on a nirenberg-type problem involving the square root of the laplacian

JID:YJFAN AID:6717 /FLA [m1+; v 1.171; Prn:27/08/2013; 17:06] P.1 (1-19)

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Journal of Functional Analysis ••• (••••) •••–•••www.elsevier.com/locate/jfa

On a Nirenberg-type problem involving the square rootof the Laplacian

Wael Abdelhedi b, Hichem Chtioui a,∗

a Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabiab Department of Mathematics, Faculty of Sciences of Sfax, 3018 Sfax, Tunisia

Received 22 January 2013; accepted 5 August 2013

Communicated by C. De Lellis

Abstract

Through an Euler–Hopf-type formula, we establish existence result to a Nirenberg-type problem involv-ing the square root of the Laplacian in sphere S2.© 2013 Elsevier Inc. All rights reserved.

Keywords: Fractional Laplacian; Critical exponent; Critical points at infinity; Nirenberg problem

1. Introduction

Let (Sn, g) be the standard sphere of dimension n, n� 2, and K be a positive function on Sn.

The classical Nirenberg problem is the following: can one change the original metric g confor-mally into a new metric g with prescribed scalar curvature equal to K for n � 3 (prescribedGaussian curvature for n = 2). This problem is equivalent to solving the following nonlinearelliptic equations:

−�g(u) + 1 = Ke2u on S2, (1.1)

Lg(u) = Kun+2n−2 on Sn, n� 3, (1.2)

* Corresponding author.E-mail addresses: [email protected] (W. Abdelhedi), [email protected] (H. Chtioui).

0022-1236/$ – see front matter © 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.jfa.2013.08.005

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2 W. Abdelhedi, H. Chtioui / Journal of Functional Analysis ••• (••••) •••–•••

where Lg := −cn�g + Rg is the conformal Laplacian, �g is the Laplace–Beltrami operator,cn = 4n−1

n−2 and Rg is the scalar curvature associated to the metric g.Recently, several studies have been performed for classical elliptic equations similar to (1.2)

and (1.1) but with the fractional conformal Laplacians instead of the Laplacian. This operatoris introduced first in [30] where Graham, Jenne, Mason and Sparling constructed a sequence ofconformally covariant elliptic operators P

gk , on Riemannian manifolds (M,g) for all positive

integers k if n is odd, and for k ∈ {1, . . . , n/2} if n is even. Moreover, Pg

1 is the well knownconformal Laplacian Lg used in (1.2), and P

g

2 is the well known Paneitz operator. Up to pos-itive constants P

g

1 (1) is the scalar curvature Rg associated to g and Pg

2 (1) is the Q-curvature.Prescribing scalar curvature problem on Sn was studied extensively in last years, see for exam-ple [7,10,11,20–23,33,36] and the references therein. Also, one group of existence results hasbeen obtained for the problem of prescribing Q-curvature on Sn, one can see [2,1,12,24–27,40].

In [19], Chang and Gonzalez generalize the above construction, using the works of Graham–Zworski [31] and Caffarelli–Silvestre [17] for fractional Laplacian on the Euclidean space Rn, todefine conformally invariant operators P

gσ of non-integer order σ . These lead naturally to a frac-

tional order curvature Rg := Pgσ (1), which will be called here σ -curvature. For the σ -curvatures

on general manifolds we refer to [19,29,31] and references therein. Corresponding to the Yamabeproblem, fractional Yamabe problems for σ -curvatures are studied in [28,29,35] and fractionalYamabe flows on Sn are studied in [32].

In this paper, we restrict our attention to the case σ = 12 . As in the Nirenberg problem associ-

ated to Pg

1 and Pg

2 , the question of prescribing 12 -curvature can be formulated as a Nirenberg-type

problem involving the square root of the Laplacian as follows: given a positive function K de-fined on (Sn, g), we ask whether there exists a metric g conformally equivalent to g such that the12 -curvature is equal to K . This problem can be expressed as finding the solution of the followingnonlinear equation with critical exponent.

P 12u = cnKu

n+1n−1 , u > 0, on Sn, (1.3)

where

P 12

= Γ (B + 1)

Γ (B), B =

√−�g +

(n − 1

2

)2

,

cn = P 12(1), Γ is the Gamma function and �g is the Laplace–Beltrami operator on (Sn, g). The

operator P 12

can be seen more concretely on Rn using stereographic projection. The stereographic

projection from Sn \ N to Rn is the inverse of F :Rn → Sn \ N defined by

F(x) =(

2x

1 + |x|2 ,|x|2 − 1

|x|2 + 1

)where N is the north pole of Sn. Then, for all f ∈ C∞(Sn), we have

(P 1

2(f )

) ◦ F =(

2

1 + |x|2)−(n+1)

2

(−�)1/2((

2

1 + |x|2) n+1

2

(f ◦ F)

)(1.4)

where (−�)1/2 is the fractional Laplacian operator (see, e.g., page 117 of [37]).


W. Abdelhedi, H. Chtioui / Journal of Functional Analysis ••• (••••) •••–••• 3

Our aim in this work is to give an Euler–Hopf-type criterion for K to find solutions for prob-lem (1.3) when n = 2, using the method of critical points at infinity (see [4]). Our choice ofthe dimension two is motivated by the special features of the non-compactness in this dimen-sion. In fact, one looking to the possible formations of blow-up points, it comes out that thestrong interaction of the bubbles in dimensions two forces all blow-up points to be single whilein dimension greater or equal to four such any interaction of two bubbles is negligible with re-spect to the self-interaction, while in dimension three there is a phenomenon of balance. Similarphenomena occur with a jump of dimension for the classical Nirenberg problem (1.2) (see [7]and [9]).

