, such that
YaaT(s,(i))(z'(s))2 2(s), s E [s; + v, st(i)] U [s 3(;), s;, - v].
Proof. We treat the two intervals separately.
Case 1. s near s;.
Let sk be as in the proof of Lemma 12. On [s, sk], we apply the generalized Sturm com-
parison theorem, comparing z with the solution of (28). We conclude that
z'(s)2 w'(s) , sE [si, ski]
Hence,
z'(s) 20W'(s) -,") - %202( a eM'~'' _s), s E [si, sl.z'(s;) w'(s;)
156
If we define v1 by the equation (Xe '' )2 = 1/(')), then
1z'(si) , S E [Si+v,S].
Consequently,
' aL(z'(s))2 aL(z;')2 = 2$(s;) , S E [S; +V1 , Ski.
Since $ is decreasing on [s;, si], it follows that
uaL(z'(s))2 Z 2$(s) , s E [s; + v 1 , Sk].
Next, consider the interval [sk, Si]. We recall from the proof of Lemma 12 that a (z')2 is
bounded below by a positive constant r, on [sk, s]. Since $(s) tends to 0 as s -+ o, it is certainly
the case that
6aL(z'(s))2 2$(s) , s E [Sk, s]J, (32)
provided i is large enough.
Case 2. s near si, 1 .
Consider the function F,
F(p)= 1- -~4p2, p 0.(1 4p)
Since F(0) = 0, there exists an e >0 such that F(E) < y. Let a be so chosen. Let sq (i)be the
(unique) point in [s.(;), s;1+] where a(s)z"'Y(s) = E. We consider the intervals [s(i), Sq(i)] and
[Sq);), s; 41] separately. We write n and q, instead of n (i) and q (i).
We begin by considering the interval [sq, si+1 ]. If z satisfies the differential equation (14),
then , defined by the expression
((t) = z(S;+1-- t), t 2 0,
is a solution of
- (+ a (s)z2J(s) - + (= 0 . (33)
(Here 'denotes differentiation with respect to t, and s = s;+1 - t.)
By taking i sufficiently large, we can certainly achieve the inequalities p/s e and
a/s2 : 12e. These observations lead us to compare the solution of (33) with the solution 71 of
the linear initial value problem
t1''- en '-4-+1 E 11 = 0 , t > 0 ; (0) = 0 , ,,'(0) = c '(0) ; (34)
157
which is
e -eal(t) =_((0) X.
Here, X1 and X are the characteristic roots of the linear differential equation (34). We observe
that X1 and X have opposite signs. The generalized Sturm comparison theorem yields the ine-
quality
('t) 11(t) _ ,1e i- heX
f(t) ST(t) = eX"'-_eX'
As t increases, the quantity in the right member approaches X, the larger root of the characteristic
equation, which is equal to - + e. There exists therefore a v2 > 0 such that2
11(t)51 + 2E , t z v2 -f(t) 2
Let v2 be so chosen. Then
z'(s) _ (t 1z(s) - f(t) 2 + 2E , S E [Sq, Si+1 - V2,
so
Z2
Zs 2 4 , S E [S, Si+1 V2] -
t(s) (1+4E)2
Given this inequality, we estimate$ as follows:
2$(s) S[(z)2 + 1a (s) /z2+21 _- z2 ay S a (z)2 1 _- _- s) z'+ 4 1+1/y z
[ (Z +1-4E
4 a(z')21- ( +4)2s'al(z)2 , s E [sq, s;+1 - v2] " (35)
On the interval [se, sq] we start from the identity
a,(z)2 = al z2 _ a sz2+v+ z2 + 2$(s)
and proceed as in the proof of Lemma 12 (the part dealing with the interval [sk, s,]), to show that
the quantity am(z)V' is bounded below by a positive constant Ti. Since $(s) tends to 0 as s -+ 00,it is therefore certainly the case that
'6aL(z'(s)) 2 2$(s), s E [sM, Sql , (36)
provided i is large enough.
158
The assertion of the lemma follows from (32), (35), and (36), with v = maxv1 ,v2). 0
We return to the expression (21) for $',
$(s) = [-(z')2 + 2+as 2+2Y + z ja (s(i)), s e (s, s;+1 ].
Because a is nonincreasing, we can ignore the middle term to obtain the estimate
$'(s) - (z)2 + az2JaT(sm(i)), s e (s, si+1 ]. (37)
We decompose the interval (s;, s,+] into a sum of two disjoint intervals,
(s;, s; 1 ] = I; uJ,
where
J; = (si, Si + v U [Sz(i), S(i)] U [Sm(i), Sp(i)] U [s+i1 - v, si1]
and
I; = (s;, s;, 1] -J;
It follows from Lemmas 11 and 15 that the length of the subinterval J remains bounded; in fact,
lime r.IJ; I = 21L + 2v. The length of the subinterval I;, on the other hand, grows beyond bounds
as i -+oo.
On I; we have the inequality
yaT(s(i))(z'(s)) 2 2$(s) , S E I;
On J;, we use the trivial estimate
yaa T(sm(i))(z(s))2 2 0 , S E J;.
We conclude that there exists a positive constant M, which does not depend on i, such that
2 M$'(s) <_- -(s)+y , s e I;, (38)
s s
M (39)$'(s) T g, S EJ/ . 9
In the following subsection we combine these inequalities for i = 0, 1, - - - into one single ine-
quality and derive an asymptotic estimate for $.
2.9 Asymptotic Behavior of$
LetI and J be the union of the intervals1i;and J;over all i = 0, l, - - -,
159
I = u; l; , J = u Ji,
and let the function f be defined by the expression
2f(s)=- , seI; f(s)=0, seJ. (40)
S
We prove the following lemma.
LEMMA 16. The functional O satisfies the inequality
$(s) 5 (s),
where
ms(s) = F (s) [sO) + MJ F 3)dt , (41)
with
F(s) = expj f (t)dt. (42)
Proof. It follows from the inequalities (38) and (39) and the definition off that
M$'(s) s -f-(s)$(s)+ -M . (43)
s
Comparing the solution of this differential inequality with the solution 0 of the initial value prob-
lem
M-f (s)(s)+- , s z so ; 0 (so) =(so) ; (44)
s
we conclude that
$(s) S (D(s) , s >_ so . (45)
The initial value problem (44) is linear and can be solved explicitly for 0. The solution is given
by (41). o
Our next task is to estimate the function 0 and thus obtain an estimate for $.
LEMMA 17. Let e be an arbitrarily small positive constant. There exists a positive constant C,
which depends on E but not on s, such that
0(s) Cs-2+e
for all sufficiently large s.
160
Proof. Let e > 0 be given. According to Lemma 16, $ is majorized by C, so it suffices to prove
the assertion of the lemma for the function CD.
Since f (s) : 2/s, we have F (s) S (sI/s o)2 and therefore
C(so) + MJ F ) dt <0(so) + -ylog- < Cos' 2 (46)so t3S2 So
for all sufficiently large s. Here Co is a positive constant which does not depend on s.
Next, we estimate F (s) from below. Without loss of generality, we make two simplifying
assumptions.
First, we recall that F is, in fact, an integral over!1, where I is the union of intervals [ai, Oil,i = 1, 2, - - - , with a1 < $ < a2 < $2< - - -. The length of each interval [a;, ;] grows beyond
bounds as i -+ co, while the length of each gap ($;, a;+1) approaches either 2 or 2v. Since F is
the integral of a positive function, we obtain a lower bound for F by assuming that the length of
each gap tends to the same constant p, where p is the larger of 2 and 2v.
Second, since we are interested in the asymptotic behavior of 0, we may assume that the
inequalities
ai+1 - 3 2p, i; (8p/c)i (47)
are satisfied for all i. If necessary, we absorb the contributions from those (finitely many) inter-
vals where the inequalities are not satisfied in the multiplicative constants and renumber the
remaining intervals.
Let s be any point in [a"+ 1, $+1i]. Then
Jf(t)dt = dt+ -- dt =2log,at at a -.- - %%+
so
F(s)= [ 1. -.- ].s 2
We write the ratio in the form
-3-... s _ s " +a;+1 - i) -
a1 .' ama"+1 a1 i=i i
It follows from (47) that (a;i, - i;)/$; E/(4i) for all i, whence we conclude, first, that
((x;1 - $;)/ ; < 1 for all i and, second, that the infinite series j((ai+,1 - $;)/i)2 converges.i=1
Then it follows from [Knopp 1956, Chap. VII, Theorem 10] that the product
H(1 + (a;4l - i)/[i) is asymptotically equivalent with the sum (ai+, - i)/aI as n -4 CO.i=1 i=1
161
There exist therefore positive constants C 1 and C2, such that
, ai+1 -~Fri " i+1 - P + ~tN
Ciexp4 ] i 1 + S C2exP ( a +-i=1 Pi i=1 +- Fis
for all sufficiently large n. Hence,
[ + a 1]C 2 exp [E- -] SC2 exp l -$i=4 i-1 iii
for all sufficiently large n.
Next, we recall the definition of Euler's constant Y,
Y = limr - -logn1;n-(a =1 1 J
cf. [Abramowitz and Stegun 1964, Section 6.1]. The definition implies that there exists a con-
stant C3 such that
exp - C3 ni=1 1
for all sufficiently large n. Therefore,
" 1+ 0+1 - Pi F/( 1+ ct5-d C4nL/i=1 P J C
for some constant C4. If s is in the interval [a,+1, $P+1], as assumed, then s > $! (8p/E)n, so
n S (E/(8p))s. There exists therefore a constant C5 such that
n u+ i+1 - Pi c/4[ 1+ 31C~
for all sufficiently large s. H':nce, there exists a constant C6 such that
r 1-101 '... P .S _ s * + i+1 - Pi -1gI(1 + ) 2 C6 s'' 4
al 'nn+ al i=1 i
for all sufficiently large s. Consequently,
A(s) Cs2 - (48)
for all sufficiently large s. This is the desired lower bound for F (s).
Combining the definition (41) of 0 with the estimates (46) and (48), we find that there
exists a constant C such that
0(s) C,-2+e
for all sufficiently large s. This proves the assertion of the lemma. 0
162
2.10 Final Contradiction
We now have all the ingredients necessary to complete the proof of the theorem.
We recall our basic supposition that the differential equation (7) has a nontrivial oscillatory
solution x. This solution gives rise to a nontrivial oscillatory solution z of (14) via the transfor-
mations (9), (11), and (13). Let so, s1, - - - be the zeros of z. If z is oscillatory, this sequence
continues indefinitely. Let the functional $ be defined in terms of z by (20).
Consider the interval (s;, s;+1) between two consecutive zeros of z. Assume that z is posi-
tive between si and s;+1, so sm(i) is the point in (s;, s;+1) where z achieves its maximum. Accord-
ing to Lemma 6, the value of az2( approaches -(1 + 1/y) as i -+ co. Let s(i) be the smallest
value of s in [s, sm()] where a(s)z211(s) = -(1 + 1/) and, similarly, let s(;) be the greatest8
value of s in [sm(i), s;+1] where a (s)z21(s) = -(1 + 1/y). In the remainder of the proof we write 1
and n, instead of 1(i) and n (i).
Integrating the expression (21) for $' from s1() to s.(;), ignoring the term involving the
derivative of a (which is nonpositive), we obtain the following estimate for $(s):
$(s) $(st) - Ja(z'(s))2 ds + Ja,(z(s))2ds . (49)SR s, ss
Let E > 0 be given. According to Lemma 17, there exists a positive constant C 1 , which depends
on E but not on i, such that
$(s)<_ C 1 s;2+c (50)
for all sufficiently large i. We assume that i has been chosen sufficiently large for (50) to hold.
Let Z be the solution of the initial value problem (25), symmetrically extended to negative
values of the argument. According to Lemma 11, there exists a positive constant p. and positive
constants C2 and C3 such that
s~
JaL(z(s))2ds 2 (Z(s))2ds = C2 , (51)Si - L
and
s,,
JaL(z (s))2 ds 2 2 J (Z(s))2ds = C3 . (52)S, -p
Combining the estimates (50), (51), and (52) with (49), we obtain the i equality
163
Ci C 2 C3$sn) S - - +C3.(53)
S1 Sn S
Clearly, as i increases, the middle term in the right member of (53) will dominate the two other
terms, so eventually the expression in the right member, and therefore 4) (s), will be negative. We
may assume that this is indeed the case on the interval under consideration; if necessary, we
increase i further.
According to Lemma 14, the functional $ is decreasing on the interval (s,s;+), so once $ is
negative at s, it remains negative on the entire interval. In particular, $ will be negative at si,+I.
But now we have a contradiction, as the definition of $ is such that $ is positive at every zero of z.
Thus, the supposition that there is an infinite sequence of zeros of z is ruled out. In other words,
if z is a nontrivial solution of (14), it must be non-oscillatory. This completes the proof of the
theorem. 0
Acknowledgments
We thank Professor F. V. Atkinson (University of Toronto; visiting Senior Mathematician
at Argonne National Laboratory) for his interest in this work and for many stimulating discus-
sions in the course of the investigation. We also thank Professor Atkinson for the hospitality
extended to us on several occasions at the University of Toronto.
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R. H. Fowler 1931. "Further studies of Emden's and similar differential equations," Quart. J.Math. (Oxford) 2, 259-288.
B. Gidas, Wei-Ming Ni, and L. Nirenberg 1979. "Symmetry and related properties via the max-imum principle," Comm. Math. Phys. 68, 209-243.
Z. Nehari 1969. "A nonlinear oscillation problem," J. Diff. Eq. 5, 452-460.
Z. Nehari 1974. Mathematical Reviews 48, no. 11661.
K. Knopp 1956. Theory and Application of Infinite Series, Dover, New York.
S. I. Pohozaev 1965. "Eigenfunctions of the equation Au + ?f (u) = 0," Dokl. Akad. Nauk.SSSR 165, 36-39; English translation Sov. Math. Dokl. 6, 1408-1411.
165
Appendix: A Generalized Sturm Comparison Theorem
In this appendix we state and prove a generalized Sturm comparison theorem that is used at
several places in the proof of Theorem 1.
THEOREM 1. Let u and v satisfy the initial value problems
u''+p (s)u'+ q(s)u = 0, s > a ; u(a) = a ; (A.1)
and
v"+P(s)v'+Q(s)v=0, s>a; v(a)= a ; (A.2)
respectively, where a z 0. Suppose that u, u', v, and v' are (strictly) positive on some interval
(a, b) of positive length. If
p(s)<-P(s), q(s) Q(s), s E [a, b], (A.3)
then the inequality
u'(a) ? v'(a) (A.4)
implies
u (s) 2V (s),'(s) '(s),u'(s) v'(s), s E [a, b] . (A.5)u(s) v(s)
Proof. We introduce the functions r = u'/u and R = v'Iv. Both r and R are (strictly) positive on
(a, b). Since u and v satisfy the equations (A.1) and (A.2), r and R are solutions of the Riccati
equations
r'= -(r2 +pr +q), s > a ; (A.6)
and
R'=- (R2 +PR+Q), s>a; (A.7)
respectively. Because of (A.3), the equation (A.7) yields the following differential inequality for
R:
R'S-(R2+pR+q), s>a . (A.8)
If (A.4) holds, then R (a) 5 r (a), so it follows from the theory of differential inequalities that
R (s) 5 r(s), s E [a, b]. (A.9)
Hence,
u'(s)/u(s) 2 v'(s)/v(s), s E [a, b]. (A.10)
This proves the third inequality in (A.5).
166
Because of the positivity of u and v and their derivatives on (a, b), we can integrate both
sides of the inequality (A. 10) over any interval [a, s ] with s e (a, b). Since u and v assume the
same initial value a at a, we obtain the inequality
log u(s) v(s) E [a, b].a a
(A.11)
Hence,
u(s) v(s), s e [a,b].
This proves the first inequality in (A.5).
Finally, the second inequality in (A.5) follows from (A.9) and (A.12),
u'(s) z v'(s) (S) Z v(s), s E [a, b] .v (s)
This completes the proof of the theorem. 0
(A.12)
(A.13)
167
CONCAVITY AND MONOTONICITY PROPERTIESOF SOLUTIONS OF EMDEN-FOWLER EQUATIONS
Hans G. Kaper and Man Kam KwongMathematics and Computer Science Division
Argonne National LaboratoryArgonne, IL 60439
Abstract
This article is concerned with concavity and monotonicity properties of solutionsof boundary value problems of the type
u''= p (x)u'+f (u), x > 0; u(0)=y, u'(0) = 0.
