nonlinear eigenvalue problems for nonhomogenous …¸tiu d. radulescu˘ “simion stoilow"...

Outline Motivation of the study Sublinear eigenvalue problems associated to the Laplace operator revisited Eigenvalue problems for the poly-harmonic operator A nonhomogeneous quasilinear eigenvalue problem Concentration of the spectrum near infinity Anisotropic quasilinear elliptic equations with variable exponent

Nonlinear eigenvalue problems fornonhomogenous differential operators

Vicentiu D. Radulescu

“Simion Stoilow" Mathematics Institute of the Romanian Academy, Bucharest, RomaniaDepartment of Mathematics, University of Craiova, 200585 Craiova, Romania

[email protected]://math.ucv.ro/∼radulescu

Isaac Newton Institute for Mathematical Sciences

Cambridge, 22 January 2014

Vicentiu D. Radulescu “Simion Stoilow" Mathematics Institute of the Romanian Academy, Bucharest, Romania

Nonlinear eigenvalue problems

Dedication

This talk is dedicated to the memory of my father,

Professor Dumitru Radulescu (1914–1982)

Collaborators

CollaboratorsP. Pucci (Univ. Perugia, Italy)M. Mihailescu (Univ. Craiova, Romania)G. Molica Bisci (Univ. Reggio Calabria, Italy)D. Repovs (Univ. Ljubljana, Slovenia)

Outline of the talk

Motivation of the studySublinear eigenvalue problems associated to the Laplace operatorrevisited

Eigenvalue problems for the poly-harmonic operator

A nonhomogeneous quasilinear eigenvalue problemSobolev spaces with variable exponentA continuous spectrum near the origin

Concentration of the spectrum near infinity

Anisotropic quasilinear elliptic equations with variable exponentA compact embedding for anisotropic Sobolev spacesCompetition effects in anisotropic equations with variableexponent

Outline of the talk

Motivation of the study

Let Ω ⊂ RN be a smooth bounded domain. Consider the problem−∆u = λu in Ωu = 0 on ∂Ω .

BASIC RESULT: there is an unbounded sequence of eigenvalues0 < λ1 < λ2 ≤ . . . ≤ λn ≤ . . ..The anisotropic case

−∆u = λa(x)u in Ωu = 0 on ∂Ω

was considered by Bocher (1914), Minakshisundaram and Pleijel(1949), Hess and Kato (1980). It was proved that there is anunbounded sequence of positive eigenvalues if a ∈ L∞(Ω), a ≥ 0 inΩ, and a > 0 in Ω0 ⊂ Ω, where |Ω0| > 0.

Case of an indefinite weight a( · ): Szulkin & Willem (1999)

Quasilinear eigenvalue problems: Anane (1987), Lindqvist (1990)

Hardy–Sobolev operator: −∆pu− µw(x)|u|p−2u (Sreenadh, 2002),with µ < (N − p)pp−p and

w(x) =

|x|−p if 1 < p < N(|x| log

if p = N.

Fully nonlinear elliptic operators: Felmer & Quaas (2004).Remark. In all these cases, the existence part is guaranteed as long asthe differential operator is positively homogeneous and is monotonewith respect to a convex cone.

w(x) =

|x|−p if 1 < p < N(|x| log

if p = N.

w(x) =

|x|−p if 1 < p < N(|x| log

if p = N.

w(x) =

|x|−p if 1 < p < N(|x| log

if p = N.

w(x) =

|x|−p if 1 < p < N(|x| log

if p = N.

Main purposes of the talk:(i) to point out some new phenomena in the case of a sublinearperturbation in the eigenvalue problem (both for the Laplace operatorand for poly-harmonic operator);(ii) to consider nonhomogeneous differential operators and to studyrelated eigenvalue problems.

Model example in case (ii): ∆p(x)u := div (|∇u|p(x)−2∇u)

The study of differential equations and variational problems involvingp(x)-growth conditions is a consequence of their applications.Materials requiring such more advanced theory have been studiedexperimentally since the middle of last century.

A quotation by Vladimir Arnold (1937–2010)All mathematics is divided into three parts: cryptography (paid for byCIA, KGB and the like), hydrodynamics (supported by manufacturersof atomic submarines) and celestial mechanics (financed by militaryand other institutions dealing with missiles, such as NASA).Cryptography has generated number theory, algebraic geometry overfinite fields, algebra, combinatorics and computers.Hydrodynamics procreated complex analysis, partial differentialequations, Lie groups and algebra theory, cohomology theory andscientific computing.Celestial mechanics is the origin of dynamical systems, linear algebra,topology, variational calculus and symplectic geometry.The existence of mysterious relations between all these differentdomains is the most striking and delightful feature of mathematics(having no rational explanation).

In1920, E. Bingham, who was surprised that some paints not run (likehoney!), studied their behaviour and described a strange phenomenon.There are fluids that flow then stop spontaneously (Bingham fluids).Within them, the forces that create flow reach a first threshold; As thisthreshold is not reached, the fluid flow without deforms as a solid.Such fluids are more common than it seems: the butter, the toothpaste...Invented in the 17th century, the “Flemish medium" makes paintingoil thixotropic: it fluidies under pressure of the brush, but freezes assoon as you leave the rest. While the exact composition of themedium Flemish remains unknown, it is known that the bonds formgradually between its components, which is why the picture freezes ina few minutes. Thanks to this wonderful medium, Rubens havepainted La Kermesse in 24 hours.