In order to state our result, we need the following non-degeneracy condition.

(nd): Assume that K is a smooth function on S2 having only non-degenerate critical pointswith

�K(y) �= 0 whenever ∇K(y) = 0.

Let

K+ = {y ∈ S2, ∇K(y) = 0, −�K(y) > 0

}.

Theorem 1.1. Let K ∈ C1,1(S2) be a positive function satisfying (nd) condition. If

∑y∈K+

(−1)2−i(y) �= 1,

then (1.3) has at least one solution. Here, i(y) denotes the Morse index of K at y.

Our approach to prove Theorem 1.1 extends the topological and dynamical methods de-veloped by Bahri [4] and Bahri–Coron [7] for the classical Nirenberg problem (1.2), to theframework of such fractional Nirenberg-type problem. To perform such an extension, we usea careful analysis of the lack of compactness of the Euler–Lagrange functional J associated toproblem (1.3) (see Section 2 for the definition of J ). Namely, we study the noncompact orbitsof the gradient flow of J , the so-called critical points at infinity (see [4]). These critical points atinfinity can be treated as usual critical points once a Morse lemma at infinity is performed fromwhich we can derive just as in the classical Morse theory, the difference of topology inducedby these noncompact orbits and compute their Morse index. Such a Morse lemma at infinityis obtained through the construction of a suitable pseudo-gradient, for which the Palais–Smalecondition is satisfied along the decreasing flow lines, as long as these flow lines do not enter theneighborhood of a finite numbers of critical points y of K such that −�K(y) > 0.

The remainder of the paper is organized as follows. Section 2 is devoted to some prelimi-nary results related to a Caffarelli–Silvestre method (see [17]), which provides a good variationalstructure to the problem (1.3). In Section 3, we perform an accurate expansion of J and its gra-dient near potential critical points at infinity. In Section 4, we characterize the critical points atinfinity associated to problem (1.3). Section 5 is devoted to the proof of the main result, Theo-rem 1.1, while we give in Section 6 a more general statement.



2. Preliminary results

First, we recall that u ∈ H12 (Sn) is a solution of (1.3) if the identity∫

Sn

P 12uϕ dx = cn

∫Sn

Kun+1n−1 ϕ dx (2.1)

holds for all ϕ ∈ H12 (Sn), where H

12 (Sn) denotes the closure of C∞(Sn) under the norm

‖u‖H

12 (Sn)

=(∫

Sn

P 12uu

)1/2

. (2.2)

Observe also that for u ∈ H12 (Sn), we have u

n+1n−1 ∈ L

2nn+1 (Sn) ↪→ H− 1

2 (Sn). We associate toproblem (1.3) the functional

I (u) = 1

2

∫Sn

uP 12u − n − 1

2n

∫Sn

Ku2n

n−1 , (2.3)

defined in H12 (Sn).

Motivated by the work of Caffarelli and Silvestre [17], several authors have considered anequivalent definition of the operator P 1

2by means of an auxiliary variable, see [17] (see also [13,

15,16,18,38]). In fact, we realize problem (1.3), through a localization method introduced byCaffarelli and Silvestre on the Euclidean space R

n, through which (1.3) is connected to a degen-erate elliptic differential equation in one dimension higher by a Dirichlet to Neumann map. Thisprovides a good variational structure to the problem. By studying this problem with classicallocal techniques, we establish existence of positive solutions. Here the Sobolev trace embeddingcomes into play, and its critical exponent 2∗ = 2n

n−1 .

Namely, let Dn = Sn × [0,∞). Given u ∈ H12 (Sn), we define its harmonic extension U =

Eσ (u) to Dn as the solution to the problem{−�U = 0 in Dn,

U = u on Sn × {t = 0}. (2.4)

The extension belongs to the space H 1(Dn) defined as the completion of C∞(Dn) with the norm

‖U‖H 1(Dn) =(∫

Dn

|∇U |2 dx dt

)1/2

. (2.5)

Observe that this extension is an isometry in the sense that

‖U‖H 1(Dn) = ‖u‖H

12 (Sn)

, ∀u ∈ H12(Sn). (2.6)

Moreover, for any ϕ ∈ H 1(Dn), we have the following trace inequality



‖ϕ‖H 1(Dn) �∥∥ϕ(.,0)

∥∥H

12 (Sn)

. (2.7)

The relevance of the extension function U = Eσ (u) is that it is related to the fractional Laplacianof the original function u through the formula

− limt→0+

∂U

∂t(x, t) = P 1

2u(x). (2.8)

Thus, we can reformulate (1.3) to the following⎧⎨⎩−�U(x, t) = 0 and U > 0 in Dn,

− limt→0+

∂U

∂t(x, t) = KU

n+1n−1 (x,0) on Sn × {0}. (2.9)

The functional associated to (2.9), is given by

I1(U) = 1

2

∫Dn

|∇U |2 dx dt − n − 1

2n

∫Sn

KU2n

n−1 dx, (2.10)

defined in H 1(Dn).Notice that critical points of I1 in H 1(Dn) correspond to critical points of I in H

12 (Sn). That

is, if U = Eσ (u) satisfies (2.9), then u will be a solution of problem (1.3).Let also define the functional

J (U) =‖U‖2

H 1(Dn)

(∫Sn KU

2nn−1 dx)

n−1n

, (2.11)

defined on Σ the unit sphere of H 1(Dn). We set, Σ+ = {U ∈ Σ/U � 0}. Problem (1.3) willbe reduced to finding the critical points of J subjected to the constraint U ∈ Σ+. The exponent

2nn−1 is critical for the Sobolev trace embedding H 1(Dn) → Lq(Sn). This embedding being con-tinuous and not compact, the functional J does not satisfy the Palais–Smale condition, whichleads to the failure of the standard critical point theory. This means that there exist a sequencesalong which J is bounded, its gradient goes to zero and which do not converge. The analysisof sequences failing PS condition can be analyzed along the ideas introduced in [7] and [39]. Inorder to describe such a characterization in our case, we need to introduce some notations.