The coefficient p may be singular near the origin. It is assumed that there exists auo such that f (u) < 0 if u > uo and f (uo) = 0. Particular cases aref(u)= l-e"; f(u) = u /(q-1) - I u I Qsign(u), where I < q < oo; andf (u) = -u /(1-q)+ Iu I sign(u), where 0 < q < 1. The results generalize andextend those presented in a 1986 preprint by Friedman, Friedman, and McLeodentitled "Concavity of solutions of nonlinear ordinary differential equations."
1. Introduction
In this article we are interested in concavity properties of solutions u of boundary value
problems of the following type:
u"=p(x)u'+f(u), x>0; u(0)=y, u'(0)=0. (1)
The coefficient p may be singular near the origin. The (nonlinear) function f is defined on
(-oo, oo), and there exists a uo such that
f(u)<0 if u>uo, f(uo)=0. (2)
We assume throughout that p E C 2((0,co)) and!f E C 2 ((-oo, oo)); other conditions are imposed as
needed. The differential equation in (1) belongs to the class of Emden-Fowler equations.
Recently, Friedman, Friedman, and McLeod [1986] considered the concavity properties of
solutions u of a special case of (1), namely,
u f-- i u'+f(u), x>0; u(0)='y, u'(0)=0. (3)
Considering first the case n = 1, these authors showed that, for a broad class of functions f, any
solution u is concave as long as it is positive. The class of admissible functions included the fol-
lowing special cases:
f (u) = I -e"; (4)
169
f (u)= u - lu I qsign(u), 1 <q <00; (5)q-l
f (u) =- u + Iu I sign(u), 0<q< 1 . (6)I -q
If n > 1, similar results are more difficult to come by. Friedman, Friedman, and McLeod estab-
lished several concavity properties when f is one of the functions (4), (5), or (6), but these proper-
ties hold only under certain restrictions on n.
The special case (6) is particularly interesting, as it arises in the study of the extinction of
solutions of the nonlinear diffusion problem
uXE- Au = -u9, xE BR, t >0; (7)
u(x,t)=0, XEaBR, t>0; (8)
u (X, 0) = $(I X I), x e BR . (9)
Here, BR is the open ball of radius R centered at the origin in R" and $ is a positive function of
the radial variable r = I x 1. The solution of this nonlinear diffusion problem is radially sym-
metric; cf. [Gidas, Ni, and Nirenberg 1979]. It was shown by Friedman and Herrero [1987] that
the origin is the only extinction point of u if $ is concave on (0,R), decreasing from a positive
value at 0 to the value 0 at R. These authors also found the asymptotic behavior of u (0,t) as
t -+ T, T being the time of first extinction. Friedman, Friedman, and McLeod [1986] extended
this result and established the asymptotic behavior of u (r,t) as (r, t) -+ (0, T) along arcs
r = 0 ((T - t)"2). They based their proof on the concavity result for (3), where f is given by (6).
The key observation was that the function w, defined in terms of u by the expression
w (p,s) = (T - )~1- )u (r, t) ,
where p = r (T - t)~12 and s = log(T - t), satisfies a parabolic equation whose stationary form is
of the type (3), namely,
wPP = 2 pn- I-Wq
An immediate consequence of the concavity result for (3) is that the only nonnegative C2 -
solution on [0,oo) that satisfies the condition w'(0) = 0 is the constant solution
w (p) = (1 - q)~"(1~ ). An alternative proof of this fact was given by Peletier and Troy [1986].
In this article we show that some of the results of Friedman, Friedman, and McLeod [1986]
can be established and even improved if one uses techniques from the classical Sturm oscillation
theory for linear ordinary differential equations. Since the oscillation theorems as used here are
not in the form commonly found in the textbooks, we have listed them, with an indication of their
proofs, in the Appendix.
170
In Section 2, we establish some concavity results for (1), assuming that p is continuouslydifferentiable on [0,oo). When applied to the special case p (x) = x/2, our results extend those of
Friedman, Friedman, and McLeod [1986] for the case n = 1.
In Section 3, we no longer assume that p is regular at the origin; p may even become very
negative there. A typical case is p (x) = x/2 - ax-Q, where a> 0 and 0 <q q 1. We establish a
monotonicity property for (1) and prove that, under specific conditions on p and f, every bounded
solution of (1) must vanish somewhere on (0,oo). Our results generalize those of Friedman, Fried-
man, and McLeod [1986] and Peletier and Troy [1986].
2. Regular Coefficients
Throughout this section we consider only functions p e C2((0,oo)) that are continuously
differentiable on [0,oo) and satisfy the conditions
p(O) 20; p"(x) 20, x>0. (10)
A standing assumption is that f e C 2(-oo, oo) and that there exists a uo > 0 such that (2) holds.
In this section we assume, in addition, that f is concave,
f "(u) 5 0 . (11)
THEOREM 1. Letp and f satisfy the conditions (10) and (11). Suppose there exists a constant a
(which may be negative) and a constant X e (0,1) such that
p'(x) a, x > 0 ; (12)
and
u [f'(u)+ 2a] z Af (u) . (13)
If y > u0 , then the solution u of (1) is concave on any interval [0,b) where u (x) > 0.
Proof. We suppose that u has its first zero at b. Then u > 0 on [0,b). We need to show that u"
satisfies the inequality u''< 0 on [0,b).
We first compare the solution u of (1) on [0,b) with the solution y of
y''= p (x)y'+ [f '(u)+ 2a]y, x > 0 ; y (0) > 0 , y'(0) =--0 . (14)
Because of (13), y oscillates less than u (cf. Theorem A.3), soy > 0 on [0,b).
The function w = - u'satisfies the differential equation
w =p(x)w'+ [f'(u)+ 2p'(x)]w -p"(x)u'-f"(u)u'2 . (15)
Furthermore, w (0) = -f (y) and w'(0) = -p (0)f (y). Because y > u0 , f (y) is negative, so w (0) is
positive; furthermore, p (0) 0, so w'(0) is nonnegative. We need to show that w is positive on
171
[0,b).
Suppose that w is not everywhere positive on [0,b). Because w(0) is positive, there must
then be a point c E (0,b), such that w(x)> 0 for all X E [0,c) and w(c) = 0. Since w = - u', it
follows that u' is decreasing on (0,c). Because u'(0) = 0, it must be the case that u'< 0 on (0,c).
We rewrite the differential equation (15) in the form
w' = p (x)w'+ [f'(u) +2p'(x)+ P (x)]w , (16)
where
P(x) =--f{p'(x)u'+f '(u)u' 2 }. (17)w
Each of the two terms inside the braces is less than or equal to 0 on (0,c), so P _0 on (0,c). The
coefficient of w in (16) is therefore at least equal to the coefficient of y in (14). Hence, w oscil-
lates less than y on [0,c); cf. Theorem A.2. Since y (c)> 0, it must therefore be the case that
w (c) > 0. But w (c) = 0 by assumption, so the hypothesis that w is not everywhere positive on
(0,b) leads to a contradiction 0
When applied to the special case p (x) = x/2, Theorem 1 gives an improvement of the
corresponding result in [Friedman, Friedman, and McLeod 1986]. Theorem 1 corresponds to the
case 0 2 1 of [Friedman, Friedman, and McLeod 1986, Theorem 2.1]. A comparison shows that
the condition (2.4) of [Friedman, Friedman, and McLeod 1986] (i.e., 1 +f'(u) < 0 if u > uo,
1 +f'(u) > 0 if u <uo), which is necessary for the proof of [Friedman, Friedman, and McLeod
1986, Theorem 2.1], is not needed.
THEOREM 2. Let p and f satisfy the conditions (10) and (11). Assume that f can be written in the
form
f (u) = k1 + k2u - $(u) , (18)
where k 1 and k2 are constants such that k1 + k 2 u > 0 when u > uo. Suppose there exist a con-
stant a 20 and a constant X E (0,1) such that
p'(x)2a, x20; (19)
and
[k l +k 2 u ]$'(u)[2a+ k 2]4(u) . (20)
If y> u~, then the solution u of (1) is concave on any interval [0,b) where $(u (x)) > 0.
Proof Let the function v be defined by the expression v (x) = $(u (x)). We suppose that v has its
first zero at b. Then v > 0 on [0,b).
Because f is twice continuously differentiable, the same is true fort. We have v'= 4'(u)u'and v''= $'(u)u"+ $'(u)u'2. If u satisfies (1), then
172
V''= P (x)v'+ .$'(u)(u)v + $'(u)u2.(21)
Furthermore, v (0) = $(y) and v'(0) = 0.
As before, we need to show that w = -- u' is such that w' is positive on [0,b). We recallthat w satisfies the differential equation (15). When f is given by (18), this equation becomes
"' = p (x )w'+ [f '(u) + 2p'(x )]w - p'(x )u'+ $'(u)u'2. (22)
Furthermore, w (0) = -f (y), which is positive, and w'(0) = -p (0)f (y), which is nonnegative.
Let z = v - w. Subtracting (22) from (21), we obtain
z= p (x)z'+ [f '(u)+ 2p(x)]z - f '(u)+2p(x)- (u) v +p '(x)u'. (23)
At x = 0, we have z (0) = k1 + k2y, which is positive, and z'(0) = p (0)f (y), which is nonpositive.
By continuity, z is positive in a right neighborhood of x = 0.
Suppose that w is not everywhere positive on [0,b). Because w(0) > 0, there must then be apoint c e (0,b), such that w (x) > 0 for x E (0,c) and w (c) = 0. We compare the solution w of
(22), which we write again in the form (16),
w'' =p(x)w'+ [f'(u)+2p'(x)+P(x)]w , (24)
where
P(x)= ".$'(u)u'2 -p"'(x)u'} , (25)
with the solution z of (23), which we write in the form
z''= p (x)z'+ [f'(u)+ 2p"(x) - Q (x)]z , (26)
where
Q (X) = [f'(u)+ 2p'(x) - '(u)u)v -1 p"(x)u'i. (27)
The first term inside the braces in (25) is nonnegative, the second term nonpositive, so P is non-
negative on [0,c ]. Now, consider the expression (27) on [0,c ]. The quantity inside the square
brackets is nonnegative, v is positive, and p''u' is nonpositive, so Q is also nonnegative on [0,c ].
Therefore the coefficient of z in (26) is less than or at most equal to that of w in (24), so z oscil-lates more than w (cf. Theorem A.2). This result implies that z must have at least one zero in
(0,c). Let d be the first zero, so z(x) >0 for all x E [0,d ] and z (d) = 0.
According to Lemma A.1 we also have z'(d) < 0, so z < 0 in a right neighborhood of d. We
claim that z cannot have a second zero in (d,c).
173
The claim is most easily verified by contradiction. Assune that z has a second zero at
T E (d, c). Then z < 0, on (d, T). We compare z with u', which satisfies the equation
(u)''=p (x)(u)'+ [f'(u) +P(x)](u),(28)
with u'(0) = 0 and (u')'(0) = f (y), which is negative. The expression inside the braces in (27) is
still nonnegative, but now z is negative, so Q is nonpositive on (d, T). Therefore,
f'(u) + 2p'(x) - Q(x) f'(u) + 2p'(x) f'(u) +p'(x)
on (d, T), so u' oscillates more than z and u' must have at least one zero in (d,r). But w is cer-
tainly positive on [0,t); hence, u' is decreasing there, which implies that u'< 0 on (0,T), so we
have a contradiction.
The claim implies that z <0 everywhere on (d,c), so z(c)< 0. Hence,
w(c) v (c) = $(u(c)) > 0. But w(c) = 0 by assumption. Thus, the hypothesis that w is not
everywhere positive on (0,b) leads to a contradiction. 0
The conditions of the theorem are satisfied for the special case (4) with $(u) = e", and for
the special case (5) with (u) = Iu Iq sign(u).
When applied to the special case p (x) = x/2, Theorem 2 reduces to [Friedman, Friedman,
and McLeod 1986, Theorem 2.3] if f is given by (4). On the other hand, although both Theorem
2 and the case 0 < 1 of [Friedman, Friedman, and McLeod 1986, Theorem 2.1] cover the case (5),
they do not seem to contain each other.
3. Singular Coefficients
In this section we consider functions p E C2 ((0,oo)) which may be singular at the origin.
The severity of the allowable singularity will be somewhat limited, but the case1
p (x) =2-x - Ox-, where $3> 0 and 0 < q 5 1, is included (cf. [Friedman and Herrero 1987]).
No assumption is made about the sign of p'. The standing assumption about f is again that f is
twice continuously differentiable and that there exists a uo > 0 such that (2) holds.
LEMMA 1. Let p satisfy the condition
lim(1/p)'(x) < 0. (29)x-O
(The limit may take the value -co.) Let y satisfy the inequality y > uo. If u is a solution of (1)
and v (x) = u'(x)/x, then
limv (x) < 0 , limv'(x) <-0 . (30)X -0x-+0
The limits may take the value - o.
174
Proof. According to l'Hospital's rule,
limv(x) = limu'(x). (31)x-+0 x -+0
We use the identity
lisp (x)u'(x) = {lim(1/p)'(x))-' limu"(x), (32)X-%( x+0x-+0
which follows from l'Hospital's rule, to derive the following identity from (1):
limu'(x) = [1 - (im(1/p)'(x)}-'f]-'f('Y) -(33)
The expression inside the brackets is positive. Because y> uo, f (y) is negative, so
limx_,0u'(x) < 0.
From the identity v'(x) = u''(x)/x - u'(x)/x2 we obtain, using l'Hospital's rule,
limv'(x) = -limu"'(x) . (34)x-+0 2 x-0
Differentiating (1) once, we obtain the equation
u"" = p (x)u"+ p'(x)u'+f'(u)u', (35)
which is satisfied for all x > 0. Because p'= - (l/p)'p2, we can rewrite this equation in the fol-
lowing form:
u"'= p (x)u'-(1/p)'(x)p2(x)u'+f '(u)u', x>0. (36)
We observe that limafX'(u (x))u'(x) = 0, because f is continuously differentiable everywhere
and u' tends to 0. Again using 1'Hospital's rule, we find
limu'(x) = -[1 - {lim(1/p)'(x))-1-'lim(1/p)'(x)1imp 2(x)u(x) . (37)x-.0 x-0 x-+0 x
The expression inside the square brackets is positive. The limit of (1/p)'is negative. As we have
seen above, u''is negative near 0, so u'is (strictly) decreasing there. Because u'(0) = 0, it follows
that u'is negative near 0. The same is then true for p2u', so the limit of p2 u' is certainly nonpo-
sitive. It follows that lim,_,0u'''(x) 0. 0
The following theorem is the analog of Theorem 1.
THEOREM 3. Let p satisfy the conditions (10) and (29). Assume that there exist a constant a,
which may be negative, and a constant X e (0,1), such that
p'(x)+p(x)/x ? 2a, x > 0 , (38)
and
u [f '(u)+ 2a] X:f (u) . (39)
175
If y> uo, then the solution u of (1) decreases monotonically on any interval [0,b) where
u(x)> 0.
Proof. Suppose u has its first zero at b. Then u > 0 on [0,b). We need to show that u' satisfies
the inequality u' <0 on (0,b).
Let v be defined by the expression v (x) = u'(x)/x, x > 0. Differentiating the identityu'(x) = xv (x) with respect to x and using the differential equation (1), we obtain
xv' = (1 + xp (x))v +f (u) .
Differentiating once more, we obtain the following differential equation for v:
V = p (x)-x] v'+ f'(u)+p'(x)+x(']v . (40)
Furthermore, lim,_Ov (x) < 0 and lim,,ov'(x) 0, according to Lemma 1. We need to show that
v is negative on (0,b).
Suppose that v is not everywhere negative on (0,b). Then there must be a point c e (0,b),
such that v (x) < 0 for all x E (0,c) and v(c) = 0. Also, because v is negative and nonincreasing
near 0, there must be a first point in [0,c) where v'vanishes. Let this point be denoted by a.
We compare the solution u of (1) on [a,c ] with the solution y of
y"= p(x)- -y'JY+ y , x>a; y(a)=v(a), y(a)=0. (41)x u
Because v is negative on [a,c), u' is negative there. Furthermore, the coefficient of y' in (41) is
(strictly) less than the coefficient of u' in (1). Hence, y oscillates less than u on [a,c ] (cf.