Thus, the painters of the 17th century invented a recipe gathers all thedesirable qualities of an artist painting:(i) spreading easily;(ii) being easily cover;(iii) dry slowly.

The first major discovery on electrorheological fluids is due to WillisWinslow in 1949. These are fluids whose flow properties differ in anyway from those of Newtonian fluids. Most commonly the viscosity ofnon-Newtonian fluids is not independent of shear rate or shear ratehistory. Electrorheological fluids have the interesting property thattheir viscosity depends on the electric field in the fluid. Winslownoticed that in such fluids (for instance lithium polymetachrylate)viscosity in an electrical field is inversely proportional to the strengthof the field. The field induces string-like formations in the fluid,which are parallel to the field. They can raise the viscosity by as muchas five orders of magnitude. This phenomenon is known as theWinslow effect. We refer to Halsey (1992) and for some technicalapplications Pfeiffer et al. (1999), Diening (2002). Electrorheologicalfluids have been used in robotics and space technology. Theexperimental research has been done mainly in the USA, for instancein NASA laboratories.

First example: image restoration

Example 1 (Chen, Levine, Rao). In image restoration, we consider aninput I that corresponds to shades of gray in a domain Ω ⊂ R2. Weassume that I is made up of the true image corrupted by noise.Suppose that the noise is additive, that is, I = T + η where T is thetrue image and η is a random variable with zero mean. Thus, theeffect of the noise can be eliminated by smoothing the input, sincethis will cause the effect of the zero-mean random variables at nearbylocations to cancel. Smoothing corresponds to minimizing the energy

E1(u) =

(|∇u(x)|2 + |u(x)− I(x)|2)dx .

Unfortunately, smoothing will also destroy the small details from theimage, so this procedure is not useful

E1(u) =

(|∇u(x)|2 + |u(x)− I(x)|2)dx .

E1(u) =

(|∇u(x)|2 + |u(x)− I(x)|2)dx .

E1(u) =

(|∇u(x)|2 + |u(x)− I(x)|2)dx .

A better approach is total variation smoothing. Since an edge in theimage gives rise to a very large gradient, the level sets around the edgeare very distinct, so this method does a good job of preserving edges.Total variation smoothing corresponds to minimizing the energy

E2(u) =

(|∇u(x)|+ |u(x)− I(x)|2)dx .

Unfortunately, total variation smoothing not only preserves edges, ilalso creates edges where there were none in the original image. Thisis the staircase effect.

A better approach is total variation smoothing. Since an edge in theimage gives rise to a very large gradient, the level sets around the edgeare very distinct, so this method does a good job of preserving edges.Total variation smoothing corresponds to minimizing the energy

E2(u) =

(|∇u(x)|+ |u(x)− I(x)|2)dx .

Unfortunately, total variation smoothing not only preserves edges, ilalso creates edges where there were none in the original image. Thisis the staircase effect.

Looking at E1 and E2, Chen, Levine and Rao suggest that anappropriate energy is

E(u) =

(|∇u(x)|p(x) + |u(x)− I(x)|2)dx ,

where 1 ≤ p ≤ 2.This function should be close to 1 where there are likely to be edges,and close to 2 where there are likely not to be edges. The approximatelocation of the edges can be determined by just smoothing the inputdata and looking for where the gradient is large.

E(u) =

(|∇u(x)|p(x) + |u(x)− I(x)|2)dx ,

E(u) =

(|∇u(x)|p(x) + |u(x)− I(x)|2)dx ,

E(u) =

(|∇u(x)|p(x) + |u(x)− I(x)|2)dx ,

Second example: electrorheological fluids

Example 2. The constitutive equation for the motion of anelectrorheological fluid is

(1) ut + div S(u) + (u · ∇)u +∇π = f ,

where u : R3,1 → R3 is the velocity of the fluid at a point inspace-time, π : R3,1 → R is the pressure, f : R3,1 → R3 representsexternal forces, and the stress tensor S : W1,1

loc → R3,3 is of the form

S(u)(x) = µ(x)[1 + |Du(x)|2](p(x)−2)/p(x)Du(x),

where Du = (∇u +∇uT)/2 is the symmetric part of the gradient of u.We observe that the highest order differential term in (1) is

(1 + |Du(x)|2)(p(x)−2)/p(x)Du(x)).

The degenerate case corresponds to the Laplace operator with variableexponent.Vicentiu D. Radulescu “Simion Stoilow" Mathematics Institute of the Romanian Academy, Bucharest, Romania

(1) ut + div S(u) + (u · ∇)u +∇π = f ,

(1 + |Du(x)|2)(p(x)−2)/p(x)Du(x)).

(1) ut + div S(u) + (u · ∇)u +∇π = f ,

(1 + |Du(x)|2)(p(x)−2)/p(x)Du(x)).

Sublinear eigenvalue problems associated to the Laplaceoperator revisited

Consider the nonlinear problem−∆u = λf (x, u), in Ωu = 0, on ∂Ω ,

where f : Ω× R→ R and λ ∈ R.We study problem (1) in the case when

f (x, t) =

h(x, t), if t ≥ 0t, if t < 0 .

f (x, t) =

h(x, t), if t ≥ 0t, if t < 0 .

f (x, t) =

h(x, t), if t ≥ 0t, if t < 0 .