For a ∈ ∂Rn+1+ and λ > 0, define the function:

δa,λ(x) = cλ

n−12

((1 + λxn+1)2 + λ2|x′ − a′|2) n−12

where x ∈ Rn+1+ , and c is chosen such that δa,λ satisfies the following equation,⎧⎨⎩

�U = 0 and U > 0 in Rn+1+ ,

− ∂U = Un+1n−1 on ∂Rn+1+ .

∂xn+1



Set

δa,λ = i−1(δa,λ),

where i is an isometry from H 1(Dn) to D1,2(Rn+1+ ); the completion of C∞c (Rn+1+ ), with respect

to the Dirichlet norm.For ε > 0 and p ∈N∗, we define

V (p, ε) =

⎧⎪⎪⎨⎪⎪⎩U ∈ Σ s.t. ∃a1, . . . , ap ∈ Sn, ∃α1, . . . , αp > 0 and

∃λ1, . . . , λp > ε−1 with ‖U −∑p

i=1 αiδai ,λi‖ < ε, εij < ε ∀i �= j, and

|J (U)n

n−1 α2

n−1i K(ai) − 1| < ε ∀i, j = 1, . . . , p,

where

εij =(

λi

λj

+ λj

λi

+ λiλj |ai − aj |2) 1−n

2

.

If U is a function in V (p, ε), one can find an optimal representation, following the ideas intro-duced in [5] and [6], namely we have

Lemma 2.1. For any p ∈ N∗, there is εp > 0 such that if ε � εp and U ∈ V (p, ε), then the

following minimization problem

min

{∥∥∥∥∥U −p∑

i=1

αiδ(ai ,λi )

∥∥∥∥∥, αi > 0, λi > 0, ai ∈ Sn

}

has a unique solution (α, λ, a). Thus, we can write U as follows:

U =p∑

i=1

αiδ(ai ,λi ) + v

where v belongs to H 1(Dn) and satisfies the following condition:

(V0) : 〈v,ϕi〉 = 0 for i = 1, . . . , p, and ϕi = δi, ∂δi/∂λi, ∂δi/∂ai,

where δi = δai ,λiand 〈., .〉 denotes the scalar product defined on H 1(Dn) by

〈U,V 〉 =∫Dn

∇U∇V dx dt.

The failure of the Palais–Smale condition can be characterized taking into account the unique-ness result of the corresponding problem at infinity, see e.g. Li–Zhu [34] following the ideasintroduced in [14,39] as follows.



Proposition 2.2. Assume that (1.3) has no solution and let (Uk) ⊂ Σ+ be a sequence satisfyingJ (Uk) → c, a positive number and ∂J (Uk) → 0. Then, there exist an integer p � 1, a positivesequence (εk)k (εk → 0) and an extracted subsequence of (Uk), again denoted by Uk such thatUk ∈ V (p, εk).

Following A. Bahri [4] and [5], we set the following definition and notation.

Definition 2.3. A critical point at infinity of J on Σ+ is a limit of a flow line U(s) of the equation:⎧⎨⎩∂U

∂s= −J (U),

U(0) = U0,

such that U(s) remains in V (p, ε(s)) for s � s0.Using Lemma 2.1, U(s) can be written as:

U(s) =p∑

i=1

αi(s)δ(ai (s),λi (s)) + v(s).

Denoting ai := lims→∞ ai(s) and αi = lims→∞ αi(s), we denote by

(a1, . . . , ap)∞ orp∑

i=1

αiδ(ai ,∞)

such a critical point at infinity.

3. Expansion of the functional and its gradient near potential critical points at infinity

First, we deal with the functional J associated to problem (1.3) for n = 2.

Proposition 3.1. For ε > 0 small enough and U = ∑p

i=1 αiδ(ai ,λi ) + v ∈ V (p, ε), we have thefollowing expansion

J (U) = Sn

∑p

i=1 α2i

(Sn

∑p

i=1 α4i K(ai))

12

[1 − 1

2β1

p∑i=1

α4i

c3�K(ai)

λ2i

logλi

− c1

γ

∑i �=j�1

αiαj εij + 1

γ

(Q1(v, v) − f1(v)

)+ O(‖v‖3)]

+ o

(∑i �=j

εij +p∑

i=1

logλi

λ2i

+ 1

λ2i

),

where



Sn = c4∫R2

dx

(1 + |x|2)2, c1 = c4

∫R2

dx

(1 + |x|2) 32

,

c3 = c4 π

2, c4 = c4

4

∫R2

|x|2 dx

(1 + |x|2)2

a positive constants independent of U and where

Q1(v, v) = ‖v‖2 − 3γ

β1

(p∑

i=1

∫S2

K(αiδi)2v2

),

f1(v) = 2γ

β1

∫S2

K

(p∑

i=1

αiδi

)3

v,

β1 = Sn

(p∑

i=1

α4i K(ai)

), and γ = Sn

(p∑

i=1

α2i

).