Theorem A.4). Therefore y does not vanish on [a,c ].
Furthermore, the solution v of (40) oscillates less than y on [a,c ], so v does not vanish on
[a, c ] either. But now we have a contradiction, since we have assumed that v (c) = 0. It must
therefore be the case that v is negative on (0,b). oIf we put one additional condition on the coefficient p, we can prove the following theorem,
which asserts that no solution of (1) can be positive everywhere on [0,oo) if the initial value y
exceeds uo.
THEOREM 4. Let the conditions of Theorem 3 be satisfied. In addition, let p satisfy the condition
JexP[JP(Y)dY dx=oo. (42)
If y > u0 , then every solution u of (1) must vanish at some point b e (0,oo).
176
Proof. Suppose that u is the solution of (1) and that u is strictly positive everywhere on [0,oo).
Then it follows from Theorem 3 that u' <0 on [0,oo), so lim ..,u (x) = 0.
Defining v as before by the identity v(x) = u'(x)/x, we have v <0 on [0,o). Also,
lim,-ov(x) <0 and lim,ov'(x) -0. Again, let a be the first point in [0,oo) where v' vanishes,
and let y be the solution of (41). We take an arbitrary point X e (a, oo) and let U be a solution of
(1) for which U(X) > 0 and U(X)/U(X) = y'(X)/y(X). Since U and u are two linearly indepen-
dent solutions of the same equation, where p, the coefficient of the derivative, satisfies (42), and u
tends to0 at infinity, it must be the case that U(x) tends to +00 or - as x -+400 (cf. Lemma A.2).
We claim that U(x) tends to +oo as x -+cc.
A comparison of y and u beyond a shows that y'(X)/y (X) > u'(X)/u (X). Hence,U'(X)/U(X)> u'(X)/u (X). Then it follows from a subsequent comparison of U and u beyond X
that U'(x)/U (x) > u(x)/u (x) for all x z X, at least as long as U (x) is positive. (Recall that u (x)
is always positive, by assumption.) Suppose that there is a point c > X where U vanishes. Then
U > 0 on [X,c), so for all x E [X,c) we would have U(x) > Eu(x), where e = U(X)/u(X) is a
positive constant. In the limit, as x - c, we would find U(c) Ecu (c), which is impossible, as
U(c) = 0and u(c) > 0. Hence, U does not vanish beyond X. But U(X) > 0, so it can only be the
case that limz,..U(x) = 00, as claimed.
Using the same arguments as above, we show that y is less oscillatory than U. Because y is
negative at a, it must be the case that lim, .. y (x) = -0o. Hence, v, which is even less oscillatory
than y and is also negative at a, has the same property. In other words, limza..u(x) = -00. Butnow we have a contradiction, since this result would imply that u (x) tends to -00 as x -4 0. It
must therefore be the case that u cannot be positive everywhere on [0,oo). This proves the asser-
tion of the theorem. 0
Theorem 4 does not say anything about solutions u of () that start with a boundary value y
less than or equal to uo. With some additional conditions on p and f, we are able to prove the fol-
lowing result.
THEOREM 5. Let p and f satisfy the conditions of Theorem 4. In addition, let there be a continu-
ously differentiable function g which satisfies the conditions
r*- dxg'(x) < 0, J = g = , (43)g (x)
such that
0 p(x) 5g(x), x>0. (44)
Furthermore, let 0 and u0 (u 0 > 0) be the only zeros off on [0,00), with f (u) > 0 for u E (0,u0 ).
If y e (0,u 0 ], then every bounded solution u of (1) must vanish at some point b E (0,00).
177
Proof. Let u be a bounded solution of (1) on [0,oo). Since u(0) is positive and u' is positive near
the origin, uis positive and increasing there. We claim that u must have a local maximum at
some point c e (0,oo).
We give a proof by contradiction. Suppose u is strictly increasing everywhere. Because u
is bounded, limn u(x) exists; we denote it by u(oo). Let F be the integral
Y.
F (u) = -Jf(s)ds , (45)0
and let yi be defined by the expression
q(x) = 2u2(x)+ F (u(x)) . (46)
Given that u satisfies (I), we conclude that i'(x) = p(x)u'2 (x) 0, so V is nondecreasing. We
have therefore F (u(oo)) = y(oo)> >-(0) = F(Y), which implies that either u(oo) e [O,y] or
u(oo) > u0 , because F is decreasing on (0,uo). The first possibility is ruled out, because u(oo) > yif u is strictly increasing. We claim that the second possibility is ruled out by (43) and (44).
We prove the claim by contradiction. Suppose that u (oo) > uo. Then f (u(x)) > 0 for all
sufficiently large x. Without loss of generality we may assume that g (x) > 0, so the second con-
dition in (43) implies that
i dx = - . (47)
o g(x)
We also know that
D(x)u'(x)Jdx ju'dx = u(oo)- y. (48)o g(x) 0
Therefore, upon division of both sides ocr(1) by g (x) and integration over x, we find that
u'(x) dx =io. " (49)o g (I
Hence, upon integration by parts,
Ju '(x)g'(x) dx = - .(50)o g2 (x)
But here we have a contradiction, as the product u'g' is nonnegative everywhere. The possibility
that u is strictly increasing and bounded is thus ruled out.
Suppose that u has a local maximum at c and that u(c) < us0 . Let d be the abscissa at the
local minimum of u immediately preceding c (d may be at the origin). Since yi is nondecreasing,
we have F(u(c)) = it(c) -I(d) = F(u(d)), which implies that either ui(c) E [0,s(d)] or
178
u (c)> u0 , because F is decreasing on (O,uo). The first possibility is ruled out by the very
definition of d: u(d) < u(c), so it must be the case that u(c) > uo.
Suppose that u has a local maximum at some point c E (0,oo). Then u(c) > uo and
u '(0) = 0. Applying Theorem 4 to u on the interval [c, oo), we conclude that u must vanish at
some point b e (c, oo). This proves the theorem. 0
Theorems 4 and 5 together generalize the results of [Friedman, Friedman, and McLeod
1986] and [Peletier and Troy 1987] for the equation (3), where f is given by (6).
References
A. Friedman, J. Friedman, and B. McLeod 1986. "Concavity of solutions of nonlinear ordinary
differential equations," Purdue University, Technical Report 47.
A. Friedman and M. Herrero 1987. "Extinction properties of semilinear heat equations with
strong absorption," J. Math. Anal. Applic. (to be published).
B. Gidas, Wei-Ming Ni, and L. Nirenberg 1979. "Symmetry and related properties via the max-
imum principle," Comm. Math. Phys. 68, 209-243.
S. Leela and V. Lakshmikantham 1969. Differential and Integral Inequalities, Vol. 1, Academic
Press, New York.
L. A. Peletier and W. C. Troy 1986. "On nonexistence of similarity solutions," preprint.
W. Walter 1970. Differential and Integral Inequalities, Springer-Verlag, New York.
179
Appendix: Auxiliary Results
In this appendix we discuss the Sturm-Picone Comparison Theorem and several of its
consequences. We also give two lemmas, which are used in the body of the paper.
A common statement of the classical Sturm-Picone Comparison Theorem reads as follows:
Let f, F, g, and G be given functions on (a,b). Assume that f and F are strictly positive, and that
f ', F', g and G are continuous on (a,b). Suppose U and V are solutions on (a,b) of the equations
(F (x)U')'+ g (x)U = 0 , (A.1)
and
(f (x)V')'+ G (x)V = 0, (A.2)
respectively. Let
f(x) F(x), g(x) G(x) x E (a,b). (A.3)
If x 1 and x2 are consecutive zeros of U on (a,b), then V must vanish at some point of (x 1,x 2).
The continuity requirements can be relaxed; indeed, it suffices for f and F to be absolutely
continuous and g and G to be integrable.
The Sturm-Picone Comparison Theorem has a stronger version which we state as a theorem.
THEOREM A.I. Let f, F, g, and G be given functions on (a,b). Assume that f (x) > 0 and
F(x) > 0, and that f', F', g, and G are continuous on (a,b). Suppose U and V are solutions on
(a,b) of the equations (A.1) and (A.2), respectively. If the inequalities (A.3) are satisfied and V
does not vanish on (a,b), then the inequality
F(a)U'(a)_ f(a)V'a)(A.4)
U(a) V(a)
implies that
F (x)U'(x) _f (x)V'(x)x) V ,(x) x e (a,b) . (A.5)
U (x x
Proof. We set r(x) = -F(x)U'(x)/U(x) and R(x) = -f(x)V(x)/V(x) on [a,b]. The functions r
and R satisfy the Riccati equations
r2r'= g (x)+ (A.6)
F (x )
and
R 2R'= G (x)+ . (A.7)
f (x)
The inequality (A.3) implies that R also satisfies the differential inequality
180
R ' g (x) + . (A.8)F (x)
Moreover, according to (A.4), R (a) z r (a). It is a well-known result from the theory of
differential inequalities (see, for example, [Leela and Lakshmikantham 1969] or [Walter 1970])
that the inequality R (x) z r (x) then holds for all x e (a, b). 0
The Sturm-Picone Comparison Theorem is a statement about the relative "degree of oscilla-
tion" of two functions that satisfy second-order linear differential equations with comparable
coefficients. We say that U oscillates less than V on [a,b] if the conclusion of the theorem holds.
A particular consequence of the theorem is that, if f (x) = F(x) and g (x) = G (x), so (A.1)
and (A.2) are identical, and U and V are two distinct solutions of the equation on (a,b) which
satisfy the inequality U'(a)/U(a) V'(a)/V(a), then U oscillates less than Von [a,b].
The next two theorems are stated in a form suitable for the equations studied in this article.
THEOREM A.2. Suppose U and V satisfy the equations
U''= p (x)U'+ Q (x)U , x e (a,b), (A.9)
V"= p(x)V'+q(x)V , x e (a,b), (A.10)
and V does not vanish in (a,b). If
Q (x) >-q (x) , x E (a,b), (A.11)
and U'(a)/U (a) V>'(a)/V (a), then U oscillates less than Von [a,b ].
Proof. Using the integrating factor exp(- Jp (x)dx), we rewrite the equations (A.9) and (A.10) in
the form (A.1) and (A.2), respectively, where
f (x) = F(x) = exp(- Jp(x)dx),
g (x) = - Q (x)exp(- Ip (x)dx),
and
G(x)=-q(x)exp(-Jp(x)dx).
The assertion of the theorem follows from Theorem A. 1. 0
THEOREM A.3. Suppose U and V satisfy the equations (A.9) and (A.JO), respectively. Suppose V
does not vanish in (a,b ). If there exists a constant X e (0,1) such that
Q (x' X-Aq(x), x E (a,b), (A.12)
and
181
U'(a) V'(a),(A.13)U(a) V (a)
then U oscillates less than V on [a,b ].
Proof The comparison is made in two steps. First, we compare U with a solution W of the equa-
tion
W''= p (x)W'+ Aq(x)W , x e (a,b) , (A.14)
which satisfies the condition
U'(a)= W(a)) W ) (A.15)U(a) W(a)
Given the inequality (A.12), we conclude from Theorem A.2 that U oscillates less than W on
[a,b].
Next, we compare W with V. Using the integrating factor exp(- J p (x)dx), we rewrite the
equations (A.17) and (A.10) in the form
(F(x)W)'+ g (x)W = 0 , x e (a,b),
and
(f (x)V')'+ G (x)V=0 , x E (a,b) ,
respectively, where
f(x)=exp - p(x)dx),
F (x) = (1/A)exp -Jp (x)dx ,
g(x)=G(x)=-q(x)exp[-p(x)dx].
Because . e (0,1), we can apply Theorem A.1 and conclude that W oscillates less than V on
[a,b].
Combining the conclusions of the two steps, we find that U oscillates less than V on [a,b ].0
The next theorem deals with the case where the coefficients of the first derivatives are
different.
THEOREM A.4. Suppose U and V satisfy the equations
U"= p (x)U'+ q (x)U , x e (a,b), (A.16)
182
V"= P(x)V'+q(x)V , x e (a,b) .
Suppose,furthermore, V does not vanish on (a,b) and
V'(x):50, x e (a,b) . (A.18)V (x)
Ifp(x) 5P(x)forall x e (a,b) and
U(a) 2V(a),(A. 19)
U(a) V(a)
then U oscillates less than V on [la,b ].
Proof. Define r = - U'/U and R = - V'/V. These functions satisfy the Riccati equations
r'=pr -q+r2 , (A.20)
R'=PR-q+R2 2pR-q+R2 . (A.21)
(The inequality in (A.21) holds because P Zp and R 0.) Moreover, R (a) z r(a), by (A.19).
The theory of differential inequalities then gives R (x) Z r (x) for x E [a,b ], from which the asser-
tion of the theorem follows. 0
Finally, we give two lemmas that are needed in the body of this paper.
LEMMA A.1. Let f and g be continuous functions on [a,b ], Let z satisfy the second-order
differential inequality
z' f (x)z'+ g (x)z , x e (a,b) . (A.22)
If z has only one zero at c E (a,b), thenz'(c) *.0.
Proof. We assume that z (x) > 0 in (c,b). It suffices to consider the special case f (x) * 0; the gen-
eral case can always be reduced to this case by means of a change of the independent variable.
Equation (A.22) implies that there exists a function P, which is nonnegative on (a,b), such
that
z"-g(x)z = - P(x), x er(a,b) . (A.23)
Let z1 and z2 be independent solutions of the homogeneous equation associated with (A.23),
satisfying the boundary conditions
z 1 (c)=0, z' 1 (c)=1;
z2(c) = 1 , z'2(c) = 0.
Suppose that the assertion of the lemma is false and that z'(c) = 0. According to the "variation
of constants" formula,
183
( A.17 )
x
z(x)= Jh(s,x)P(s)ds, (A.24)C
where
h (s,x) = z I(s)z 2(x) - z I(x)z:(s).The partial derivative hx, being the negative of the Wronskian of zI and z 2, is equal to -1 at the
origin. By continuity, h= is therefore negative in a neighborhood of (0,0). Furthermore,
h (s,s) = 0; hence, h (s,x) < 0 for x > s and x sufficiently close to s. It then follows that the
integrand in (A.24) is negative for x E (c,c + e) for some sufficiently small e. This implies that
z (x) 0 for all x e (c,c + e). But now we have a contradiction, as we assumed z to be positive in
(c,b). 0
LEMMA A.2. Let f be such that
7exp[I f (y)dyjdx = . (A.25)
Suppose z satisfies the differential equation
z"= f (x)z'+g(x)z, x > a . (A.26)
If z has only a finite number of zeros and lim, z (x) = 0, then any solution of (A.26) that is
independent of z must approach either + « or -oo as x -+4o.
Proof. Let y be a solution of (A.26) which is independent of z. Define $(x) = exp( f (x)dx). We
may assume without loss of generality that y'(x)z (x) - y (x)z'(x) $f(x). (Use Abel's identity.)
We may also assume that z(x)> 0 for x > a, so that tan~1(y /z) is defined. Now,
(tany/z)'= 2 (A.27)y 2+ z
If (A.25) holds and both y (x) and z (x) remain bounded as x -+ c, the expression in the left
member is integrable, while that in the right member is not, and we have a contradiction, so it
must be the case that z is unbounded as x -+ o. 0
184
UNIQUENESS FOR A CLASS OF NON-LINEAR INITIAL VALUE PROBLEMS
Hans G. Kaper and Man Kam KwongMathematics and Computer Science Division
Argonne National LaboratoryArgonne, IL 60439
Abstract
This article is concerned with non-negative solutions of the non-linear initialvalue problem x'= p (t)xa + q(t), t > 0; x(0) = 0. Here, a is a constant,a E (0, 1); p and q are real-valued functions that are integrable and not identicallyzero near 0, with p (t) 0. The function q may assume positive, as well as nega-tive values. The following result is shown. Let q t denote the integral of q,
q1(t)= )fq (s) ds, t a G. Let x be a solution of the initial value problem which ispositive on (0, T). Then x is unique if (i) q 1 (t) 0 for all t E[0, T, and (ii)there exists a sequence (tj)jE N of points tj e(0, T) which converges to 0, suchthat q1(t) > 0 forj = 1, 2, - -".
1. Statement of the Theorem
In this article we are concerned with non-negative solutions of the non-linear initial value
problem
x '= p (t)x G+ q(t), t >0; x (0) =0. (1)
Here, a is a constant, a E(0, 1); p and q are real-valued functions that am integrable and not
identically zero near 0, with p (t) 0. The function q may take positive, as well as negative
values.