Assume h : Ω× [0,∞)→ R is a Carathéodory function satisfying thefollowing hypotheses(H1) there exists C ∈ (0, 1) such that |h(x, t)| ≤ Ct for any t ≥ 0 anda.e. x ∈ Ω;(H2) there exists t0 > 0 such that H(x, t0) :=

∫ t00 h(x, s) ds > 0, for

a.e. x ∈ Ω;(H3) limt→∞

h(x,t)t = 0, uniformly in x.

Examples.1. h(x, t) = sin (t/2), for any t ≥ 0 and any x ∈ Ω;

2. h(x, t) = k log(1 + t), for any t ≥ 0 and any x ∈ Ω, wherek ∈ (0, 1) is a constant;

3. h(x, t) = g(x)(tq(x)−1 − tp(x)−1), for any t ≥ 0 and all x ∈ Ω, wherep, q : Ω→ (1, 2) are continuous functions satisfyingmaxΩ p < minΩ q, and g ∈ L∞(Ω) satisfies 0 < infΩ g ≤ supΩ g < 1.

∫ t00 h(x, s) ds > 0, for

Define

λ1 = infu∈H1

0(Ω)\0

∫Ω|∇u|2 dx∫Ω

u2 dx. (3)

TheoremAssume that f is given by relation (2) and conditions (H1), (H2) and(H3) are fulfilled.Then λ1 defined in (3) is an isolated eigenvalue of problem (1) and thecorresponding set of eigenvectors is a cone. Moreover, anyλ ∈ (0, λ1) is not an eigenvalue of problem (1) but there existsµ1 > λ1 such that any λ ∈ (µ1,∞) is an eigenvalue of problem (1).

Define

λ1 = infu∈H1

0(Ω)\0

u2 dx. (3)

Define

λ1 = infu∈H1

0(Ω)\0

u2 dx. (3)

Sketch of the proof.Step 1: any λ ∈ (0, λ1) is not an eigenvalue of problem (1).Step 2: λ1 is an eigenvalue of (1). Moreover, the set of eigenfunctionscorresponding to λ1 is a cone.Step 3: λ1 is isolated in the set of eigenvalues of problem (1).Step 4: the associated energy functional Eλ is bounded from belowand coercive.Step 5: there exists λ∗ > 0 such that for all λ ≥ λ∗,

0(Ω)Eλ < 0 .

Conclusion. Steps 4 and 5 show that for any λ > 0 large enough, Eλhas a negative global minimum. Thus, any λ > 0 large enough is aneigenvalue of problem (1). Combining this fact with Steps 1, 2 and 3,we conclude the proof.

Let B be any ball of Rn centered at the origin and of fixed radiusR > 0. Consider the linear eigenvalue problem

(−∆)Ku = λu in Bu = Du = · · · = DK−1u = 0 on ∂B,

where K is a positive integer.Then the lowest eigenvalue λ1 of problem (4) is simple, that is, theassociated eigenfunctions are merely multiples of each other.Moreover they are radial, strictly monotone in r = |x| and neverchange sign in B.

Consider the nonlinear eigenvalue problem(−∆)Ku = λf (x, u) in Bu = Du = · · · = DK−1u = 0 on ∂B,

where λ is a positive parameter and the nonlinear function f is givenby

f (x, t) =

t, if t < 0h(x, t), if t ≥ 0,

where h : B× R+0 → R is a Carathéodory function,

H(x, t) :=∫ t

0 h(x, s)ds, and the following conditions are fulfilled:

(H1) There exists c ∈ (0, 1) such that |h(x, t)| ≤ ct for all t ∈ R anda.a. x ∈ B;

(H2) There exists t0 > 0 such that H(x, t0) > 0 for a.a. x ∈ B;

(H3) limt→∞

h(x, t)t

= 0 uniformly in B \ O, with µ(O) = 0.

f (x, t) =

t, if t < 0h(x, t), if t ≥ 0,

H(x, t) :=∫ t

(H3) limt→∞

h(x, t)t

f (x, t) =

t, if t < 0h(x, t), if t ≥ 0,

H(x, t) :=∫ t

(H3) limt→∞

h(x, t)t

f (x, t) =

t, if t < 0h(x, t), if t ≥ 0,

H(x, t) :=∫ t

(H3) limt→∞

h(x, t)t

The following result establishes the existence of a continuousspectrum in a neighbourhood of +∞.

TheoremSuppose that f is of type (6) and that hypotheses (H1)–(H3) arefulfilled. Then the first eigenvalue λ1 of (4) is an isolated eigenvalueof problem (5) and the corresponding set of eigenfunctions is a cone.Moreover, any λ ∈ (0, λ1) is not an eigenvalue of (5), while thereexists µ1 > λ1 such that any λ ∈ (µ1,∞) is an eigenvalue of problem(5).

A nonhomogeneous quasilinear eigenvalue problem

Let Ω ⊂ RN be a bounded domain with smooth boundary. Considerthe problem

−div(|∇u|p(x)−2∇u) = λ|u|q(x)−2u, x ∈ Ωu = 0, x ∈ ∂Ω ,

where p, q are continuous on Ω and ∆p(x)u := div (|∇u|p(x)−2∇u).