Proof. Let us recall that

J (U) = ‖U‖2

(∫S2 KU4)

12

.

We need to estimate

N(U) = ‖U‖2 =∥∥∥∥∥

p∑i=1

αiδi + v

∥∥∥∥∥2

and D2 =∫S2

K(x)U4.

Using the fact that v satisfies (V0), we derive

N(U) =p∑

i=1

α2i ‖δi‖2 +

∑i �=j

αiαj 〈δi, δj 〉 + ‖v‖2. (3.1)

Arguing as in [4], we can easily have the following estimates

‖δi‖2 =∫S2

δi4 dx = c4

∫R2

dx

(1 + |x|2)2= Sn, (3.2)

〈δi, δj 〉 = c1εij + o(εij ). (3.3)

Thus, we derive that

N(U) =p∑

α2i Sn + c1

∑αiαj εij + ‖v‖2 + o

(∑εij

). (3.4)

i=1 i �=j i �=j



For the denominator, we have

∫S2

K

(p∑

i=1

αiδi + v

)4

=∫S2

K

(p∑

i=1

αiδi

)4

+ 4∫S2

K

(p∑

i=1

αiδi

)3

v

+ 6∫S2

K

(p∑

i=1

αiδi

)2

v2 + O(‖v‖3). (3.5)

We also have

∫S2

K

(p∑

i=1

αiδi

)4

=p∑

i=1

α4i

∫S2

Kδ4i + 4

∑i �=j

α3i αj

∫S2

Kδ3i δj

+ O

(∑i �=j

∫S2

δ2i inf(δiδj )

2)

. (3.6)

A computation similar to the one performed in [4] shows that,∫S2

δ2i inf(δiδj )

2 = O(ε2ij log ε−1

ij

), (3.7)

∫S2

Kδ3i δj = c1K(ai)εij + o

(εij + 1

λ2i

), (3.8)

∫S2

Kδ4i = K(ai)Sn + c3�K(ai)

λ2i

logλi + o

(logλi

λ2

). (3.9)

Combining (3.2)–(3.9) and the fact that J (U)2α2i K(ai) = 1 + o(1), for each i, the result fol-

lows. �Proposition 3.2. For any U =∑p

i=1 αiδi ∈ V (p, ε), we have the following expansion,

⟨∇J (U),λi

∂δi

∂λi

⟩= 2J (U)

[−c1

∑j �=i

αjλi

∂εij

∂λi

+ J (U)2 α3i

2

c3�K(ai)

λ2i

logλi

+ o

(p∑

j=1

logλj

λ2j

)]+ o

(p∑

j=1

1

λ2j

+∑j �=i

εij

). (3.10)

Proof. We have

⟨∇J (U),h⟩= 2J (u)

[〈U,h〉 − J (U)2

∫2

KU3h

].

S



Thus,

⟨∇J (U),λi

∂δi

∂λi

⟩= 2J (U)

[⟨p∑

j=1

αj δj , λi

∂δi

∂λi

⟩− J (U)2

∫S2

K

(p∑

j=1

αj δj

)3

λi

∂δi

∂λi

].

Observe that,

∫S2

K

(p∑

j=1

αj δj

)3

λi

∂δi

∂λi

=p∑

j=1

α3j

∫S2

Kδ3j λi

∂δi

∂λi

+ 3∑i �=j

∫S2

K(αiδi)2λi

∂δi

∂λi

(αj δj )

+ O

(∑j �=i

∫S2

δ2j inf(δj , δi)

2)

. (3.11)

A computation similar to the one performed in [4] shows that,⟨δi, λi

∂δi

∂λi

⟩= 0, (3.12)⟨

δj , λi

∂δi

∂λi

⟩= c1λi

∂εij

∂λi

+ o(εij ), (3.13)∫S2

Kδ3i λi

∂δi

∂λi

= −1

4

c3�K(ai)

λ2i

logλi + o

(logλi

λ2i

), (3.14)

∫S2

Kδ3j λi

∂δi

∂λi

= c1K(aj )λi

∂εij

∂λi

+ o(εij ) + o

(1

λn+1

2i λ

n−12

j

), (3.15)

3∫S2

Kδj δ2i λi

∂δi

∂λi

= c1K(ai)λi

∂εij

∂λi

+ o(εij ) + o

(1

λ32i λ

12j

). (3.16)

Using (3.7), (3.11)–(3.16), and the fact that J (U)2α2i K(ai) = 1+o(1), for each i, the proposition

follows. �Proposition 3.3. For any U =∑p

i=1 αiδi ∈ V (p, ε), we have the following expansion,

⟨∇J (U),

1

λi

∂δi

∂ai

⟩= 2J (U)

[−c1

∑j �=i

αj

λi

∂εij

∂ai

− J (U)2c2α3i

∇K(ai)

λi

]

+ O

(∑i �=j

εij +p∑

j=1

1

λ2j

), (3.17)

where c2 = c4 ∫2

dx2 2 .
4 R (1+|x| )



Proof. As in the proof of Proposition 3.2, we have

⟨∇J (U),

1

λi

∂δi

∂ai

⟩= 2J (U)

[⟨p∑

j=1

αj δj ,1

λi

∂δi

∂ai

⟩− J (U)2

∫S2

K

(p∑

j=1

αj δj

)31

λi

∂δi

∂ai

].