We use the symbols p 1 and q 1 to denote the integrals of p and q, respectively,
p1i(t) =JpP(s) ds, q1 (t) =Jq(s) ds, t -a0. (2)0 0
Thus, p 1 is positive near 0 and non-decreasing for all t > 0; q,1 may be increasing, as well as
decreasing.
We shall prove the following theorem.
THEOREM. Assume that there exists a solution x of (1) that is positive on (0, T) for some T > 0.
If q 1 satisfies (i) q1 (t) z 0 for all i 6r[0, T], and (ii) there exists a sequence (t)jE N of points
tj E (0, T) which converges to 0, such that yI (tj) > O for j = 1, 2, ."-" , then x is unique.
This result does not seem to fall under any of the uniqueness criteria given in the literature;cf. [Hartman 1964, Section 1.6], [Piccinini, Stampacchia, and Vidossich 1984, Section III.3.2],
185
[Bownds 1970], and [Bernfeld, Driver, and Lakshmikantham 1975/76]. Murakami [1966] con-
sidered a special case of (1), where a = 1/2 and both p and q are constant and equal to one; but, as
was shown by Bownds and Diaz [1973], his uniqueness result is an immediate consequence of a
theorem in [Bownds 1970]. The hypotheses underlyng the uniqueness criteria of [Bemfeld,
Driver, and Lakshmikantham 1975/76] cover (1) only for non-negative functions q; it is not clear
whether the technique can be modified to allow for functions q that change sign.
The motivation for this investigation stems from some recent work on the uniqueness of
non-negative solutions of semi-linear elliptic boundary value problems [Peletier and Serrin 1986].
A special case of the theorem (a = 1/2) is used to prove the uniqueness result in our forthcoming
article [Kaper and Kwong 1988].
2. Proof of the Theorem
We begin by establishing a lower bound for solutions of (1). Throughout this section, ine-
qualities (equalities) of the type x > y (x = y) involving two non-negative functions x and y are
understood to hold pointwise everywhere on the common domain of definition of x and y.
LEMMA. Let the conditions (i) ana (ii) of the theorem be satisfied. Then any non-negative solu-
tion x of (1) on [0, T ] satisfies the inequality
x ((1 - a) P1)1(~--.)(3)
Proof. Let K denote the integral expression
K(x)(t) = q 1(t) + fP (s)x (.) (ds, t z 0. (4)0
K is monotone, in the sense that K(x);K(y) if x zy k0. We use K to define a sequence of
functions (x).
x.+1 = K(x.), n =1, 2,.--..;x1 =q1 .(5)
Clearly, x1 10 on [0, T]. Furthermore, x2 z x 1 and, by induction, x+ 1 zx, for n = 1, 2, -- -.
Hence, the sequence (x,.) is monotonically increasing. Since it is also bounded from above, it
converges as n -4 co to an element x... The limit satisfies the integral equation x = K(x) on
[0, T ] and is therefore a non-negative solution of the initial value problem (1).
Let x be another nbn-negative solution of (1). Then x = K (x) z x 1 . Because K(x) = x and
K is monotone, it follows that x Zx. for n = 1, 2, - - -. Hence, r2x.., so x.. is the minimalnon-negative solution of (1). It suffices therefore to establish (3) for x...
186
Next, consider the "homogeneous" form of the initial value problem (1),
x'=p(t)xa, t > 0; x(0)=0; (6)
and the corresponding integral expression
L (x)(t) = Jp(s)xa(s) ds, t z 0. (7)0
Like K, L is monotone. We use L to define another sequence of functions (yJ, which shall con-
verge to a non-negative solution of (6). The choice of the initial function y is critical.
Let E be any of the points tj of the sequence mentioned in condition (ii) of the theorem. Let
p1,E be defined by
p 1 ,E(t) = 0, 0:5 t:5 E; p 1,E(t) = Jp(s) ds, t e. (8)
Because q 1(e) > 0 and p i(E) = 0, there certainly exists a r > e such that q 1 > p 1 ,E on (E, T).
Taking such a r, we define the initial element y 1 of the sequence {y,} by
y (t) = (( - a)p 1,E(t))1 1/(l~), 0 S t S 'r; y1 (t) = 0, t z ; (9)
and subsequent elements by
y.+1 = L(y.), n = 1, 2, ---. (10)
The sequence {y.) converges as n - oo to a limit function y.., which is a non-negative solution of
(6).
A direct computation shows that y coincides with ((I - a)p1,E)1 1(-) on [0, r] for
n = 1, 2, - - -, so the same must be true for y... Notice, however, that ((1 -a)p,)1 04 is the
unique non-trivial solution of the initial value problem (6) beyond e. Hence, y.. and
((1 - a)p ae)1/(1~) must be one and the same function.
Clearly, x 1 y1i 0 on [0, T]. Because K and L satisfy the ordering relation K(x) L (y)
whenever x z y z 0, it follows 'hat x z y~ on [0, T ] for n = 1, 2, - - -. In the limit n -+ oO we
have, therefore, x.. 2y.. on [0, T].
We complete the proof of the lemma by taking E successively equal to any of the points tj in
the sequence. Because the sequence converges to 0 and p1 ,E converges pointwise to p as e -+.0,
the inequality (3) follows. 0
Proof of the theorem. Suppose u and v are two distinct solutions of the initial value problem (1),both strictly positive on (0, T). Without loss of generality we may assume that u > v. The
difference w = u - v satisfies the differential equation
187
(11)
and the initial condition w (0) = 0.
According to the mean value theorem, u - v a -= (ap/4'-) w for some 4 E (v, u). (We
omit the argument t.) Estimating 4 from below by v and applying the lower bound (3) to v, we
find that the positive function w satisfies the following differential inequality:
vi ' (1 - a) (t) K . (12)
Hence, upon integration,
w(12 ) [p1 (t2 ) (1-a)I , (13)
w(11 ) p1(ti )
for any two points:t 1 and t2 in (0, T).
For the remainder of the proof we distinguish three cases.
(i)0<a< 1/2.
Because u and v are continuous, with u (0) = 0 and v (0) = 0, we can choose t1 sufficiently close
to 0 that u a c I on [0, t ij]. It follows from (11) that w ' Spua(1 - (v /u)a) pu a 5p and, hence,
w 5 p 1 on [0, t 1 ]. Using (13) with any t2 E (t 1 , T), we find
w (02):5 (p 1(12))a 1-)(P 1(t11))1-2cy1-a) (14)
Since (1 - 2a)/(1 - a)> 0 and p1i(t) -+0 as t -+0, the right hand side of (15) tends to zero as
t1 -+0. Thus, w (t2) can be made arbitrarily small. In the limit, t2 can be anywhere in the inter-
val (0, T). It follows that w vanishes identically on (0, T).
(ii) a = 1/2.
For any E > 0, there exists a t 1 sufficiently close to 0 that u e on [0, t ]. As under (i) above,
we find that
w (2) 5 E(p 1 (1 2 ))(1~)(P 1(t ))t-2ay(1-) = ep 1(2), (15)
for any t2 e (, T). Here, we can make the right hand side arbitrarily small by taking e
sufficiently close to 0. In the limit E -+0, t 1 tends to 0 and t2 can be taken anywhere in the inter-
val (0, T). Again, it follows that w vanishes identically on (0, T).
(iii) 1/2 < a< .
This time, the argument is more delicate, as the choice of t requires some care.
Suppose that, for any e > 0, there exists a tj such that u(t1 ) e(p(t:1))w(1~). Then also
w (t1) S E(p 1(t 1))w(t1~). Using (13) with any 12 E(:j, T), we find
w ( 2 ) E(pI(2))d(1-) (16)
188
w'=p (t )(u C- v C,
Since E is arbitrarily small, it follows again that w vanishes identically.
Now suppose that u(t) co(p 1 (t))w(I) for all t E (0, T), for some E0 > 0. Then u1 -
_ E1-apa = (Eo/p 1)I-cp 1. Since a < 1, we may assume that (c/p 1)I- > 2a everywhere on
(0, T); if necessary, we simply reduce T until the inequality is achieved everywhere. Thus,
u 1-a ? 2ap1i on [0, T]. But then
w '= p(t)(ua -v a) = ((u -("4)I~4v) 1UW < P2 W (1)U-a v1-a 2 cpl(t)W
on [0, TI. Upon integration, this inequality gives
w(t 2 ) p1(t 2) 1/(2a)
<I I (18)w(1 ) p1(t1)(1
for any pair t 1, t2 E (0, T). The desired conclusion follows by the same arguments as under (i)
above. First, we choose t 1 sufficiently close to 0 that w 5 p 1 on [0, t1]. Then, using (18) with
any t2 E (t1 , T), we find
w(t 2) 5 (p (t 2))1/(2a(P 1 (t i ))0"~***2). (19)
Here, (2a - 1)/(2a) > 0, so by letting t 1 tend to 0, we can make the right hand side arbitrarily
small. In the limit, t2 can be anywhere in (0, T), so w vanishes identically on (0, T). 0
3. Remarks
The condition (i) of the theorem cannot be relaxed, unless the condition (ii) is strengthened.This may be illustrated with the following example, which concerns the initial value problem
x'=xI 2n+q(t), t >0; x(0)=0. (20)
Let 1 be any continuous function defined on [0, T] that is (strictly) positive on (0, T). Without
loss of generality we may assume that 1 is differentiable and strictly increasing, with
1(t) <_ (1/100)12.
Let L be any function defined on [0, T], not necessarily differentiable, that satisfies the ine-
qualities
0 <L(t) 54(1(t))2, tE(0, T). (21)
Without loss of generality we may also assume that
L (t)5 t2, t E [0, TI; (22)100
otherwise, we take E(t) = min (L (t), t2/100). Then
189
JL1/2(s)'ds -'s(t),(23)
because of (21), and
JL 2(s) ds 5 1 t2, (24)
0 20
because of (22).
Let {ti)he N be a sequence of points in the interval (0, T) which converges to 0. Let x be a
function that is constructed from L by adding spikes at the points t, which reach to (1/10)t2 . The
widths of the spikes is chosen such that
Fx1/2(s)ds <2JLla(s)ds. (25)0 0
The function x thus constructed satisfies (20) with q 1, the integral of q, given by
q1 (t) =x(t)-Jx112(s)ds. (26)0
At any tj, we have x(tj) = (1/10)t2 , so
2t2 2 tqi(t) = 10-jJx'2(s)ds > -l 2JL112(s)ds > 0, (27)
0
while at any other t e [0, T ],
r t
q1i(t) 2 - x 12(s) ds-2-2 L 12(s) ds 2-1l(t). (28)
0 0
So q, is bounded below by the function 1, which is strictly negative, albeit arbitrarily small in
absolute value, while the condition (ii) of the theorem is satisfied.
Now, consider the sequence (y),
y(t)= Jya'3(s) ds + q i(t); -y (t) = t2. (29)
A subsequence of [y) converges to a solution of (20), which we call y.
Now,
y(t) y Si (s) ds -l(t) 2 .1Y i (s) ds - t2. (30)0b y s0s
Obviously, y 1(t) z -0t 2 . Suppose that y~ - (t) z 2l-t2 for some integer n. Then
190
yt) 1 2 t 2 2 t2,()
2 10 100 10
so, by induction, y(t)> 2it2 for all n. The same is then true for the limit function y. Thus,10
y (t) 2 t2, so y must be different from the solution x.10
Thus, relaxing the condition (i), while retaining the condition (ii), leads to non-uniqueness.
References
P. Hartman 1964. Ordinary Differential Equations, Wiley & Sons, Inc., New York.
L. C. Piccinini, G. Stampacchia, and G. Vidossich 1984. Ordinary Differential Equations in R",Springer-Verlag, New York.
J. Bownds 1970. "A uniqueness theorem for y'= f (x, y) using a certain factorization," J. Diff.Eq. 7,227-231.
S. R. Bernfeld, R. D. Driver, and V. Lakshmikantham, 1975/76. "Uniqueness for ordinarydifferential equations," Math. Sys. Theory 9, 359-367.
H. Murakami 1966. "On nonlinear ordinary and evolution equations," Funkcial. Ekvac. 9, 151-162.
J. M. Bownds and J. B. Diaz 1973. "On restricted uniqueness for systems of ordinary differentialequations," Proc. Amer. Math. Soc. 37, 100-104.
L. A. Peletier and J. Serrin 1986. "Uniqueness of non-negative solutions of semilinear equationsin RR," J. Diff. Eq. 61, 380-397.
H. G. Kaper and Man Kam Kwong 1988. "Uniqueness of non-negative solutions of semilinearelliptic equations," in Nonlinear Diffusion Equations and Their Equilibrium States II, Wei-Ming Ni, L. A. Peletier, and J. Serrin (eds.), Springer-Verlag, New York, 1-17. Also, Proc.1986-87 Focused Research Program on "Spectral Theory and Boundary Value Problems,"ANL-87-26, vol. 4, Hans G. Kaper, Man Kam Kwong, and Anton Zettl (eds.), ArgonneNational Laboratory, Argonne, IL.
191/f
OSCILLATION OF EMDEN-FOWLER SYSTEMS
Man Kam KwongMathematics and Computer Science Division
Argonne National LaboratoryArgonne, IL 60439
James S. W. WongDepartrrent of Mathematics
The University of Hong KongHong Kong
Abstract
The oscillation theory of a certain form of systems of two first-order nonlineardifferential equations is studied. This form includes in particular the classicalEmden-Fowler equations. The well-known oscillation criteria of Atkinson,Beiohorec, and Waltman are generalized.
1. Introduction
In D. D. Mirzov's papers [1973, 1974, and 1980], he studies the Emden-Fowler system
u' = a (t0lU2 Xsign U2
u'2 =-a2(t)Iu1 I signu1 , (1.1)
with a 1 (t) z 0 or a2 (t) 0. A solution is said to be continuable if it exists on the whole half-
infinite interval [0,oo). A continuable solution is said to be oscillatory if it has an infinite number
of zeros with c as the only accumulation point. The system (1.1) is said to be oscillatory if every
pair of continuable solutions, x (t) and y (t), are oscillatory.
When a1 (t)> 0 and X = 1, the system reduces to the classical Emden-Fowler equation:
( 1 +a2 (t)Iu1 I sign uI = 0. (1.2)aci(t)
Mirzov generalizes many of the well-known oscillation criteria for (1.2) to cover (1.1).
In this paper we consider the more general nonlinear system
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193
x'=a 1()f1(y)
Y= -a2(t)f2(x) (1.3)
and show how similar generalizations can be achieved. We assume that the functions f; are con-
tinuous and that
uf;(u) > 0 for u *0. (1.4)
Further conditions will be imposed in the appropriate sections.
In Sections 2 and 3 we assume that both a 1 and a2 are non-negative, and we establish gen-
eralizations of Atkinson's theory for superlinear equations and Belohorec's theorem for sublinear
equations respectively. In Section 4 we assume only that one of the a8 's is non-negative and gen-
eralizes Waltman's theorem using techniques of integral inequalities. In Section 5 we give a
counterexample to show that Waltman's result is no longer true if both coefficients a 1 and a2 can
assume negative values for arbitrarily large t.
Notice that if a1 (t) z 0, the oscillation of y follows from that of x. Indeed, if x oscillates,
then x' assumes both positive and negative values for large t. By the first equation in (1.3), f (y)must assume both positive and negative values for large t. It follows that y oscillates.
2. Superlinear Case: Atkinson's Theorem
Atkinson [1955J has shown for the equation (1.2), if a1 (t) 1, a2 (t) > 0, and a2> 1, then a
necessary and sufficient condition for oscillation is that
Jta2()dt = . (2.1)0
We will generalize this result to the system (1.3).
In this section we assume that
a1 (t) 0 a2(t) !0, (2.2)
but neither one vanishes identically in any half-line [a, oo). When one of the a;'s does vanish in a
neighborhood of oo, then either x or y becomes a constant for largest. Obviously that is not a very
interesting case.
Besides condition (1.4), we also assume that fi and f2 are C' functions and that theysatisfy the following "superlinear" conditions:
f'1 (u)>0 for u 20 (2.3)
f'2(u)20 for u20 (2.4)
194
<
du
(2.5)f2(u)
Define
A(t) = Jai(s)ds. (2.6)0
THEOREM 1. Suppose that conditions (1.4) an; (2.2)-(2.5) are satisfied. Suppose further ther
limA (t) =_** (2.7)
and
JA(t)a2 (t)dt = 0 . (2.8)0
Then the system (1.3) is oscillator.