Previous result. Case p(x) = q(x) was considered by Fan, Zhang &Zhao (2005): there is an unbounded sequence of eigenvalues.

Sobolev spaces with variable exponent

Assume p ∈ C(Ω) and p > 1 in Ω. Set

C+(Ω) = h; h ∈ C(Ω), h(x) > 1 for x ∈ Ω.For h ∈ C+(Ω), define

h+ = supx∈Ω

h(x) and h− = infx∈Ω

For any p ∈ C+(Ω), define the variable exponent Lebesgue space

Lp(x)(Ω) = u;

∫Ω|u(x)|p(x) dx <∞.

Luxemburg norm:

|u|p(x) = inf

µ > 0;

∣∣∣∣u(x)

∣∣∣∣p(x)

dx ≤ 1

Then Lp(x)(Ω) is separable and reflexive Banach space.Vicentiu D. Radulescu “Simion Stoilow" Mathematics Institute of the Romanian Academy, Bucharest, Romania

h+ = supx∈Ω

Lp(x)(Ω) = u;

∫Ω|u(x)|p(x) dx <∞.

Luxemburg norm:

|u|p(x) = inf

µ > 0;

∣∣∣∣u(x)

∣∣∣∣p(x)

dx ≤ 1

h+ = supx∈Ω

Lp(x)(Ω) = u;

∫Ω|u(x)|p(x) dx <∞.

Luxemburg norm:

|u|p(x) = inf

µ > 0;

∣∣∣∣u(x)

∣∣∣∣p(x)

dx ≤ 1

h+ = supx∈Ω

Lp(x)(Ω) = u;

∫Ω|u(x)|p(x) dx <∞.

Luxemburg norm:

|u|p(x) = inf

µ > 0;

∣∣∣∣u(x)

∣∣∣∣p(x)

dx ≤ 1

h+ = supx∈Ω

Lp(x)(Ω) = u;

∫Ω|u(x)|p(x) dx <∞.

Luxemburg norm:

|u|p(x) = inf

µ > 0;

∣∣∣∣u(x)

∣∣∣∣p(x)

dx ≤ 1

Lp′(x)(Ω): the dual space of Lp(x)(Ω), where 1/p(x) + 1/p

′(x) = 1.

For u ∈ Lp(x)(Ω) and v ∈ Lp′(x)(Ω) the Hölder inequality∣∣∣∣∫

Ωuv dx

∣∣∣∣ ≤ ( 1p−

p′−

)|u|p(x)|v|p′ (x) (7)

holds true.MODULAR of the Lp(x)(Ω): ρp(x) : Lp(x)(Ω)→ R defined by

ρp(x)(u) =

∫Ω|u|p(x) dx.

′(x) = 1.

Ωuv dx

∣∣∣∣ ≤ ( 1p−

p′−

)|u|p(x)|v|p′ (x) (7)

ρp(x)(u) =

∫Ω|u|p(x) dx.

′(x) = 1.

Ωuv dx

∣∣∣∣ ≤ ( 1p−

p′−

)|u|p(x)|v|p′ (x) (7)

ρp(x)(u) =

∫Ω|u|p(x) dx.

If un, u ∈ Lp(x)(Ω) then

|u|p(x) > 1 ⇒ |u|p−

p(x) ≤ ρp(x)(u) ≤ |u|p+

|u|p(x) < 1 ⇒ |u|p+

p(x) ≤ ρp(x)(u) ≤ |u|p−

min|u|p−

p(x), |u|p+

p(x) ≤ ρp(x)(u) ≤ max|u|p−

p(x), |u|p+

|un − u|p(x) → 0 ⇔ ρp(x)(un − u)→ 0.

|u|p(x) > 1 ⇒ |u|p−

p(x) ≤ ρp(x)(u) ≤ |u|p+

|u|p(x) < 1 ⇒ |u|p+

p(x) ≤ ρp(x)(u) ≤ |u|p−

min|u|p−

p(x), |u|p+

|un − u|p(x) → 0 ⇔ ρp(x)(un − u)→ 0.

|u|p(x) > 1 ⇒ |u|p−

p(x) ≤ ρp(x)(u) ≤ |u|p+

|u|p(x) < 1 ⇒ |u|p+

p(x) ≤ ρp(x)(u) ≤ |u|p−

min|u|p−

p(x), |u|p+

|un − u|p(x) → 0 ⇔ ρp(x)(un − u)→ 0.

|u|p(x) > 1 ⇒ |u|p−

p(x) ≤ ρp(x)(u) ≤ |u|p+

|u|p(x) < 1 ⇒ |u|p+

p(x) ≤ ρp(x)(u) ≤ |u|p−

min|u|p−

p(x), |u|p+

|un − u|p(x) → 0 ⇔ ρp(x)(un − u)→ 0.

Variable exponent Lebesgue spaces do not have the mean continuityproperty: if p is continuous and nonconstant in an open ball B, thenthere exists a function u ∈ Lp(x)(B) such that u(x + h) 6∈ Lp(x)(B) forall h ∈ RN with arbitrary small norm (Kovacik & Rakosnik, 1991).

Most of the problems in the development of the theory of Lp(x) spacesarise from the fact that these spaces are virtually never translationinvariant. The use of convolution is also limited: the Young inequality

‖f ∗ g‖Lp(x) ≤ C ‖f‖Lp(x) ‖g‖L1

holds if and only if p is constant.