Observe that,

∫S2

K

(p∑

j=1

αj δj

)31

λi

∂δi

∂ai

=p∑

j=1

α3j

∫S2

Kδ3j

1

λi

∂δi

∂ai

+ 3∑i �=j

∫S2

K(αiδi)2 1

λi

∂δi

∂ai

(αj δj )

+ O

(∑j �=i

∫S2

δ2j inf(δj , δi)

2)

. (3.18)

A computation similar to the one performed in [4] shows that,⟨δi,

1

λi

∂δi

∂ai

⟩= 0, (3.19)⟨

δj ,1

λi

∂δi

∂ai

⟩= c1

λi

∂εij

∂ai

+ o(εij ), (3.20)∫S2

Kδ3i

1

λi

∂δi

∂ai

= c2∇K(ai)

λi

+ O

(1

λ2i

), (3.21)

∫S2

Kδ3j

1

λi

∂δi

∂ai

= c1K(aj )

λi

∂εij

∂ai

+ o(εij ), (3.22)

3∫S2

Kδ2i

1

λi

∂δi

∂ai

δj = c1K(ai)

λi

∂εij

∂ai

+ o(εij ) + o

(1

λ32i λ

12j

). (3.23)

Combining (3.7), (3.18)–(3.23), and the fact that J (U)2α2i K(ai) = 1 + o(1), for each i, the

proposition follows. �In the rest of this section, we deal with the v-part of U , in order to show that it can be neglected

with respect to the concentration phenomenon.Set

Eε = {v ∈ H 1(Dn)/v satisfying (V0), and ‖v‖ < ε

}.

Proposition 3.4. For any U = ∑p

i=1 αiδi ∈ V (p, ε), there exists a unique v = v(α, a,λ) whichminimizes J (U + v) with respect to v ∈ Eε . Moreover, we have the following estimates:

‖v‖� c

[p∑

i=1

( |∇K(ai)|λi

+ 1

λ2i

)+∑i �=j

εij log ε−1/2ij

].



Before giving the proof of this result, we need the following lemma which is proved in [3] forthe prescribed mean curvature problem and the proof still works in our case here.

Lemma 3.5. The quadratic form Q1(v, v) defined in Proposition 3.1 is a positive definite formin Eε .

Proof of Proposition 3.4. First, we focus on the estimate of v. Let v ∈ Eε where ε is a fixedsmall positive constant depending only on p. Using Lemma 3.5, we derive that

‖v‖ < c‖f1‖ (3.24)

where c is a positive constant. It remains now to estimate ‖f1‖. We have

f1(v) = 2γ

β1

∫S2

K

(p∑

i=1

αiδi

)3

v

=p∑

i=1

2α3i γ

β1

∫S2

Kδ3i v + O

(∑i �=j

2α2i αj γ

β1

∫S2

δ2i inf(δi, δj )|v|

).

Expanding K around ai , we obtain

f1(v) = O

(p∑

i=1

∣∣∇K(ai)∣∣ ∫S2

|x − ai |δ3i |v| + ‖v‖

λ2i

+∑i �=j

∫S2

δ2i inf(δi, δj )|v|

)

� c‖v‖[

p∑i=1

( |∇K(ai)|λi

+ 1

λ2i

)+∑i �=j

εij log ε−1/2ij

]. (3.25)

Using (3.24) and (3.25), the estimate of ‖v‖ follows. �Proposition 3.4 implies the following consequence whose proof is very easy so we omit it.

Corollary 3.6. For the optimal v defined in Proposition 3.4, there is a change of variables

v − v → V

such that

J (U) = Sn�p

i=1α2i

(Sn

∑p

i=1 α4i K(ai))

12

[1 − 1

2β1×

p∑i=1

α4i

c3�K(ai)

λ2i

logλi

− c1

γ

∑i �=j�1

αiαj εij + o

(∑i �=j

εij +p∑

i=1

logλi

λ2i

)]+ ‖V ‖2.



4. Characterization of critical points at infinity

We give here, the characterization of the critical points at infinity associated to problem (1.3)for n = 2. This characterization is obtained through the construction of a suitable pseudo-gradientat infinity for which the Palais–Smale condition is satisfied along the decreasing flow lines.Namely, first, we prove that there are no critical points at infinity in V (p, ε) with p � 2. Second,we construct a special pseudo-gradient W for which the Palais–Smale condition is satisfied alongthe decreasing flow lines, as long as these flow lines do not enter in the neighborhood of criticalpoints yi of K such that �K(yi) < 0. Finally, we characterize the critical points at infinity in thiscase.

Proposition 4.1. For p � 2, there exists a pseudo-gradient W such that there is a constant c > 0independent of u such that:

1. 〈−∇J (U),W 〉� c(∑

i �=j εij +∑p

i=1|∇K(ai)|

λi+ 1

λ2i

),

2. 〈−∇J (U + v),W + ∂v∂(αi ,ai ,λi )

(W)〉 � c(∑

i �=j εij +∑p

i=1|∇K(ai )|

λi+ 1

λ2i

),

for all U =∑p

i=1 αiδi ∈ V (p, ε). Furthermore, |W | is bounded and the λi ’s decrease along theflow lines.

Proof. For the sake of simplicity, we can assume without any inconvenient that λ1 � λ2 �· · ·� λp . Let N = {i/λi |∇K(ai)| � 1} and we set

Z1 =p∑

i=2

2iαiλi

∂δi

∂λi

, Z2 =∑i∈N

1

λi

∂δi

∂ai

∇K(ai)

|∇K(ai)| .