Proof. As pointed out in Section 1, the assumption (2.2) allows us to infer the oscillation of x
from that of y and vice versa. Suppose that (1.3) is not oscillatory, so that x and y are eventually
of one sign. Without loss of generality we may assume that x (t) 0 for t 0, and that y (t) is
either 5 0 or 0 for t 0.
Let us consider the former case first. The second equation of (1.3) implies that y is decreas-
ing. Since y (t) 5 0, it approaches either -o or a finite negative value as t - o. It follows that,
likewise, f, (y (t)) approaches either -* or a negative value as t -+ o. Together with (2.7), this
implies that
Jai(t)fdy(t))dt = - .0
Now integrating the first equation in (1.3) gives
T
x(T) =x(0)+JaI(t)fI(y(t))dt -+-o.,0
contradicting the assumption that x (t) 0.
To handle the second case, we define w (x) = f, (y)/f2(x). The following equation holds as
a consequence of (1.3):
w'+a1 (t)f'2(x)w2 +a 2(t)f'1 (y) = 0 . (2.9)
By (2.4) the second term is non-negative. The assumption that x(t) 20 and y (t) 0 implies thaty (t) decreases to a non-negative constant. By (2.3) and compactness, there is a positive constantc such that
f' 1 (u) c >0 for u E [O,y (0)].
We thus have
w'(t) :5 -ca 2(t) . (2.11)
Multiplying the two sides of (2.11) by A (t) and integrating from t = 0 to t = T, and applying the
integration by parats formula to the left-hand side, we obtain the inequality
T T
A (T)w (T)- Ja1()w(t)dt 5 -cJfA (t)a2(t)dtc. (2.12)0 0
Notice that the right-hand side tends to - as T -+ >o, while the first term on the left-hand side is
positive. Thus if we can show that the second tern is bounded, we have a contradiction and our
theorem is proved. Using the first equation in (1.3), we see that the second term can be evaluated
as follows and the required assertion is hence a consequence of (2.5).
T T __ d x ~ <o
Jaj(t)w(t)dt = x(t) di<s .(2.13)0 0 f2(X) o f 2(u)
0
If A (t) 2 kt for some constant k > 0 (for instance, if a1 (t) is bounded from below by a posi-
tive constant), then condition (2.8) reduces to the Atkinson condition (2.1).
3. Sublinear Case: Belohorec's Theorem
Belohorec [1961] has shown that for equation (1.2), if a1 (t) * 1, a2 (t) > 0 and 0 < 2 < 1,
then a necessary and sufficient condition for oscillation is th ':
fta2(t)dt = @ . (3.1)0
This result is generalized to the system (1.3) in this section.
Again we assume that a 1(t) and a2 (t) are non-negative and nontrivial in any half-line, and
that (2.7) holds. The functions f;, on the other hand, satisfy the following "sublinear" conditions:
f'(u) 20 i=1,2 (3.2)
f2 (uv) zf2(u)f2(v) for all v 2!1, u> 0 (3.3)
J u < , (3.4)0 f 2 0 h (u)
where f2 o f denotes the composite function f2 o f1(u) = f2(f1 (u)).
196
(2.10)
Examples of functions satisfying (3.3) aref2(x) = Ix I x, O < y < 1, and f2 (x) = 1 - e".
THEOREM 2. Suppose that (1.4), (2.2), (2.7), and (3.2)-(3.4) are satisfied. Suppose further that
Jf2 (A(t))a2 (t)dt = 00(3.5)0
where A (t) is defined as in (2.6). Then the system (1.3) is oscillatory.
Proof Let us first show that if to > 0 is any fixed value, then
Jf2 (A (t) - A (to))a2(t)dt = . (3.6)to
Since A (t) -+ oa as t -+co, thereis a t 1 > t0 such that
A (t) max(1,2A (t0 )) for t > t1 . (3.7)
Then
A(t)-A(t 0 ) 4A(t) for t > t . (3.8)
By (3.2) and (3.3), fort > t 1 ,
f 2(A (t) - A (to)) f2 (%A (t))
f2(A)f 2 (A(t)) . (3.9)
Now (3.6) follows from (3.9) and (3.5).
Suppose now that (1.3) is not oscillatory, so that x(t) and y(t) are of one sign for ta> to.
Using (3.6), we may do a translation to shift to to 0. This way we may assume without loss of
generality that x(t)z0 and y(t) is of one sign for all t 20. The case when y(t)S0 can be
disposed of in exactly the same way as in the proof of Theorem 1.
We now assume that y (t) 0 for t > 0. Analyzing (1.3) shows that x is an increasing and y
is a decreasing function. Integrating the first equation in (1.3) yields
r
x(t) = x(0)+JaI(s)fj(y (s))ds0
z Ja,(s)fi(y(t))ds0
= A (t)f1 (y (t)). (3.10)
It follows that, for all t large enough such that A (t) 1,
197
> f 2 (A (t))f2(f 1(y (t)))". (3.11)
Using the second equation in (1.3), we obtain
y'(t) < -a2(t)f2 (A ())f2 0 f1(y (t)) (3.12)
or
- y') z a2 (t)f2 (A (t))-. (3.13)A f (y(t))
Recall that y (t) is decreasing. Integrating the above inequality, we obtain
Y(,) du Tj- d Ja2(t)f2(A (t))dt . (3.14)
y(T) 2 o f 1 () o
The right-hand side tends to o as T - cc while the left-hand side remains bounded, by (3.4), as
y (T) -+.0. This contradiction completes the proof of the theorem. 0
4. Generalization of Waltman's Theorem
Waltman [1965] has shown that for the equation (1.2), if a 1 (t) > 0 and 2 > 1, then a
sufficient condition for oscillation is that
Ja2 (q)dt = o. (4.1)0
Notice that the function a2 (t) need not be non-negative for this result to hold. Later Wong [1966]
improves the result by replacing the nonlinear term I u1 I ' sign u 1 with any non-decreasing func-
tion f(u 1) (as well as with more general forms involving the first derivative of the unknown func-
tion u 1). In [Kwong and Wong 1982] we obtain the same result using the techniques of integral
inequality. In this section we would like to show that the same technique can be employed to
extend the result to (1.3).
We assume that
a1 (t) Z 0, (4.2)
but that a2 (t) can assume both positive and negative values.
For the functions f, we assume besides (1.4)
liminfIf1 (u)l *0 (4.3)
f'2(u)20 for u z0. (4.4)
198
f2(x(t)) f2(A (r)fI (Y (f)))
THEOREM 3. Suppose that (1.4) and (4.2)-(4.4) are satisfied. Suppose further that
T
limJ a(t)dt = 00 i = 1,2. (4.5)T-+o
Then the system (1.3) is oscillatory.
Proof. Without loss of generality we may assume that x (t) 20 for t 0. Dividing the second
equation in (1.3) by f2 (x) and integrating over [0,t ], we obtain
j r
- ds = faJa2 (s)ds =B(t). (4.6)o f2(x(s)) o
Using integration by parts on the first term, we derive
-- (t = B (t) + - Q + y J f2(~'s ds . (47)
f2(x(t)) f 2 (x(0)) J f (x)
The last term is non-negative as f'2 0 and y(s)x'(s) = a 1 (s)y(s)f1(y) 0 by (1.4) and (4.2).
By (4.5), there exists a value to such that the sum of the first two terms on the left-hand side of
(4.7) is at least 1, for all t ? to. We thus have the integral inequality
(-Y (t)) 1 + f f2(x)x(s)
(x(t)) 1 + (-y (s))ds for all t 2to . (4.8)
In particular, we infer that y(t) <0 for t to. The theory of integral inequalities allows us to
compare -y (t) with the solution U (t) of the following integral equation:
U t)= f'2(x)x'(s)= 1 +1f - (U(s))ds for all t Z to , (4.9)
f2 (x(t)) 1 A(x)
to conclude that
-y(t) U(t) for all t to. (4.10)
By differentiating (4.9), we see easily that U'(t) = 0, from which follows that U (t) = 1. Thus
(4.10) reduces to
y (t):5 -1 for all t z to. (4.11)
From the hypothesis (4.3), we have
fi(Y(t)) 5 s upf(y) = k < 0. (4.12)
Thus, by (4.5),
199
I r
faj(s)f1 (y(s))ds 5 k aj(s)ds -4-o. (4.13)t o'
Integrating the first equation in (1.3) gives
x(t) -x(to) = Jaj(s)fi(y(s))ds . (4.14)to
The fact that the right-hand side tends to -o contradicts our assumption that x remains positive
for all t. This completes the proof the theorem. 0
5. A Counterexample and Remarks
In all the theorems proved in this paper, some positivity condition has to be imposed on one
of the coefficients a;(t). We would like to show with an example that some such condition is
inevitable.
Let us just look at the linear case, namely, when f;(u) = u for i = 1,2. The new variable
r (t) = -y (t)/x(t) satisfies the Riccati differential equation
r(t)= a2 (t)+a1(t)r 2 (t) , (5.1)
or, upon integration, the Riccati integral equation
r
r (t) = Q@(t)+Ja(s)r2(s)ds , (5.2)0
where
Q (t) = r(0)+Ja2 (s)ds . (5.3)0
The question whether (1.3) has a solution x of one sign (let us say positive) is equivalent to the
question whether (5.2) has a continuous solution on [0,oo).
In the following we are going to construct step functions Q (t) and a2(t) such that (5.2) does
have a solution on [O,oo), if we allow a; to change sign, even though condition (4.5) is satisfied.
Although our smoothness condition on ai precludes such step functions, and the solution has
jumps at each of the points 3k, 3k + 1, modification of our example by smoothing out the abrupt
jumps can easily lead to an acceptable counterexample.
We use the following a1 :
al(t)=f-1 t e [3k, 3k + 1)I) t e [3k + 1, 3k + 3),
Obviously,
200
T
lim fa1(t)dt=oo.T-o
Let us now construct our Q (t). In each [3k, 3k + 1), we choose Q (t) to be a constant a = a(k) so
large that, for instance,
3k
a + fa1(s)r2 (s)ds = k . (5.4)0
3k
Later, we are going to show that the number Ja (s)r2 (s)ds is negative for all k. Hence,0
a(k) k -+oo ask -+oo. (5.5)
In the interval [3k, 3k + 1), equation (5.2) reduces to
r
r(t) = k - Jr2(s)ds3k
which has the simple solution
r (t) = -[t-3k + /kft1
Note that the right-hand side of the above formula is well defined, as the denominator does not
vanish in [3k, 3k + 1). From (5.2), we have
3k+1
Skl(s)r2(s)ds=r(3k+ )-a=-k -a< . (5.6)
In [3, +1, 3k+3), we take Q (t) to be the negative of the number in (5.6). Taking into account
equation (5.5), we see that
lim Q (t) = co.
In [3k + 1, 3k + 3), (5.2) reduces to the simple equation
r(t) = .fr2 (s)ds ,3k+1
which has the trivial solution
r(t) = 0 for t E [3k + 1, 3k + 3).
Thus,
3(k+1) 3k+1
Ja 1 (s)r2 (s)ds = Ja 1 (s)r2 (s)ds <0. (5.7)0 0
We have started with the assumption that the above inequality is true when the upper limit of
201
integration is 3k and conclude that the same is true for 3(k+1). This induction step allows us to
infer that (5.7) holds for all k > 0.
Continuing our process over each interval, we obtain a solution r (t) defined on the whole
half-line [0,oo), and so we have an example of a non-oscillatory system (1.3) for which (4.5) is
satisfied.
It is natural to ask whether some relaxation on the positivity requirement is possible. Is it
sufficient to assume only that at each t one of the a;'s is non-negative?
As seen above, the Riccati equation is an important tool in the study of the oscillation of
linear equations. Another case in which this is true is the study of (1.1) when X4 = 1. This is
the so-called half-linear case. With the introduction of the variable r () = -u2 / I u1 I 'sign u1, we
get the Riccati equation
r'(t) = a2 (t)+ 42 a 1 (t)rY(t) , (5.8)
where y = (X + 1)/k. All Stunnian comparison-type theorems continue to hold, and oscillation
criteria can easily be obtained via the traditional methods.
The sufficiency part of Atkinson and Belohorec's theorems continues to hold even when the
coefficient is allowed to change sign, although the proofs are much more complicated. Further-
more, extensions have been obtained by various authors. See, for instance, the references quoted
in the recent papers [Kwong and Wong 1982, 1983a, and 1983b]. It will be nice if the same facts
can be extended to the system (1.3).
References
F. V. Atkinson 1955. "On second order nonlinear oscillation," Pacific J. Math. 5, 643-647.
S. Belohorec 1961. "Oscillatory solutions of certain nonlinear differential equations of second
order," Mat. Fyz. tasopis Sloven Akad. Vied. 11, 250-255.
M. K. Kwong and J. S. W. Wong 1982. "An application of integral inequality to secouid ordernonlinear oscillation," J. Duff. Eq. 46, 63-77.
M. K. Kwong and J. S. W. Wong 1983a. "On an oscillation theorem of Belohorec," SIAM J.Math. Anal. 14,474-476.
M. K. Kwong and J. S. W. Wong 1983b. "Linearization of second-order nonlinear oscillationtheorems," Trans. Am. Math. Soc. 279, 705-722.
D. D. Mirzov 1973. "Oscillation of solutions of a system of nonlinear differential equations,"Duff. Uravn. 9, 581-583.
202
D. D. Mirzov 1974. "The oscillation of solutions of a system of nonlinear differential equa-tions," Mat. Zametki 4, 571-576.
D. D. Mirzov 1980. "Oscillation properties of solutions of a nonlinear Emden-Fowler differentialsystem," Diff. Uravn. 16, 1980-1984.
P. Waltman 1965. "An oscillation criterion for a nonlinear second order equation," J. Math.Anal. Apple. 10, 439-441.
J. S. W. Wong 1966. "On two theorems of Waltman," SIAM J. Apple. Math. 14, 724-728.
2O3/o 0
LOCAL AND GLOBAL PROPERTIES OF SOLUTIONSOF QUASILINEAR ELLIPTIC EQUATIONS*
Mohammed Guedda and Laurent VeronDdpartement de Mathdmatiques
University de ToursFaculty des Sciences et Techniques
Parc de Grandemont37200 Tours, France
Abstract
We study the positive radial solutions of the equationdiv(IuIp-2Du)+uQ =0for0<p-l<q. Whenq<Np/(N-p)-1,we give the complete classification of the isolated singularities of thisequation.
1. Introduction
In this paper we study the global behavior and the isolated singularities of the positive radial
solutions of the following doubly nonlinear equation in an N-dimensional space:
div(IDu IP- 2Du)+u4 =0. (1.1)
When p = 2, (1.1) is called Emden-Fowler's equation and plays an important role in astrophysics.
The properties of its positive and radial solutions have been studied by Emden [1907], Fowler
[1914, 1931], Chandrasekhar [1967], and many other authors. They noticed the existence of two
critical values for q in (1,+oo) which are N/(N-2) and (N+2)/(N-2), and they gave precise
asymptotics for radial positive singularities. Later, positive, not-necessarily-radial solutions of
Emden-Fowler's equation were studied by Gidas and Spruck [1980], Aviles [1983], and Lions
[1980] in the case I < q < (N+2)/(N-2).
The first results concerning the general case of (1.1) (0 <p- 1 < q) have been obtained by
Ni and Serrin [1986a, 1986b], who gave a priori estimates near a singularity. In particular, they
showed that q = N (p -I)/(N-p) is a critical value. They also obtained nonexistence results for
positive solutions in an exterior domain for p-1 < q < N(p-1)/(N--p). Their methods were
based on a general Pohozaev identity associated with (1.1), and another critical value of q,
Np /(N-p)-l, corresponding to the imbedding of Wo 1P (RN) into LP(RN), played an important
role.
*This work was partially supported by NSF grants DMS 8600710 and DMS 8501397. I is reprinted with
Acadenvc Press' permission from J. Diff. Eq. 76 (1988), pp. 159-189.