‖f ∗ g‖Lp(x) ≤ C ‖f‖Lp(x) ‖g‖L1

If p ∈ C+(Ω), let W1,p(x)(Ω) be the variable exponent Sobolev spaceconsisting of functions u ∈ Lp(x)(Ω) such that∇u ∈ [Lp(x)(Ω)]N . Thisspace is endowed with the norm

‖u‖ = |u|p(x) + |∇u|p(x) .

As shown by Zhikov (1987), the smooth functions are in general notdense in W1,p(x)(Ω). If p is logarithmic Hölder continuous (notation:p ∈ C0, 1

| log t| (Ω)), that is,

|p(x)− p(y)| ≤ C| log |x− y| |

∀ x, y ∈ Ω, |x− y| ≤ 1/2,

then the smooth functions are dense in W1,p(x)(Ω) and so the Sobolevspace W1,p(x)

0 (Ω) with zero boundary values is the closure of C∞0 (Ω)under the norm ‖ · ‖. Edmunds and Rakosnik (1992) derived thesame conclusion under a local monotony condition on p.

‖u‖ = |u|p(x) + |∇u|p(x) .

|p(x)− p(y)| ≤ C| log |x− y| |

∀ x, y ∈ Ω, |x− y| ≤ 1/2,

‖u‖ = |u|p(x) + |∇u|p(x) .

|p(x)− p(y)| ≤ C| log |x− y| |

∀ x, y ∈ Ω, |x− y| ≤ 1/2,

‖u‖ = |u|p(x) + |∇u|p(x) .

|p(x)− p(y)| ≤ C| log |x− y| |

∀ x, y ∈ Ω, |x− y| ≤ 1/2,

Since Ω is bounded and p ∈ C+(Ω) is logarithmic Hölder continuous,then

|u|p(x) ≤ C |∇u|p(x) ∀ u ∈ W1,p(x)0 (Ω) [Poincaré inequality],

where C = C(p, |Ω|, diam (Ω),N). Poincaré’s inequality holds undera much weaker assumption on p than the Sobolev inequality andembedding, namely if the exponent p is not too discontinuous.

Remarks. 1. If Ω is bounded then

C0,1(Ω) ⊂ W1,q(Ω) (q > N) ⊂ C0, 1| log t| (Ω).

2. If Ω is unbounded, p is said logarithmic Hölder continuous if

|p(x)− p(y)| ≤ C| log |x− y| |

∀ x, y ∈ Ω, |x− y| ≤ 1/2

|p(x)− p(y)| ≤ Clog(e + |x|) |

∀ x, y ∈ Ω, |y| ≥ |x|.

In such a case we cannot require p ∈ W1,q(Ω) (since∫Ω |p(x)|qdx =∞).

C0,1(Ω) ⊂ W1,q(Ω) (q > N) ⊂ C0, 1| log t| (Ω).

|p(x)− p(y)| ≤ C| log |x− y| |

∀ x, y ∈ Ω, |x− y| ≤ 1/2

|p(x)− p(y)| ≤ Clog(e + |x|) |

∀ x, y ∈ Ω, |y| ≥ |x|.

C0,1(Ω) ⊂ W1,q(Ω) (q > N) ⊂ C0, 1| log t| (Ω).

|p(x)− p(y)| ≤ C| log |x− y| |

∀ x, y ∈ Ω, |x− y| ≤ 1/2

|p(x)− p(y)| ≤ Clog(e + |x|) |

∀ x, y ∈ Ω, |y| ≥ |x|.

LetW1,(∞,q(·))(Ω) := u ∈ L∞(Ω); |∇u| ∈ Lq(·)(Ω),

where N < q− ≤ q+ <∞.Conclusion. If Ω is unbounded then the hypotheses(i) p ∈ C0,1(Ω);(ii) p ∈ W1,(∞,q(·))(Ω) with N < q− ≤ q+ <∞;

(iii) p ∈ C0, 1| log t| (Ω)

are independent each other.

LetW1,(∞,q(·))(Ω) := u ∈ L∞(Ω); |∇u| ∈ Lq(·)(Ω),

where N < q− ≤ q+ <∞.Conclusion. If Ω is unbounded then the hypotheses(i) p ∈ C0,1(Ω);(ii) p ∈ W1,(∞,q(·))(Ω) with N < q− ≤ q+ <∞;

(iii) p ∈ C0, 1| log t| (Ω)

are independent each other.

A striking example. A function u ∈ W1,p(x)(0, 1) is a minimizer withboundary values 0 and a > 0 if u(0) = 0, u(1) = a, and∫ 1

0|u′(y)|p(y)dy ≤

0|v′(y)|p(y)dy,

for all v ∈ W1,p(x)(0, 1) with v(0) = 0, v(1) = a.If p is constant, then the minimizer is linear, namely u(x) = ax. Let usassume that p(x) = 3χ(0,1/2) + 2χ(1/2,1). Assume that u is aminimizer and denote u(1/2) = b. Then u|(0,1/2) is the solution of theclassical energy integral problem with values 0 and b, and u|(1/2,1) isthe solution with boundary values b and a. Thus, these functions arelinear. This u has Dirichlet energy 4b3 + 2(a− b)2. The functionb 7−→ 2b3 + (a− b)2 has a minimum at b = (

√12a + 1− 1)/6,

which determines the minimizer of the variable exponent problem. Acomputation shows that the minimizer is convex if a > 2/3, concaveif a < 2/3 and linear if a = 2/3.