Using Propositions 3.2 and 3.3, we derive that

⟨−∇J (U),Z1⟩� c

∑i �=j

εij + O

(p∑

i=2

logλi

λ2i

)+ o

(logλ1

λ21

)+ o

(∑i �=j

εij

), (4.1)

⟨−∇J (U),Z2⟩� c

∑i∈N

|∇K(ai)|λi

+ O

(∑i �=j

εij

)+ O

(∑i∈N

1

λ2i

). (4.2)

For any critical point y of K , we set μ > 0 such that if d(a, y)� 2μ then |�H(a)| > c > 0. Twocases may occur:

Case 1. λ2 � λ21 or d(a1, y) > μ for any critical point y of K .

In this case, we set W1 = MZ1 + Z2 where M is a large constant. Observe that in the casewhere d(a1, y) > μ, we can appear 1/λ1 in the lower bound of (4.2) and therefore all the 1/λi ’s.Combining (4.1) and (4.2), we derive

⟨−∇J (U),W1⟩� c

(p∑( |∇K(ai)|

λi

+ 1

λ2i

)+∑

εij

). (4.3)

i=1 i �=j



In the other case, that is λ2 � λ21, we can prove easily that 1/λ2

1 = o(ε12). Therefore we can alsoobtain (4.3) in this case.

Case 2. λ2 � λ21 and d(a1, y) � 2μ for any critical point y of K .

In this case, we set Z3 = sign(−�K(y))λ1∂δ1∂λ1

, that is, we increase λ1 if −�K(y) > 0, oth-erwise we decrease it. Observe that

⟨−∇J (U),Z3⟩� c

(1

λ21

+ O

(∑i �=j

εij

)). (4.4)

We define W2 = MZ1 + Z3 + mZ2 where M is a large constant and m is a small constant, wederive that

⟨−∇J (U),W2⟩� cM

∑i �=j

εij + o(1/λ2

1

)+ c

λ21

+ O

(∑i �=j

εij

)+ m

∑i∈N

|∇K(ai)|λi

+ O

(m∑i �=j

εij

)+ O

(m

λ21

)

� c

(p∑

i=1

( |∇K(ai)|λi

+ 1

λ2i

)+∑i �=j

εij

).

The pseudo-gradient W will be built as a convex combination of W1 and W2 and W will satisfythe first estimate. Arguing as in [5], we derive that⟨

−∇J (U + v),W + ∂v

∂(αi, ai, λi)(W)

⟩

�⟨−∇J (U),W

⟩+ o

(p∑

i=1

( |∇K(ai)|λi

+ 1

λ2i

)+∑i �=j

εij

)(4.5)

and therefore Proposition 4.1 follows under (4.5). �Corollary 4.2. Let K ∈ C1,1(S2) be a positive function satisfying (nd) condition. Then for eachp � 2, there are no critical points at infinity in V (p, ε).

Proof. Using Proposition 4.1, we can see that all the λi ’s are bounded in V (p, ε), p � 2. Theflow lines are compact and hence there are no critical points at infinity. �Proposition 4.3. For any U = αδ(a,λ) + v ∈ V (1, ε), there exists a pseudo-gradient W ′ such thatthere is a constant c > 0 independent of u such that:

1. 〈−∇J (U),W ′〉 � c(|∇K(a)|

λ+ 1

λ2 ),

2. 〈−∇J (U + v),W ′ + ∂v∂(α,a,λ)

(W ′)〉 � c(|∇K(a)|

λ+ 1

λ2 ),3. W ′ is bounded,4. the only region where λ increases along the flow lines of W ′ is the region where a is near a

critical point y of K with −�K(y) > 0.



Proof. Let ρ > 0 be such that, for any critical point y of K , if d(x, y) � 2ρ then |�K(x)| >

c > 0. Three cases may occur:Case 1. d(a, y) > ρ for any critical point y. In this case we have |∇K(a)| > c > 0. Set

W1 = 1

λ

∂δ

∂a

∇K(a)

|∇K(a)| .

From Proposition 3.3, we have

⟨−∇J (U),W1⟩� c

|∇K(a)|λ

+ O

(1

λ2

)� c

|∇K(a)|λ

+ c

λ2. (4.6)

Case 2. d(a, y)� 2ρ where y is a critical point of K with −�K(y) < 0. Set

W2 = −λ∂δ

∂λ+ mϕ

(λ∣∣∇K(a)

∣∣)W1

where m is a small constant and ϕ is a C∞ function which satisfies ϕ(t) = 1 if t � 2 and ϕ(t) = 0if t � 1. Using Propositions 3.2 and 3.3, we derive that

⟨−∇J (U),W2⟩� c

λ2+ cm

( |∇K(a)|λ

+ O

(1

λ2

))� c

|∇K(a)|λ

+ c

λ2.

Case 3. d(a, y)� 2ρ where y is a critical point of K with −�K(y) > 0. Set

W3 = λ∂δ

∂λ+ mϕ

(λ∣∣∇K(a)

∣∣)W1.

We obtain the same equality as in Case 2.Hence, W ′ will be built as a convex combination of W1, W2 and W3. The proof of (1) is

thereby completed. Claims (3) and (4) can be derived from the definition of W ′. The claim (2)

can be obtained using the claim (1.3) and arguing as in [5]. �Next, we derive from the above results the characterization of the critical points at infinity.