205
In this paper we first give Ni and Serrin's estimates under a slightly weaker hypothesis with
a simpler method settled upor. an adequate change of variables and concavity properties. As a
consequence, we get the following:
Assume 1 < p < N; then
(i) If p-1 < q < Np /(N-p)-1, there exists no positive radial solutions of (1.1) in RN;
(ii) If q = Np /(N -p)-1, the only positive radial solutions of (1.1) in RN are the functions
(N-pYp2
Ya(x) = Na( N-p )P1 (a + Ix IP/(1))(-NYP, (1.2)
where a is any positive real number.
If we look for solutions of (1.1) under the form u(r) = oar, then we get I= -p/(q+1-p)
(we always assume q > p -l; otherwise (1.1) falls into the scope of Serrin's works [1964, 1965])
and
1/(q+1-p)a = XN,p,q = (-1 -)P1 (N _ ) , (1.3)
q+1-p q+1-p
and it is clear that .N,p,q exists only when p < N and q > N(p -1)(N-p). If we set (x) the fun-
damental solution of the p-Laplace equation, that is,
(- 1(Np( IxI(-NY -')iI < p < N ,(x) = R( I x I) = N )-1(N- log(1/ Ix I) ifp = N ,
where mN is the volume of the unit ball 81(0) in RN, we prove the following in the subcritical
case.
Assume 1 < p S N, p -l < q < N (p -l)/(N-p) (p -1 < q if p = N) and u is a positive radial
solution of (1.1) in B 1(0)\ (0). Then either u is regular in B1(0) or there exists a > 0 such that
lim u(x)/ (x) = a. (1.5)x-+0
Moreover, u satisfjies
-div (I Du I)'-2 Du)-u = f~- 8 (1.6)
in D'(B 1(0)), and such solutions truly exist for a E (0,ao).
This result is essentially prove with the change of variable mentioned earlier and with
Serrin's results [1965). In the supercritical case we can transform (1.1) into an autonomous sys-
tem in R2; and thanks to Poincard-Bendixon theory, we get the following.
206
Assume I < p < N, N(p-1)/(N-p) < q < Np/(N-p) - I and u is a positive radial solutionof (l. I) in B 1(0) \ (0). Then either u is regular in B 1(0) or
lir I x Ip/(q+1-p) u (x) =NP. (1.7)
In the critical case we introduce Ix I(N-p)(p-1) u (x) as a new variable transforming (1.1)
into a nonautonomous first-order equation, and we prove the following.
Assume I < p < N, q = N (p-1)/(N-p), and u is a positive radial solution of (1.1) inB 1(0) \ (0). Then either u is regular in B, (0) or
(N-p)/(p'-p)
lim (Ix I (log(l/I x I))1'P)(N-PY(P-1) u-(x) = p) -p,(1.8)
This paper is organized as follows: Section 2, the subcritical case; Section 3, global solu-
tions; Section 4, the supercritical case; and Section 5, the critical case. In the Appendix we givethe proof of an unpublished result from Serrin and Veron concerning local existence and unique-
ness of solutions of
div (I Du IP-2 Du) = f(u), (1.9)
when p 2 2 (he case I < p 5 2 has been proved by McLeod and Serrin [1987]).
2. The Subcritical Case
Assume C is an open subset of RN containing 0 and n'= \ (0). Our first result is the fol-
lowing.
THEOREM 2.1. Assume 1 < p 5N, p - I < q < N (p -l)/(N-p ), and u E C'(0') is a positiveradial solution of (1.1) in W'. Then
(i) either u can be extended to f2 as a C' solution of (1.1) in 0, or
(ii) there exists a > 0 such that
limu(x)/ (x) = a. (2.1)X-0O
Moreover, u satisfies
-div (IDu IP-2 Du) - u9 = a~ -So (2.2)
in D'(fl).
We first start with the following a priori estimates of the singularity which have been
obtained in a different way by Ni and Serrin [1986b].
207
PROPOSITION 2.1. Assume I < p N, q > p -I and u is a positive radial solution of (1.1) in s2'.
Then we have the following:
(i) if p - 1 < q < N (p -1)/(N -p), u (x)/p(x) remains bounded near 0;
(ii) if 1 <p <N and q = N (p -1 )/(N-p ), u(x)/(g(x)(log(1/Ix ))(-N)Y(p 2-p) remains bounded
near 0; and
(iii) if I < p < N and q > N (p -l)/(N -p), x xI '9(+1-p) u (x) remains bounded near 0.
Proof. Without any loss of generality we can assume C 1 B 1(0), and we first assume I < p < N.
The function u (x) = u (r), r = I x I satisfies
(rNI -Iur Ip-2 ur)r+rN-1 q-o0(2.3)
in (0,1]. We then introduce the following change of variable:
s = r(p-NY(p-1) u (r) = v (s) ; (2.4)
then
(I vI~2 v,), +( ~ s-p(N-'(N-p) q = 0 (2.5)N-p
holds in [1,+oo). Hence v is concave, v, is decreasing, and either lim v(s) is finite and u is regu-
lar in S2 [Serrin 1964], or lim v (s) = +o. Moreover, v(s)/s remains bounded, which implies
that u(x)/p(x) is bounded in B 1(0) and (i). When q z N(p-l)/(N-p), we can improve this esti-
mate. From concavity and lim v(s) = +=0, we have v(s) z sv,(s)(1+o(1)) and, as v,(s) > 0, (2.5)
becomes
(v~17),+ CS-p-1Np)(2.6)
for some c > 0 and s large enough. If we set or(s) = v~1 (s), we get
Ws, + cs-p(N-lY(Np>+, q I(p-1 ) 0. (2.7)
If we integrate (2.7), we deduce
(s)-(q+1=P)) (P-1 C-) s Js, N(p Y(Np) for q > N(p-1)/(N -p) ,logs for q = N(p -1)/(N-p), (2.8)
or equivalently
sIN(p-Y(N-p -qv(q+1-p) for q > N(p-1)/(N -p),0 S v,(5s) S c (log s)4i(NPP-P) for q = N(p-1)/(N-p) . (2.9)
Integrating this last relation yields
vs rP(p-1Y[(N-pXq+1-p)1 for q > N(p-1)/(N-p) ,v(s ) 5 c s (log s)-(P-1(NP-P) for q = N(p-1)/(N-p) , (2.10)
which is the desired estimate.
In the case N = p the previous change of variable is replaced by the following one:
1s = log -, u(r) = v(s), (2.11)
r
and v satisfies
( IvI1N-2 vs)s + e vIq =0 (2.12)
on [0,+oo). Then v is concave, and v(s)/s remains bounded, which implies
u(x) 5 c log (2.13)
for 0 < lx I < 1/2.
Proof of Theorem 2.1. We assume 1 <p 5 N, and from the concavity of v (from (2.5) or (2.12))
either rn u(x) is finite and u is a regular solution of (1.1) in n [Serrin 1964] or lim u(x) = +o.z-,O x-.O
If we write (1.1) under the form
div (IlDu IP-2 Du)+ d(x)uP-1 = 0 (2.14)
with d(x)= uq+1-p, then for e > 0 small enough, d e LN -E (a). We then deduce from Serrin
[1965] that if is is singular at 0, there exists 0> 0 such that
lhp(x)u(x) -1(x) (2.15)
for 0 < lx I < 1/2. If wereturn to v, we have
$:5 v (s)/s 5- (2.16)
for s large enough. As s-+ v(s)-v(1) _ v(s) 1-v(1)/v(s) ) is decreasing and boundeds -1s 1-1/s
below by f/2 for large s, it admits a positive limit a as s tends to +oo, which implies (2.1). As v is
concave and lim v (s)/s = a > 0, we getS -++"e
lim v,(s) =a, (2.17)
which implies
lim u,(r)/,(r) = a. (2.18)r-.O
Fore > 0 small enough, set QE = 01 \ BE(0). If cE CA (S2) we have
209
-j IDuIp-2Du.D(dx+Ju dx= j Du"~2 Du ( X da. (2.19)
If we use (2.18), we get (2.2) as E -+0.
It must be noticed that there truly exist solutions of (2.2) for some a > 0. In fact, if we con-
sider the following equation
- div ( I Du I p-2 Du) - uQ = a-1 8 in D(B 1(0))
u =0 on B 1(0), (2.20)
we have the following proposition.
PROPOSITION 2.2. Assume 1 < p N, p-1 < q < N(p-1)/(N -p). Then there exists a* > 0 such
that (2.20) admits at least one positive radial solution for 0 < a < a* and no such solution for
a> a*.
Proof. We shall treat only the case 1 < p < N, the ideas being the same when p = N. The proof
is divided into three steps.
Step 1. The subset T of a > 0 such that there exists a solu don of (2.20) is not empty.
To see that we consider the change of variable (1.4) and for y > 0 , let vy be the solution of
(I v Ip-2 v,), + (-)P s -P(N-ly(N -P) Y = 0 fors z1 (,
vN )-p v ( =y(2.21)
v. is defined on some maximal interval [1,T) where it stays positive; and from the concavity of
v. on [1,T] we have three possibilities:
(i) T <+oo and v(T) = 0,
(ii) T = +oo and vy admits a finite limit at infinity,
(iii) T =+oo and lim v 7 (s) =+oo.
In cases (i) and (ii) there exists t e (1,T] such that v,(t) = 0. As v Y(s) < ys by concavity we get
(v+1),+(P ) - p(-y(-p)> (2.22)N -p
on (1,T), which implies (even if r = +oo in case (ii))
C(N,p,q)-f(l - '-N -1(Np)) > -1(2.23)
with
210
C (N,p,q) = p-1 N-pN -p N (p-l)-q(N-p )
which clearly is impossible for 0 < y5 C(N,p,q)-l/(q+'-P). Hence for y e (0, (C (N,p,q))-~/"(ql-P)
the function vy satisfies (iii) and from Theorem 2.1 we have (2.1) and (2.2). Moreover, a < y.
Step 2. For a large enough, (2.20) admits no positive radial solution.
Assume the contrary, and set y. = v?,-1 for y > 0. From concavity v7(s) 2 (s-1)v,(s).
Then from (2.21) we get
yy~4'lp-1)+ (p-1)Py s-P(-1N~ (s -1)q <0 . (2.24)N-p
If we set a = a(y) = lim v(s)/s = lir vy(s), then
- , 1 -(q+1P) + ( Jy-1 -P)(s-1)(ds 5 o~q+1P)(P-1) . (2.25)q+1-p N -p q+1-p
If a is such that
1-1(+-P)
a [+i ) YJ s - (N~ Y"~p)(s-1) d ,(2.26)p-1 N-p (
(2.25) is impossible.
Step 3. If as e r, then (0,&0] c r. In fact, we shall prove that if j >0 is such that
lim v7(s)/s = a, then for any a E (0,4) there exists at least one y e (0,Y) such that
lim vy(s)/s = a and vt(s) < vj(s) for s > 1. To see that, we define three sequences (a),.),
{(' J. n) [w ) eo such that
Ian=a(1- ), n0, (2.27)
n+2
wO(s) = _ --1 (No)Nf-s , s z 1 (2.28)2 N -p
"/(p-I)
S= ,Ia~1 + (N-p +sP -((s)ds-n)I2.29p -i 1
(w? r ), + (N p(- 1 _ ( _1y -1 = 0 , s 2 1,
w()=0, w,,(1)=y . .(2.30)
Clearly lim we(s)/s = lim w,~(s) = a, The sequences (7).2 and (w),o are well defined,S -++- 5 -+.GO
211
and if u,(r) = w(s), then u~ satisfies
-div((I Du Ip- 2DuN) - unI = a' 0 in D'(B(0)),U. = 0 on SB (0). (2.31)
Moreover,
limu,(x)/p(x) = a,. (2.32)r-+O
From classical comparison principles we have
u~_1 5 u u (2.33)
for n z 1, where Wi(r) = vj(s). Hence the sequence (u) converges in CI (B1(0 )(0)) to some u
satisfying (2.20) and (2.1).
In the next result we extend Theorem 2.1 to more general solutions.
THEOREM 2.2. Assume 1 < p S N, p -I < q < N (p-l)/(N -p), and u E C' (2') is a positive
solution of (1.1) such that
u(x)/(x) S C (2.34)
for some C > 0 and Ix I small enough. Then the conclusions of Theorem 2.1 still hold.
Proof. We write (1.1) in the form (2.14), and we still have Serrin's alternative:
(i) either u can be extended as a C' solution of (1.1) in fl, or
(ii) there exists $3> 0 such that
13 (x) S U(X) (x) (2.35)
for Ix I small enough. We still assume ->81(0), and we define
y = lim sup . (2.36)z-.O p(x)
Then there exists a sequence (x) converging to 0 such that y = lim u(x)/s(xN). If we set
= Ix I, , = xN/.8 we can assume that Yn = u(x~)/(x) = supu(x)/(x) and we define
now
u ( ) = u(S01)/t(S) (2.37)
for 0 < I1I < 1/SN. The function ug satisfies
div(IDu& I- 2Du&)+ C(SN)u4 = 0 (2.38)
in B ,(0)\(0) with
212
C(8)= (s())2.-p
Moreover, as u satisfies (2.34), we also have classically
IlDu(x)l 5 C IxVI~' (x) (2.40)
IDu(x)-Du(x')I 5 C I x-x'I I I x I-1a(x) (2.41)
for 0 < lx I S lx'I 5 1/2, for some C > 0 and a x (0,1). Hence for any compact subset K of
RN\f0) there exists CK such that
Ilu8 .1c'--(K) CK (2.42)
for S, small enough and the set of functions (ui) is relatively compact in the Ci-topology of
RN\(0}. As a consequence there exists a p-harmonic function w in RN\(0) and a subsequence
[(8~) of (5,} such that (us,.) converges to w in the C a-topology of RN\t0). Moreover, w Z 0 in
RN\(0); hence if p = N, w is constant [Kichenassamy and Veron 19861 and we get from (2.37)
w = Y= liu ug(E) for 141 = 1. For E > 0, there exists n0 such that y. > y-e for n 1t0 and,
comparing u and (L-e) in (x : S, < Ix I < 8., p > n zn o) implies u(y-e). As a conse-
quence we get our result as
lim inf u y. (2.43)x-0 (x)
When 1 < p < N we deduce from (1.37) that
S'(x)S w (x) 5 $'s(x) (2.44)
for some 1'>0 and any x e RN. Whence w = At for some A > 0 [Kicherhassamy and Veron1986] and A = y/ (1). Comparing again u and ('y-e) in a sequence of annulus converging to 0
implies (2.43).
3. Global Solutions
We study here some properties of positive radial solutions of (1.1) in an exterior domain.
Some of the results presented here have already been obtained by Ni and Serrin [1986a] under a
slightly stronger hypothesis with a completely different method.
PROPOSMON 3.1. Assume 1 < p 5 N, q > p -l, and G = (x e RN : xI X?1). Then we have the
following:
(i) If p -l < q S N(p -l)/(N-p) when l <p < N or q > p -l when p = N, there exist no posi-
tive radial solutions of (1.1) in G;
(ii) If 1 < p < N and q > N (p-1)/(N-p) and if u is a positive radial solution of (1.1) in G,
there exist two positive constants X and v such that
213
(2.39)
Xk(x) u(x)SV Ix I -p(*-P) (3.1)
forx e G.
Proof.
Step 1. Assume 1 <p N, q >p-1 and u is a positive radial solution of (2.1) in G. Then
lim u (x) = 0. To prove it, we shall consider only the case 1 < p < N and make the change ofIx I -++
variable (2.4). As v is concave in (0,1], it admits a nonnegative limit a at 0. Let us assume that a
is nonzero; then we deduce from (2.5) that the following relation
-C ISp(N-1Y(Np) < (I vsIp-2vs) -- Sp(N-1y(NP) (3.2)C'
holds for s small enough and c1 > 0, which implies
ciS-N(p-lY(N-p) s I v5 Ip2Vs 1s-N(p-IY(N-p)
for0<s sso <1. Hence v,>0on(0,so]and
clS -N INp S- s5 ,-NI(N-p)(34
which implies lim v(s) = -c, a contradiction.s-O
Step 2. Assume 1 <p 5 N and p -1 <q 5 N(p -1)/(N-p); then (i) holds. As v(0) = 0 and v is
concave, v(s) sv,(s) for s > 0; (2.5) becomes
(vi-1 ),+ (1N-p) sq-p(N-1Y(N-p)Vq 0 p-(3.5)p-i
(we have assumed 1 < p < N). If we set W(s) = v~1 (s), we gets)+1~ (pI) -'sq-N(p-1YN-p) for q < N(p-1)/(N-p),
(yt(5 )y.{+l-log(l/s) for q = N(p-1)/(N-p), (3.6)
for s < 1 and c '> 0 As 4t( 0) > 0, we get a contradiction. When p = N, we just set t = log(1/r),
w(t) = u(r).