0|u′(y)|p(y)dy ≤

0|v′(y)|p(y)dy,

√12a + 1− 1)/6,

0|u′(y)|p(y)dy ≤

0|v′(y)|p(y)dy,

√12a + 1− 1)/6,

0|u′(y)|p(y)dy ≤

0|v′(y)|p(y)dy,

√12a + 1− 1)/6,

0|u′(y)|p(y)dy ≤

0|v′(y)|p(y)dy,

√12a + 1− 1)/6,

0|u′(y)|p(y)dy ≤

0|v′(y)|p(y)dy,

√12a + 1− 1)/6,

0|u′(y)|p(y)dy ≤

0|v′(y)|p(y)dy,

√12a + 1− 1)/6,

0|u′(y)|p(y)dy ≤

0|v′(y)|p(y)dy,

√12a + 1− 1)/6,

A continuous spectrum near the origin

Consider the problem−div(|∇u|p(x)−2∇u) = λ|u|q(x)−2u, x ∈ Ωu = 0, x ∈ ∂Ω .

TheoremAssume that q(x) < p?(x) for all x ∈ Ω and

1 < minx∈Ω

q(x) < minx∈Ω

p(x) < maxx∈Ω

Then there exists λ? > 0 such that any λ ∈ (0, λ?) is an eigenvaluefor problem (8).

1 < minx∈Ω

q(x) < minx∈Ω

p(x) < maxx∈Ω

1 < minx∈Ω

q(x) < minx∈Ω

p(x) < maxx∈Ω

1 < minx∈Ω

q(x) < minx∈Ω

p(x) < maxx∈Ω

Jλ(u) =

|∇u|p(x)dx−λ∫

|u|q(x)dx u ∈ W1,p(x)0 (Ω).

Remarks. 1. Jλ is not coercive, since q+ > p−.2. If maxx∈Ω p(x) < minx∈Ω q(x) and q(x) < p?(x) then any λ > 0 isan eigenvalue of problem (8).3. If maxx∈Ω p(x) < maxx∈Ω q(x) then ∃φ ∈ W1,p(x)

0 (Ω) such that

Jλ(tφ)→ −∞ as t→∞. Thus, ∃ (un) ⊂ W1,p(x)0 (Ω) such that

Jλ(un)→ c > 0 J′λ(un)→ 0.

However, in this case, we are not able to show that (un) is bounded inW1,p(x)

0 (Ω).

Jλ(u) =

|u|q(x)dx u ∈ W1,p(x)0 (Ω).

0 (Ω) such that

Jλ(un)→ c > 0 J′λ(un)→ 0.

0 (Ω).

Jλ(u) =

|u|q(x)dx u ∈ W1,p(x)0 (Ω).

0 (Ω) such that

Jλ(un)→ c > 0 J′λ(un)→ 0.

0 (Ω).

Sketch of the proof.

1. ∃λ∗ > 0 such that ∀λ ∈ (0, λ∗), ∃ a, ρ > 0:

Jλ(u) ≥ a ∀ u ∈ W1,p(x)0 (Ω), ‖u‖ = ρ.

This follows from p+ > q−.2. ∃φ ∈ W1,p(x)

0 (Ω), φ ≥ 0 such that Jλ(tφ) < 0 for t > 0 smallenough (since p− < q−).3. For all λ ∈ (0, λ∗) we have

inf∂Bρ

Jλ > 0 and −∞ < c = infBρ

Jλ < 0.

4. Fix 0 < ε < inf∂Bρ Jλ − infBρ Jλ and apply Ekeland’s principle toJλ : Bρ → R. Then ∃ uε such that

1. ∃λ∗ > 0 such that ∀λ ∈ (0, λ∗), ∃ a, ρ > 0:

Jλ(u) ≥ a ∀ u ∈ W1,p(x)0 (Ω), ‖u‖ = ρ.

inf∂Bρ

Jλ < 0.

1. ∃λ∗ > 0 such that ∀λ ∈ (0, λ∗), ∃ a, ρ > 0:

Jλ(u) ≥ a ∀ u ∈ W1,p(x)0 (Ω), ‖u‖ = ρ.

inf∂Bρ

Jλ < 0.

1. ∃λ∗ > 0 such that ∀λ ∈ (0, λ∗), ∃ a, ρ > 0:

Jλ(u) ≥ a ∀ u ∈ W1,p(x)0 (Ω), ‖u‖ = ρ.

inf∂Bρ

Jλ < 0.

Jλ(uε) < infBρ

Jλ + ε

Jλ(uε) < Jλ(u) + ε ‖u− uε‖

(⇒ uε ∈ Bρ). Thus, uε is a minimum point of

Iλ(u) = Jλ(u) + ε ‖u− uε‖.

This implies ‖J′λ(uε)‖ ≥ ε. Thus, ∃ (wn) ⊂ Bρ such that

Jλ(wn)→ c J′λ(wn)→ 0.

Thenwn w in W1,p(x)

0 (Ω) wn → w in Lq(x)(Ω).

We conclude that wn → w in W1,p(x)0 (Ω).