Corollary 4.4. Let K ∈ C1,1(S2) be a positive function satisfying (nd) condition. The only criti-cal point at infinity of J in V (1, ε) for ε small enough corresponds to

(y)∞ := 1

K(y)12

δ(y,∞),

where y is a critical point of K such that −�K(y) > 0. Moreover, such a critical point at infinityhas a Morse index equal to 2 − ind(K,y).



Proof. Using Proposition 4.3, Corollary 3.6, and arguing as in [5], we can find a change ofvariables when the concentration point is near a critical point y of K such that −�K(y) > 0,leading to the following normal form.

(a,λ, v − v) → (a, λ,→ V ),

such that

J (u) = Sn12

K(a)12

(1 − η�K(a)

λ2log λ

)+ ‖V ‖2,

where η is a small positive constant and v is the optimal value defined in Proposition 3.4.From Proposition 4.3, we know that the only region where λ increases along the pseudo-

gradient W defined in Proposition 4.3, is the region where the concentration point a is near acritical point y of K such that y ∈K+. The above normal form yields a splitting of the variablesa and λ, thus it is easy to see that if a = y, only λ can move. To decrease the functional J ,we have to increase λ, thus we obtain a critical point at infinity only in this case and our resultfollows. �5. Proof of Theorem 1.1

Using Corollary 4.4, the only critical points at infinity associated to problem (1.3) are inone-to-one correspondence with the elements of the set K+. That is,

(y)∞ := 1

K(y)12

δ(y,∞), y ∈ K+.

The Morse index of such critical point at infinity (y)∞ is:

i(y)∞ = 2 − ind(K,y).

Like a usual critical point, it is associated to a critical point at infinity (y)∞, stable and unstablemanifolds, Ws(y)∞ and Wu(y)∞ (see [5, pages 356–357]). Recall that if y is a critical point of K ,then Ws(y) denotes the stable manifold of y defined by the set of all points x ∈ S2 attracted byy through the decreasing flow η(., x) of (−∂K): Ws(y) = {x ∈ S2, s.t. η(s, x) → y when s →+∞} and the unstable manifold of y; Wu(y) = {x ∈ S2, s.t. η(s, x) → y when s → −∞}. Werecall also (see [5]), that in V (1, ε), the unstable manifold Wu(y)∞ for the critical point at infinity(y)∞, along the flow lines of −∂J behaves as Ws(y). We have dimWu(y) = ind(K,y) and

dimWs(y) = 2 − ind(K,y) = i(y)∞ = dimWu(y)∞.

We order the critical values of K :

K(yi )� K(yi ) � · · ·� K(yi ), yi ∈K+, 1 � j � �.
1 2 � j



Let cr = 1

K(yir )12

be the critical value at infinity. For the sake of simplicity, we can assume that

ci ’s are different. Then, we have

b1 < minΣ+ J = c1 < b2 < c2 < · · · < b� < c� < b�+1.

Recall that we already built in the previous section (see Propositions 4.1 and 4.3) two vector fieldsW and W ′ defined in V (p, ε),p � 1. Outside

⋃p�1 V (p, ε

2 ), we will use −∇J and our globalvector field Z will be built using a convex combination of W,W ′ and −∇J . Now, according toCorollary 4.4, there is no critical value above the level b�+1. Let Jc = {U ∈ Σ+/J (u) < c}.

Arguing by contradiction, we suppose that J has no critical points in Σ+. Using the vectorfield Z, we have

Jbr+1∼= Jbr ∪ Wu(yir )∞,

where ∼= denotes retracts by deformation (see Sections 7 and 8 of [8]). Then, denoting by χ theEuler–Poincaré characteristic, we have

χ(Jbr+1) = χ(Jbr ) + (−1)2−i(yir ).

It is easy to see that χ(Jb1) = χ(∅) = 0 and χ(Σ+) = 1, therefore

1 =∑

y∈K+(−1)2−i(yir ) (5.1)

which contradicts the assumption of Theorem 1.1.

6. A general existence result

In the last section of this paper, we give a generalization of Theorem 1.1. Namely instead ofassuming that σ = 1

2 and n = 2, we assume that σ ∈ (0,1) and 2 � n < 2 + 2σ . Following thesame scheme as in Theorem 1.1, we state here an existence result to the following problem,

Pσ u = cn,σ Kun+2σn−2σ , u > 0, on Sn, (6.1)

where

Pσ = Γ (B + 12 + σ)

Γ (B + 12 − σ)

, B =√

−�g +(

n − 1

2

)2

,

cn,σ = Pσ (1), Γ is the Gamma function and �g is the Laplace–Beltrami operator on (Sn, g). Weare now ready to state the following existence result:

Theorem 6.1. Let K ∈ C1,1(Sn) be a positive function satisfying (nd) condition with 2 � n <

2 + 2σ and σ ∈ (0,1). If



∑y∈K+

(−1)n−i(y) �= 1,

then (6.1) has at least one solution. Here, K+ = {y ∈ Sn, ∇K(y) = 0, −�K(y) > 0}.

Proof. The proof goes along with the proof of Theorem 1.1, therefore we will only sketch thedifferences. Keeping the notation of the proof of Theorem 1.1 and regarding the expansion ofthe functional J , we observe that the strong interaction of the bubbles in dimensions n < 2 + 2σ

forces all blow-up points to be single. Thus, the only critical points at infinity associated toproblem (6.1) correspond to

(y)∞ := 1

K(y)n−2σ

n

δ(y,∞), y ∈K+.