Step 3. Assume 1 < p < N and q > N (p-1)/(N-p); then (ii) holds. We shall distinguish accord-
ing to whether v~1 (0) is finite. In the first case it is clear that u (r)= p (r) near 0 for some
p > 0. In the second case we deduce from (3.5) that
v,() s te s-tq-N(p - sY(N-p)/(q+1-p) s ( 7)
near 0 for some c ' > 0, which implies the right-hand side of (3.1). As for the left-hand side, it is
214
(3.1)
just the consequence of v(s) z cs near 0.
REMARK 3.1. When 1 < p < N and q > N (p -1)/(N--p), there truly exist solutions u of (0,1) in G
satisfying either
lim Ix I P'(+1-P)u(x)= A.,pq (3.8)
or
lim Ix I(NPY(P- 1)u(x)=aIX I -i+« 3.9
for any a > 0. The solutions satisfying (3.9) can be obtained through the local fixed-point rela-
tion
v (s) = J -(3.1)P(N-Y(N-p) 10
The existence of such a fixed point for s small enough is essentially the same as for Theorems A. 1
and A.2 and thus will be omitted.
As a consequence of Proposition 3.1, we have the following theorem.
THEOREM 3.1. Assume 1 < p < N. Then
(i) if p - < q <Np/(N-p)-1, there exist no positive radial solutions of (1.1) in RN.
(ii) If q = Np /(N -p)-1, the only positive radial solutions of (1.1) in RN are the functions
(N-pYp2
y(x) = Na( N -p )i (a + Ix I P1(P-1))(p-NYP, (3.11)
where a is any positive i eal number.
Proof. In the case N(p-1)/(N-p) < q < Np/(N-p)-1, (i) is the consequence of a still well-
known first integral of Pohozaev type [Ni and Serrin 1986a; Pohozaev 1965; Pucci and Serrin1986]. As u~r(r) exists whenever u,(r) * 0 (at most one point), we have from the Gauss theorem
j div((Du.x) I Du I P-2 Du)dx = JIlDu IP-2 (Du.x)u da, (3.12)B,(0) W,(0) (.2
where uv = (Du.xl/ Ixl). But
div((Du.x) I Du I P~2Du) = (Du.x) div(I Du IP~2 Du) + IDu I P~2(l u.D (Du.x))
and I Du I P-2 (Du.D (Du.x)) = IDu I P + - (x.D I Du I P) . Hencep
f (Du.x)div(I Du I P- 2Du)dx + J IDu IPdxBa(O) B,(O)
215
+- Jf(x.D I Du IP)dx = JIDuIP-2 (Du.x)uvda . (3.13)P B,(0) W8,(0)
But
J (x.D I Du I )dx =- N JfIDu I dx+ J IDu I (x.v)daBR(0) B,(0) a,(0)
and
J (Du.x)div(I Du I P- 2 Du)dx = - Ju(Du.x)dx = - +1 J(x.Du+1)dx,Ba(0) Ba(0) q+ B(0)
and finally we get
N f u9+'dx + (1 -N) JIDu Ildx = 1-- u+(x.v)daq+1 B,(0) P Ba(0) q+1 MR(O)
- 1 Jf IDu I(x.v)da+ Jf IDu IP- 2 (Du.x)uda . (3.14)P a8(O) Ma(O)
As
fIDu I'dx = uQ+'dx +f u IDu IP-2u wda ,
B(0) B(0) a a(0)
we get
N + -1 IJuQ dx =f uQ+(x.v)daq+1 p B,() q+1 a,(0)
- Jf IDu I (x.v)da + fIDu IP- 2(Du.x)uvdaP aB.(0) a8(O)
-( 1 - N) If u IDu IP-2uvda. (3.15)P aB8(0)
N NAs q+1 < Np /(N-p), -- +1-->0. Moreover, from (3.1) I Du I = IurI =q+1 p
O(x~(*+'y )). Hence, for a positive radial solution of (2.1) in RN, the right-hand side of
(3.15) goes to0 as R goes to +o0 , a contradiction.
For (ii) it is clear that y satisfies Dya(0) = 0
(Na-piP)
ya(0) = NN-p p_ = b
a i )" and
216
div(I Du Ip- 2Du)+ uN/(N--p)-6
in RN. From a result of McLeod and Serrin [1987] in the case 1 <p 52 and Serrin and Veron
[1987] in the case of p > 2, there exists a unique local solution of
rN- 1 (IUr I P-2jrr + rN-1 uNPl(N-pp- = 0 in [Orm)-((0) = 0, u(0) = b * 0. (3.17)
Moreover, as u, < 0 in (0,rmax), u is bounded, and (3.17) never degenerates for r > 0. Hence
rma = +00, and there is no bifurcation along the trajectory of u so u = ya.
REMARK 3.1. When p = 2, Theorem 2.1 is still true even if the solutions are not radial [Lions
1980]. It is likely that this hypothesis of radiality is unnecessary in any case.
4. The Supercritical Case
We still assume that 0 is an open subset of RN containing 0, '== ( \ (0). We start with
the following extension of a result from Gidas and Spruck [1980].
THEOREM 4.1. Assume 1 < p < N, q > N (p-1)/(N-p), and u E C 1 (f) is a positive solution of
(1.1) such that
uaq1-p P e L Lo, (l). (4.1)
Then u can be extended to gas a C 1 solution of (1.1) in Q.
Proof. We follow Gidas and Spruck's method, which is based on the Nash-Moser iterative
scheme and Serrin's results. We define
p* = Np/(N-p) cco = (q+1-p)N/(pp*) (4.2)
and for a z a and I > 0
F (u) = 1u1 if 0<u l, (4.3)--- (lua- , a'+ (ao-a)la) if u 1,
and G(u) = F(u)(F'(u))P-1 - aP-l; F is C' on R+ and G continuous and piecewise-regular.
Moreover,
F(u) S la-a ,
uF'(u) 5 aF(u), (4.4)1 G (u) 15 F (u)(F '(u ))P~-1
217
(3.16)
and
- (F '(u))P if 0 < u <1l,
G'(u) - ( f > (4.5)-(F(u))P if u > 1 ,
with y = ap -p + 1, yo = aop -p + 1. Without any loss of generality, we can assume
B 1 (0) c- (, and let k and i be two nonnegative C functions in i2, 4 with compact support in
B i(0), t vanishing in some neighborhood of 0. We get
J (a) IDu I PG-(u)dx +p J fIDu I p-2y''-1G (u)(Du.D (4))dxB1(0) B1(0)
= u9()G(u)dx (4.6)B1(0)
from (1.1). Using (4.5) we deduce
f(4)P I DF(u) I Pdx S C(a)f (I Du IP-1(t)P~1 I F'(u) Ip- F(u) I D(tE) I)dx
BI(O) B1(0) ,
+ f ()PuQ+1-PFP(u)}. (4.7)B(0)
From Young's inequality
IDu l?~1 F '(u )I~1 F (u)(4)P~1 ID( )I S ,IDu Ip IF'(u)I" ( p
p
+ - FP(u)ID( )I P, (4.8)OP
with 0 > 0 and p'= p/(p-1). Taking 0 = ( )1'', we get2C (a)
J ID( F(u))IP C'(a) {J(ID( )IPFP(u)+ ( )Pu4+1-PFP(u))dxJ. (4.9)B1 B1(O)
From [Serrin 1964, Lemma 8] there exists a sequence ( of ncnnegative C functions in B 1(0)
vanishing in some neighborhood of 0 converging to I in Bi(0) \ /0) and such that (D C} con-
verges to 0 in LN(B 1 (0)). From previous estimates
5 FP(u)(dx C(l,a) 1 u dx = C(, a) fu(+1-PWpdx (4.10)Ba(0) B(0)NHn r ()
as aop* = (q+1-p )N Ip. H ence from (4.2)
218
jFP(u)dx 5 KC(l, a) (411)B,(0)
for some K > 0. From Hlder's inequality
J P1Dt 1 FP(u) 5 it D IIl g(B,(o)) II(F(u) II ['(B,() - (4.12)B, (0)
If we replace { by ( in (4.12) and use (4.11), we get
lim j tP I DCI'FP(u) = 0 (4.13)k-++*B, (0)
for any 1 > 0 and a > ao, so (4.9) becomes
J (V(u ))Pdx S C2(a) J (I D4I + u+~'-P')FP(u)dx. (4.14)B, (o) B,(0)
Moreover, j u *+'.i''FIP(u)dx ,j((F(u)Y' dx] [J u( 1 dx p If we
take supp({) such that C2(a) u P~'' dxj / y, we get (by letting I go to infinity)
j (ua*dx 5 2C2 (a) J (I DtIua)Pdx . (4.15)B,(O) B,(O)
If we take a = (q+1-p)N/p2 , we deduce that u e LW (Q). Iterating this process, we deduce that
u e La(1) for any t E [1,00). If d(x) = uq+1-p, then d E L () for any t <+o+, and as (2.1)
can be written as
div (IDu IP- 2Du) = d(x)u-'-1 = 0, (4.16)
then from [Serrin 1965] either there exists 0 > 0 such that
Ix l -Nyp-')5 x S I (-Ny('-)417
or u can be extended to n as a C1 solution of (1.1) in n. But it is clear that (4.17) is not possible,
which ends the proof.
The main result of this section is the following.
THEOREM 4.2. Assume I < p < N, N (p -1)I(N-p) < q < Np/(N -p)-1 and u is a positive radial
solution of (1.1) in C '. Then we have the following:
(i) either lim I x IP/(q+P~J)u(x) = kNp.q, or
(ii) u can be extended to S2 as a C' solution of (2.1) in Q.
Proof The proof is divided into four steps and is based upon a phase plane diagram analysis. Let
219
us define i(r)= rP'(+l-P)u(r). From Proposition 2.1 y is bounded for 05 r 5 1. If we use the
same scaling method as in [Friedman and Veron 1986], ry, is also bounded on [0,1]. Moreover,
w satisfies (whenever (1.1) is not degenerated)
I ryr - P VP-2 (p -1)r2yrr + N-1-2 p l)i ry,q+1-p q+1p
+ p2q-Np (q+1-p)(q+1-p)2 +19=0. (4.18)
We classically set t = log(1/r), w(t) = y(r), which transforms (4.18) into the following auto-
nomous equation:
1w, + Sw IP-2((p-1)w,, -(N -6(q+p-1))w, - (N - Sq)w) + w = 0, (4.19)
where 6 =p/(q+1-p) and t >0.
Step 1. We claim that w, + Sw > 0. We write (4.19) as
(p-1) Iw, + Sw Ip~2 (w~ +6w,) - (N-Sq)lw, + Sw IP-2(w, + w) q = 0 . (4.20)
As w and w, are bounded and N - Sq > 0, we get
1w, + Sw IP- 2 (w, +-w) = 1 yfe tw'(s)ds . (4.21)
As a consequence w, + Sw > 0 and (4.20) can be written as
(p-1)w,, - (N-6(q+p-1))w, - 6(N-Sq)w + w'(w, +86w)2-p = 0 , (4.22)
and we consider (4.22) as a nonlinear autonomous system in R~xR with the unknowns (w,w,).
Step 2. The point (71,p,q,0) is asymptotically stable. For simplicity, set . = kNp,q; by lineariza-
tion at (A.,0) we get
(p -1)x~ - [N-(q+p-1)+(p-2)X.+R-P6~.PJx,
- [6(N-&q) - qq+I-P62 1 +(p-2)q+'1-P62P-x =0, (4.23)
and the characteristic equation is
(p-1)p2 - (p-)[N-5(q+1)]p + S(N-q)(q+1-p) = 0. (4.24)
As p-I < q <Np /(N-p)-1, (4.24) admits two roots with negative real part, which proves theasymptotic stability.
Step 3. Assume w (t) decreases to 0 as t tends to +oo; then
220
w (t) cea', (4.25)
for t z0 and some c > 0. In fact, from (4.21) we have
(w,1+ 6w)~1(t): 5w< t , (4.26)N-Sq
which implies
w1 +6w -cwq/(p-1) 0 (4.27)
and
w(t) 5 (ae ( 1~' A-~* + c )-(p-l **q - ) (4.28)
for some a > 0, for t large enough, which implies (4.25). Moreover, Theorem 4.1 implies that if
w satisfies (4.25), then u can be extended to f2 as a regular solution of (1.1) in Q.
Step 4. End of the proof. Let T be the trajectory of (w,w,) in the phase plane R~xR for t z 0, and
let F'(T) be its 0)-limit set at +oo. From the boundedness of T, 1*(T) is a nonempty connected
compact subset of R~xR. We shall distinguish two cases:
Case 1: (GN,p,q,0)E 1'+(T,
Case 2: (XN.p,q,0) 4 -
In Case 1, Step 2 implies that
lim w(t) = lim y(r) = -Npq.- (4.29)I-++- r -+O
So let us assume that we are in Case 2 and there exists ao> 0 such that
(w(t),w,(t)) 4 a((.N p,q,O)) = B (4.30)
for any t .20. Again we are left with two possibilities:
Case 2-1: w (t) is monotone for t large enough,
Case 2-2: w(t) is not asymptotically monotone.
In Case 2-1, it is clear that w (t) converges to some 0 0 when t tens to +o. In that case we get
lim e(N-q)rJ(8->?w (s)ds= 0g4++.- N-q(4.31
From (4.21) it implies that w,(t) admits a limit at infinity. This limit is necessary, 0, and
221
dp-19p-1 = O . (4.32)N-Sq
Hence 0 = 0 or 0 = AN,p,q- As (.N,p,q,0) 4 1+(T), we deduce that w(t) decreases to 0 when t
tends to +ao and u is regular in C from Step 3. In Case 2-2 we are again left with two possibili-
ties:
Case 2-2-1: (0,0)4 FI*(T),
Case 2-2-2: (0,0) e 1'(T). (See Fig. 1.)
In Case 2-2-1, there exists &> 0 such that
(w(t),w,(t)) 4 Bg((0,0)) (4.33)
for t 0. As w (t) is not asymptotically monotone, there exists a sequence {t,} -++oo such that
w,(t~) = 0 and w(t:) is a local minimum for w(t). Hence (4.33) implies that
w (t) >_ Q(4.34)
for t 0, and (4.21) implies that
(w, + 6w)p~1 N (4.35)
for t 0, and the equation never degenerates. Applying the Poincard-Bendixon theorem in
A = {(w,w,) e R~xR: (4.30), (4.33), (4.35) are satisfied), (4.36)
we deduce that T*(T) is a cycle in A. To prove the nonexistence of such a cycle, we introduce
y = (w, + 6w)P-1 and (w,y) satisfies
w, = y 0-Sw = P(w,y )
y = (N-&q)y -W q = Q(w,y) .(4.31)
Let X be the corresponding (w,y) domain, F* the corresponding o)-limit set, and D the bounded
domain of d with boundary F*. As P and Q are regular in d, we deduce
JJ(Qy +P)dydw = J (Qdw - Pdy). (4.38)D P
But the right-hand side of (4.38) is clearly 0 from (4.37). As for the left-hand side we get
Qy + Pw = N - S(q+1) (4.39)
which is nonzero as q < Np/(N -p)-l. Hence, Case 2-2-1 is impossible (this method is called the
Bendixon criterion), and we are left with Case 2-2-2. Then there exists a sequence (i,) such that
w,(~) = 0. w(2.) is a local minimum, w(2,~+ 1 ) is a local maximum, and (from the equation)
222
0 <W(U2,) <kNp,q <w(2n+1) (40
Moreover, (w(2.)) decreases to 0 and (w(72n+ 1)) increases to some A as n tends to infinity. Let
z be the solution of
(p -1)I z,+zI p-2(zu+&zf) - (N -&q)Iz1+z I p-2(z,+6z) + zq = 0z (O) = A, z, (0) = 0 (441)
and P its trajectory in the phase-plane. It is clear that
P = J7(T) (4.42)
as (w 2n+ 1), wa 2(,+ 1)) converges to (A,0) and P c R~xR. Moreover, as
lim (w(2,), w(2,,)) = (0,0), we getn -++eo
lim z(t) = 0. (4.43)
and z,(t) < 0 for t large enough. Hence the function
u(x) = 1x P/(q+1-p)z(log(1/Ix I)) (4.44)
is a positive radial solution of (1.1) in RN (from Step 3), which is not possible from Theorem 3.1,and we are left with Case 1 or Case 2-2, which ends the proof.
wt
F a
0 B~N
Fig. 1. Case 2-2-2
223
(4.40)
REMARK 4.1. From the asymptotic stability of (XN,p,q,0) it is clear that there exist many solutions
of (1.1) satisfying (i). As for regular solutions, they can easily be obtained by minimization of
1/p lDu ldx among the positive and radial C' functions on B(0) satisfying u(1) = 0,B,(O)
(u0 dx = 1. It is also clear that the case q 2 Np/(N-p)-l is substantially more difficult tofB1(0)
handle: when q = Np/(N-p)-1, the roots of (4.23) are purely imaginary; and when
q > Np/(N-p)-1, this equation admits two roots with positive real part and in that case the only
positive and radial solutions of (1.1) satisfying (i) are the restrictions to S' of the function
a'N,p~q IXIp~ -
REMARK 4.2. In case (i) of Theorem 4.2, there exists q* such that the two roots of (4.23) are real
for N (p-1) < q 5 q* and complex for q* < q < Np /(N-p)-l. In that second case the solutionsf P
of (1.1) satisfying (i) are such that I x I Pl'-P)u(x) oscillates around XN,p,q when x tends to 0.