Consider the problem−div((|∇u|p1(x)−2 + |∇u|p2(x)−2)∇u) =

λ|u|q(x)−2u, x ∈ Ωu = 0, x ∈ ∂Ω ,

where1 < p2(x) < min

y∈Ωq(y) ≤ max

y∈Ωq(y) < p1(x) ;

maxy∈Ω

q(y) < p?2(x), ∀ x ∈ Ω ,

with p?2(x) := Np2(x)N−p2(x) if p2(x) < N and p?2(x) = +∞ if p2(x) ≥ N.

Consider the problem−div((|∇u|p1(x)−2 + |∇u|p2(x)−2)∇u) =

λ|u|q(x)−2u, x ∈ Ωu = 0, x ∈ ∂Ω ,

where1 < p2(x) < min

y∈Ωq(y) ≤ max

y∈Ωq(y) < p1(x) ;

maxy∈Ω

q(y) < p?2(x), ∀ x ∈ Ω ,

with p?2(x) := Np2(x)N−p2(x) if p2(x) < N and p?2(x) = +∞ if p2(x) ≥ N.

Define

λ1 := infu∈W1,p1(x)

0 (Ω)\0

1p1(x)

|∇u|p1(x) dx +

1p2(x)

|∇u|p2(x) dx∫Ω

1q(x)|u|q(x) dx

TheoremAny λ ∈ [λ1,∞) is an eigenvalue of problem (5). Moreover, thereexists a positive constant λ0 such that λ0 ≤ λ1 and any λ ∈ (0, λ0) isnot an eigenvalue of problem (5).

Sketch of the proof.STEP 1: λ1 > 0.STEP 2: λ1 is an eigenvalue of problem (5).STEP 3: any λ ∈ (λ1,∞) is an eigenvalue of problem (5).Define

J1(u) =

∫Ω|∇u|p1(x) dx +

∫Ω|∇u|p2(x) dx, I1(u) =

∫Ω|u|q(x) dx

λ0 = infv∈W1,p1(x)

0 (Ω)\0

I1(v)> 0.

STEP 4: any λ ∈ (0, λ0) is not an eigenvalue of problem (5).Open problem: what about the relationship between λ0 and λ1? Isλ0 = λ1 or λ0 < λ1? In the latter case an interesting questionconcerns the existence of eigenvalues of problem (5) in the interval[λ0, λ1).

J1(u) =

∫Ω|∇u|p1(x) dx +

∫Ω|∇u|p2(x) dx, I1(u) =

∫Ω|u|q(x) dx

0 (Ω)\0

I1(v)> 0.

J1(u) =

∫Ω|∇u|p1(x) dx +

∫Ω|∇u|p2(x) dx, I1(u) =

∫Ω|u|q(x) dx

0 (Ω)\0

I1(v)> 0.

J1(u) =

∫Ω|∇u|p1(x) dx +

∫Ω|∇u|p2(x) dx, I1(u) =

∫Ω|u|q(x) dx

0 (Ω)\0

I1(v)> 0.

J1(u) =

∫Ω|∇u|p1(x) dx +

∫Ω|∇u|p2(x) dx, I1(u) =

∫Ω|u|q(x) dx

0 (Ω)\0

I1(v)> 0.

J1(u) =

∫Ω|∇u|p1(x) dx +

∫Ω|∇u|p2(x) dx, I1(u) =

∫Ω|u|q(x) dx

0 (Ω)\0

I1(v)> 0.

Remark. It is natural to consider in the proof functionals of the type∫Ω|∇u|p(x) dx and

1p(x)|∇u|p(x) dx.

Indeed, for a constant q ∈ (1,∞), the Dirichlet energy integral is∫Ω|∇u|qdx.

If we minimize this functional over all Sobolev functions with givenboundary data, then the corresponding Euler-Lagrange is theq-Laplace equation

div (|∇u|q−2∇u) = 0.

1p(x)|∇u|p(x) dx.

div (|∇u|q−2∇u) = 0.

1p(x)|∇u|p(x) dx.

div (|∇u|q−2∇u) = 0.

1p(x)|∇u|p(x) dx.

div (|∇u|q−2∇u) = 0.

If we change the exponent in the minimization problem from q top(x), we arrive at the energy∫

Ω|∇u|p(x)dx,

with Euler-Lagrange equation div (p(x) |∇u|p(x)−2∇u) = 0.

Replacing q with p(x) in the differential equation instead leads to theminimization of the energy∫

|∇u|p(x)dx,

with Euler-Lagrange equation div (|∇u|p(x)−2∇u) = 0.

If we change the exponent in the minimization problem from q top(x), we arrive at the energy∫

Ω|∇u|p(x)dx,

with Euler-Lagrange equation div (p(x) |∇u|p(x)−2∇u) = 0.

Replacing q with p(x) in the differential equation instead leads to theminimization of the energy∫

|∇u|p(x)dx,

with Euler-Lagrange equation div (|∇u|p(x)−2∇u) = 0.

A compact embedding for anisotropic Sobolev spaces

Denote by→p : Ω→ RN the vectorial function

→p = (p1, ..., pN). We

define W1,→p (·)

0 (Ω), the anisotropic variable exponent Sobolev space,as the closure of C∞0 (Ω) with respect to the norm

‖u‖→p (·) =

N∑i=1

|∂xiu|pi(·) .

If pi ∈ C+(Ω) are constant functions for any i ∈ 1, ..,N, the

resulting anisotropic Sobolev space is denoted by W1,→p

0 (Ω), where→p

is the constant vector (p1, ..., pN). Then W1,→p

0 (Ω) is a reflexiveBanach space for any

→p ∈ RN with pi > 1 for all i ∈ 1, ..,N.

→p = (p1, ..., pN). We

define W1,→p (·)

‖u‖→p (·) =

N∑i=1

|∂xiu|pi(·) .

0 (Ω), where→p

→p = (p1, ..., pN). We

define W1,→p (·)

‖u‖→p (·) =

N∑i=1

|∂xiu|pi(·) .

0 (Ω), where→p

→p = (p1, ..., pN). We

define W1,→p (·)

‖u‖→p (·) =

N∑i=1

|∂xiu|pi(·) .

0 (Ω), where→p

Set→P+,

→P− ∈ RN as

→P+ = (p+

1 , ..., p+N ),

→P− = (p−1 , ..., p

−N ),

and P++, P+

−, P−− ∈ R+ as

P++ = maxp+

1 , ..., p+N , P+

− = maxp−1 , ..., p−N ,

P−− = minp−1 , ..., p−N .

Assume thatN∑

1p−i

and define P?− ∈ R+ and P−,∞ ∈ R+ by

P?− =N∑N

i=1 1/p−i − 1, P−,∞ = maxP+

−,P?− .

TheoremAssume Ω ⊂ RN (N ≥ 3) is a bounded domain with smoothboundary. Assume relation (8) is fulfilled. For any q ∈ C(Ω) verifying

1 < q(x) < P−,∞ for all x ∈ Ω, (10)

then the embedding

W1,→p (·)

0 (Ω) → Lq(·)(Ω)

is continuous and compact.

Important reference: I. Fragalà, F. Gazzola, B. Kawohl, Existenceand nonexistence results for anisotropic quasilinear equations, Ann.Inst. H. Poincaré Anal. Non Linéaire 21 (2004), 715-734.

TheoremAssume Ω ⊂ RN (N ≥ 3) is a bounded domain with smoothboundary. Assume relation (8) is fulfilled. For any q ∈ C(Ω) verifying

1 < q(x) < P−,∞ for all x ∈ Ω, (10)

then the embedding

W1,→p (·)

0 (Ω) → Lq(·)(Ω)

is continuous and compact.

Important reference: I. Fragalà, F. Gazzola, B. Kawohl, Existenceand nonexistence results for anisotropic quasilinear equations, Ann.Inst. H. Poincaré Anal. Non Linéaire 21 (2004), 715-734.

Competition effects in anisotropic equations with variable exponent

Competition effects in anisotropic equations with variableexponent

Consider the problem −N∑

(|∂xiu|

pi(x)−2 ∂xiu)

= λ|u|q(x)−2u

u = 0 on ∂Ω .

TheoremAssume that q ∈ C(Ω) verifies

P++ < min

x∈Ωq(x) ≤ max

x∈Ωq(x) < P?−. (12)

Then for any λ > 0 problem (11) has a nontrivial weak solution.Vicentiu D. Radulescu “Simion Stoilow" Mathematics Institute of the Romanian Academy, Bucharest, Romania

(|∂xiu|

pi(x)−2 ∂xiu)

= λ|u|q(x)−2u

u = 0 on ∂Ω .

P++ < min

x∈Ωq(x) ≤ max

x∈Ωq(x) < P?−. (12)

(|∂xiu|

pi(x)−2 ∂xiu)

= λ|u|q(x)−2u

u = 0 on ∂Ω .

P++ < min

x∈Ωq(x) ≤ max

x∈Ωq(x) < P?−. (12)

TheoremAssume q ∈ C(Ω) satisfies

1 < minx∈Ω

q(x) ≤ maxx∈Ω

q(x) < P−− . (13)

Then there exists λ?? > 0 such that for any λ > λ?? problem (11) hasa nontrivial weak solution.

TheoremAssume q ∈ C(Ω), with

1 < minx∈Ω

q(x) < P−− and maxx∈Ω

q(x) < P−,∞ . (14)

Then there exists λ? > 0 such that for any λ ∈ (0, λ?) problem (11)has a nontrivial weak solution.

TheoremAssume q ∈ C(Ω) satisfies

1 < minx∈Ω

q(x) ≤ maxx∈Ω

q(x) < P−− . (13)

Then there exists λ?? > 0 such that for any λ > λ?? problem (11) hasa nontrivial weak solution.

TheoremAssume q ∈ C(Ω), with

1 < minx∈Ω

q(x) < P−− and maxx∈Ω

q(x) < P−,∞ . (14)

Then there exists λ? > 0 such that for any λ ∈ (0, λ?) problem (11)has a nontrivial weak solution.

CorollaryAssume q ∈ C(Ω) verifies

1 < minx∈Ω

q(x) ≤ maxx∈Ω

q(x) < P−−.

Then there exist λ? > 0 and λ?? > 0 such that for any λ ∈ (0, λ?)and λ > λ?? problem (11) has a nontrivial weak solution.

nonlinear eigenvalue problems for nonhomogenous …¸tiu d. radulescu˘ “simion stoilow"...

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