5. The Critical Case
We assume that n is as in Section 3, and we prove the following results, which extend pre-
vious results from Fowler [1931] and Aviles [1987] in the case p = 2.
THEOREM 5.1. Assume I < p < N and u E C'1(s') is a positive radial solution of
div(I Du I P-2 Du) + N(p-1Y(N-p) = 0 (5.1)
in S'. Then
(i) either u can be extended to 0 as a C'1 solution of (5.1) in n, or
(ii)
N - p N -p
lim Ix I P-1 (log(l/ Ix I))P(P-') u(x) = ( N -p) N -)~ -. (5.2)x-[o p p-1
We use the same change of variable as in the proof of Theorem 4.2. If we set
w(t) = r p~1 u(r), t = log(1/r), (5.3)
then w satisfies
(I w, + Sw IP~2(w1 +5&w)),+w = 0 (5.4)
with
224
q = (-1N- = ,(5.5)N-p P
and from Proposition 2.1
W (t ) 5 ct*' ,~'p ) (5.6)
for t 0.
LEMMA 5.1. The function w is strictly decreasing with limit 0 at infinity and
w1 +6w> 0 (5.7)
on [0,+o).
Proof. From (5.6) there exists a sequence {t) converging to +o such that lim w,(t.) = 0. From
(5.4) we have
r,
[(w, +6w)Iw+6wlp-2J + Jw9(s)ds = 0 (5.8)r
fort~ 2t z 0. As w e L'(0,+oo)we deduce from (5.8) that
I wt + Sw I- 2(w, +6w) = J w9(s)ds (5.9)r
for t 0, which implies (5.7). If we assume now that w is not strictly decreasing on [0,+oo), there
would exist 0 < to < t1 such that
w,(to) = w,(t1 ) = 0,w, 0 on (to,t1 ). (5.10)
Hence [(wj'-1]' = - Jw (s)ds and w(t1 ) < w(to), a contradiction, which ends the proof.to
From this result the function t -4 w (t) is a C 1 diffeomorphism from [0,+.0.) onto (0,w (0).
We then set
w(t)=a,(5.11)w,(t) = y(a).
Hence y satisfies
y((y +6ay'-1) 0 + a = 0 (5.12)
on (0,w(0)]. The main point is the following.
LEMMA 5.2. The function y(a)/a admits a limit as a tends to 0, and this limit is either 0 or -6.
225
Proof. We still know that
-S < y < o, (5.13)
and we sety = az; -S< z < 0 and z satisfies
z[oP-1za(z + $p-2+2 2(z + 8f-1]+ !11o"~1 = 0. (5.14)
That is,
z(z+S8~1 + 11 *q~1
zz = -z )2 =F(a,z). (5.15)az(z+5 -2
Step 1. We claim that z 0a(a) has a constant sign for a small enough. Let us assume the contrary.
Then there exists a sequence (t,} such that
ya(ri) / Iya(T,,) = (-1)" . (5.16)
If a e (0,w(0)] is such that z 0 (a) = 0, we have
z(a)(z(a) + 8P-1 + 1 aq+1~ = 0 , (5.17)p-1
and
z o(a) = F0 (a,z(a))+F.(a,z(a))z0 (a) = F0 (Qzz(a)). (5.18)
But
F (a,z) = - 1 [ ~ g aq+1-P - 1 q+-p - z(z+6r -202 z(z+8f- 2 _1-P_
and using (4.17), we get
z0 0(a)= -( ) 2>0.(5.19)P - az(a)(z(a)+8)-2
But from (5.16) there exists a sequence (a,.) converging to 0 such that y(a.) is a local maximum
for y and ya,(a.) S0, a contradiction. Hence z admits a finite limit y at 0 and
-8 y 0. (5.20)
Step 2. Either y = 0 or y = - S. Let us assume that
-8< Y< 0.(5.21)
Then we have
226
lim w,(t)/w(t) = Y , (5.22)
and for T large enough and E > 0 small enough,
e()' 5 w(t) 5 e( E") (5.23)
for t T. We then deduce from (5.23) that
(w, +w)-' 2( (8+y- E)~'w-' ce( XJ'-')' (5.24)
wQ 5 c 'e(re), (5.25)
and from (5.9)
ceI I"- C e re" (5.26)y+C q
for t T. If E is chosen such that
0< E < ,(5.27)q+p-1
the relation (5.26) is impossible.
Proof of Theorem 5.1. We hall distinguish according to y = -6 or'y = 0.
Case): y=-8. Wethenget
lim w,(t)/w(t)=- S , (5.28)
which implies
w (t) 5 e--6(5.29)
for any e > 0 small enough and t > T and u can be extended to n as a C' solution of (5.1) in n[Serrin 1964].
Case 2: y = 0. If we set ir(t) = Iw (s)ds, then (5.9) implies
lim 8P-1(-Wr(t)) ~-1l,/(t) = 1 , (5.30)I-
or equivalently
lInea(t)/Iin-g((t)5=3-1/8yel(5.31)
Integrating (5.31) yields
227
lim te(~l(q+l-p) W(t) = ( g+1p )-(p-1Y(q+1-p) , (5.32)r- (p-1)Si
and, as w (t) = (-W,(t)) lq, we get from (5.30)
lim tllq+1~p)w) = (q+l-p ,_q+pS- _,+ ,p(5.33)r- -p -1
which implies (5.2) as q+1-p = (N-p)/(p2_
References
P. Aviles 1983. "On isolated singularities in some nonlinear partial differential equations," Indi-ana Univ. Math. J. 35, 773-791.
P. Aviles 1987. "Local behaviour of solutions of some elliptic equations," Comm. Math. Phys.108, 177-192.
R. Bellman 1953. Stability Theory of Differential Equations, McGraw-Hill, New York.
S. Chandrasekhar 1967. An Introduction to the Study of Stellar Structure, Dover Publ. Inc., NewYork.
R. Emden 1907. Gaskugeln, Anwendungen der mechanischen Warmentheorie auf Kosmologieund meterologische Probleme, Chap. XII, B. G. Teubner, Leipzig.
L. C. Evans 1980. "A new proof of local C 1a regularity for solutions of certain degenerate ellip-
tic P.D.E.," J. Diff. Eq. 50, 315-338.
R. H. Fowler 1914. "The form near infinity of real continuous solutions of a certain differentialequation of the second order," Quart. J. Math. 45, 289-350.
R. H. Fowler 1931. "Further studies on Emden's and similar differential equations," Quart. J.Math. 2, 259-288
A. Friedman and L. Veron 1986. "Singular solutions of some quasilinear elliptic equations,"Arch. Rat. Mech. Anal. 96, 259-287.
B. Gidas and J. Spruck 1980. "Global and local behaviour of positive solutions of nonlinearelliptic equations," Comm. Pure Appl. Math. 34, 525-598.
S. Kichenassamy and L. Veron 1986. "Singular solutions of the p-Laplace equation," Math.Ann. 275, 599-615.
J. L. Lewis 1983. "Regularity of the derivatives of solutions to certain degenerate elliptic equa-tions," Ind. Univ. Math. J. 32, 849-858.
228
P. L. Lions 1980. "Isolated singularities in semilinear problems," J. Diff. Eq. 38, 441-450.
K. McLeod and J. Serrin 1987. Private communication.
W. M. Ni and J. Serrin 1986a. "Existence and nonexistence theorems for ground states of quzsil-inear partial differential equations: the anomalous case," Acad. Naz. dei Lincei 77, 231-257.
W. M. Ni and J. Serrin 1986b. "Nonexistence theorems for singular solutions of quasilinear par-tial differential equations," Comm. Pure Appl. Math. 38, 379-399.
S. I. Pohozaev 1965. "Eigenfunctions of the equation Au + kf(u) = 0," Soviet Math. 5, 1408-1411.
P. Pucci and J. Serrin 1986. "A general variational identity," Ind. Univ. Math. J. 35, 681-703.
J. Serrin 1964. "Local behaviour of solutions of quasilinear equations," Acta Math. 111, 247-302.
J. Serrin 1965. "Isolated singularities of solutions of quasilinear equations," Acta Math. 113,219-240.
J. Serrin and L. Veron 1987. Private communication.
P. Tolksdorf 1983a. "On the Dirichlet problem for quasilinear equations in domain with conicalboundary points," Comm. P.D.E. 8, 773-817.
P. Tolksdorf 1983b. "Regularity for a more general class of quasilinear elliptic equations," J.Diff. Eq. 51, 126-150.
N. N. Ural'ceva 1968. "Degenerate elliptic systems," Sem. Math. V. A. Steklov Math. Inst. 7,83-99.
L. Veron 1987. "Singularities of some quasilinear equations," in Nonlinear Diffusion Equationsand Their Equilibrium States, W. M. Ni, L. A. Peletier, and J. Serrin, eds., MSRI Publ.,Springer-Verlag (to appear).
229
Appendix: Local Existence Results
Let A be a continuous function from (0,+c) into R and
0(p) = pA(Ip I) (A-1)
for p e R. In the sequel C is supposed to be continuous and strictly increasing on R, and we con-
sider the following equation in RN:
div (A (I Du I)Du)+f(u) = 0. (A-2)
Then the following result has been proved by McLeod and Serrin [1987].
THEOREM A-1. Assume f and S-' are locally Lipschitz continuous on R. Then for any a E R
there exists ra > 0 such that (A-2) admits a unique C' radial solution in
B. (0) = (x E RN: Ix I < ra) such that u (a) = 0.
This result in particular applies in the case of equation (1.1) when I < p S2. The following
result of Serrin and Veron [1987] applies to equation (1.1) when p > 2.
THEOREM A-2. Assume f is locally Lipschitz continuous and A is nondecreasing on [0,+oo).
Then for any a e R such that f (a) * 0, there exists ra > 0 such that (A-2) admits a unique C'
radial solution in B,.(0) satisfying u (0) = a.
Proof.
Step 1: fIfIB I S I1B 2I > 0, we claim that
A- (B2)i -' (B1)-f -(B52)) IB, -B 2 1 . (A-3)B2
In fact, if B1 and B2 have the same sign, both positive, for example, (A-3) is equivalent to
A~' (B2)a-'(B1) - A-(B2)5 1B)B(B1 - B2),
B2
or
Bf
B2
and finally,
S2~'(B S) f-'(B2)(A-4)
B 1 B2
If we then set Ii' (B;) = b; (i = 1,2), then b, > b2 and (A-3) is equivalent to
230
A (b1) A (b2),(A-5)
which is satisfied. If B 1 and B2 have opposite sign, B2 < 0 < B 1 for example, we set B2 = - B2and (A-3) is equivalent to
A~r'(Bj)a~' (B 1) + -1(B2) 5 , 2(B 1+ B2), (A-6)
or
f2~1(B1)S n-'(Bj)(A-7)
B1 B2
which is tnie.
Step 2: Local uniqueness. If u is a radial solution of (A-2), that is, of
(rN-1 (ur))r + rN-lf(u) = 0 , (A-8)
such that u (0) = a, then ur(0) = 0 and
u (r) = a + J~Q {1-s 1-NJ o-1f(u (a))da ds (A-9)
on [0,ra). If v is another radial solution of (A-2) with the same initial data, then
I u (r) - v(r) I 5f0a-1 s 1-NJ 5 3 N-1_f(u (a))c 5 2-1 {s 1NJ0N-1f(y (a))daJ(s
Set $(s) = min 1-NJNv-1f(u(a))dJ 1-NJ 0 N-If (v (a))doj. Then (A-3) implies that
I u(r) - v(r)1I J -(1( s)1-Nl JIf(u(a)) - f(v(a))IaN-1'dads. (A-I)o $(s) o
For m > 0, let Km be the Lipschitz constant of f on [a-rn, a+m ]. For 0 5 r r0 small enough,we have both
max max(Iu(r)-aI, Iv(r)-a I)rm, (A-12)OSrSr 0
max min (If(u(r))I1, If(v(r))I)2 If(a) I. (A-13)
Hence $(s) - If (a)I and
w u (r) - v(r)ihaddK, ef (A -1 4
with P= I f (a) I /2N. As the right-hand side of (A-14) is equal tc
231
Ki r N-1tu(a)-v(a)I [Jr 1-N )dsldo0
and
0 2-Nfor N > 2,
sl -_Nda N-2 (A-15)
log - for N = 2,Pa a
Kru(r) -v(r)I < Kf Iu(a) -v(a)I a(a)da ,(A-16)
where
) (1/(N-2) for N > 2,a(a) =f-()k)j(-7
(log(1/a) for N = 2.
By the Cronwall inequality we deduce a = v on (0,r0[
Step 3: Local existence. Let m and K. be as in Step 2, and m 1 = min(m, If(a) /2K ). We set
ra = max r 0:f~rn( If(a) I)ds m 1 If(a)IK(A-18)0 2N 2 1f(a)I+2m 1KJ
Ba =({$ E C 0([0,ra ];R): I$(s)-aI _m 1,VO _S <_ra)- (A-19)
If $ e Ba, we define
(T($))(r) = a + f ~p-s1-N J -1f(u(a))d ds , (A-20)
and we have
If($(a)) -f(a) 5m 1 K,,. (A-21)
As
S I 1-N JoN-1f((a))daJ
Q-1 -s 1-N o-1f ~)d s1-N f cN-1 f$a)Id.sJ1-N f oN-1NJ 0 N (()I00
From the choice of m 1 and (A-21), f(4()) has constant sign on [0,ra], and from the hypothesis
we get
232
S( sIf(a)I)if-1 f-s 1 N N-f($)(a))da 5 2N s 1-N N-1 ())I a. A-2
- If(a)I2N
Hence
Q-1( s If(a)I)
I T($)(r)-a I _r 2N S 1-N (v-1 If((a))Idads
If(a)I
Q~1s If(a)t)sdsI T($))(r) -a I<fr 2N falmK)
2 If(a)I N
and finally
I (T(4)(r)-a I S [J~f-1( s I If(a) (I)ds.f(f+m1 K,)(A-23)0 2N If(a) I
From the definition of r it implies
I T ($)(r)-a I S m i , $ E B., V r E [0, r ] . (A-24)
The existence of a solution u to (A-8) is then equivalent to the existence of a fixed point u to T in
Ba. As in Step 2 we have
K r
I T($))(r) - T(1i)(r) I f 1 4$(s) - or(s) I a(s)ds , (A-25)
So
where a is defined in (A-17). By iteration it is then classical that there exists k e N* such that
max I Tk(4 )(r) - Tk(y)(r) 15 (1-E) max I $(r) -- i(r) I (A-26)osrsr. osrsr.
for some E> 0 and any $, y in Ba. If Ba is endowed with the uniform convergence distance, it is
a complete metric space; hence, Tk and T admit a unique fixed point in Ba
REMARK A-1. In the case of the following type of equation,
(rN-1 IUrIp-2ur)r - 1u , (A-27)
it is clear that the hypothesis u(0) # 0 is necessary for uniqueness when p > 2 as (A-27) admits
the trivial solution and the solution
-1/(p-2)u(r) = 1( )(N + ) rp ~-2) .(A-28)
P-2 P2
233