The Morse index of such critical point at infinity (y)∞ is: i(y)∞ = n − ind(K,y). Now theremainder of the proof is identical to the proof of Theorem 1.1. �References

[1] W. Abdelhedi, On a fourth-order elliptic equation involving the critical Sobolev exponent: the effect of the graphtopology, Nonlinear Anal. 82 (2013) 82–99.

[2] W. Abdelhedi, H. Chtioui, On the prescribed Paneitz curvature problem on the standard spheres, Adv. NonlinearStud. 6 (2006) 511–528.

[3] W. Abdelhedi, H. Chtioui, The prescribed boundary mean curvature problem on standard n-dimensional ball, Non-linear Anal. 67 (2007) 668–686.

[4] A. Bahri, Critical Point at Infinity in Some Variational Problems, Pitman Res. Notes Math. Ser., vol. 182, LongmanSci. Tech., Harlow, 1989.

[5] A. Bahri, An invariant for Yamabe-type flows with applications to scalar curvature problems in high dimensions,in: A celebration of J.F. Nash Jr., Duke Math. J. 81 (1996) 323–466.

[6] A. Bahri, J.M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of topologyof the domain, Comm. Pure Appl. Math. 41 (1988) 255–294.

[7] A. Bahri, J.M. Coron, The scalar curvature problem on the standard three dimensional spheres, J. Funct. Anal. 95(1991) 106–172.

[8] A. Bahri, P. Rabinowitz, Periodic orbits of hamiltonian systems of three body type, Ann. Inst. H. Poincaré Anal.Non Linéaire 8 (1991) 561–649.

[9] M. Ben Ayed, Y. Chen, H. Chtioui, M. Hammami, On the prescribed scalar curvature problem on 4-manifolds, DukeMath. J. 84 (1996) 633–677.

[10] R. Ben Mahmoud, H. Chtioui, Existence results for the prescribed scalar curvature on S3, Ann. Inst. Fourier 61(2011) 971–986.

[11] R. Ben Mahmoud, H. Chtioui, Prescribing the scalar curvature problem on higher-dimensional manifolds, DiscreteContin. Dyn. Syst. 32 (5) (2012) 1857–1879.

[12] A. Bensouf, H. Chtioui, Conformal metrics with prescribed Q-curvature on Sn, Calc. Var. Partial Differential Equa-tions 41 (2011) 455–481.

[13] C. Brandle, E. Colorado, A. de Pablo, U. Sánchez, A concave–convex elliptic problem involving the fractionalLaplacian, Proc. Roy. Soc. Edinburgh Sect. A 143 (01) (2013) 39–71.

[14] H. Brezis, J.M. Coron, Convergence of solutions of H-systems or how to blow bubbles, Arch. Ration. Mech. Anal. 89(1985) 21–56.

[15] X. Cabré, Y. Sire, Nonlinear equations for fractional Laplacians I: regularity, maximum principles, and Hamiltonianestimates, preprint, arXiv:1012.0867, 2011.

[16] X. Cabré, J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math. 224(2010) 2052–2093.

[17] L. Caffarelli, L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial DifferentialEquations 32 (2007) 1245–1260.

http://refhub.elsevier.com/S0022-1236(13)00310-8/bib6132s1






















http://refhub.elsevier.com/S0022-1236(13)00310-8/bib626E6331s1

http://refhub.elsevier.com/S0022-1236(13)00310-8/bib626E6331s1













[18] A. Capella, J. Davila, L. Dupaigne, Y. Sire, Regularity of radial extremal solutions for some non local semilinearequations, Comm. Partial Differential Equations 36 (8) (2011) 1353–1384.

[19] A. Chang, M. Gonzalez, Fractional Laplacian in conformal geometry, Adv. Math. 226 (2) (2011) 1410–1432.[20] A. Chang, M. Gursky, P. Yang, The scalar curvature equation on 2 and 3 spheres, Calc. Var. 1 (1993) 205–229.[21] C.C. Chen, C.S. Lin, Prescribing the scalar curvature on Sn , I. Apriori estimates, J. Differential Geom. 57 (2001)

67–171.[22] H. Chtioui, Prescribing the scalar curvature problem on three and four manifolds, Adv. Nonlinear Stud. 3 (4) (2003)

457–470.[23] H. Chtioui, R. Ben Mahmoud, D.A. Abuzaid, Conformal transformation of metrics on the n-sphere, Nonlinear

Anal. 82 (2013) 66–81.[24] H. Chtioui, A. Rigan, On the prescribed Q-curvature problem on Sn, J. Funct. Anal. 261 (2011) 2999–3043.[25] Z. Djadli, E. Hebey, M. Ledoux, Paneitz-type operators and applications, Duke Math. J. 104 (2000) 129–169.[26] Z. Djadli, A. Malchiodi, M. Ould Ahmedou, Prescribing a fourth order conformal invariant on the standard sphere,

Part I: a perturbation result, Commun. Contemp. Math. 4 (2002) 375–408.[27] Z. Djadli, A. Malchiodi, M. Ould Ahmedou, Prescribing a fourth order conformal invariant on the standard sphere,

Part II: blow up analysis and applications, Ann. Sc. Norm. Super. Pisa 5 (2002) 387–434.[28] M. Gonzalez, R. Mazzeo, Y. Sire, Singular solutions of fractional order conformal Laplacians, J. Geom. Anal. 22

(2012) 845–863.[29] M. Gonzalez, J. Qing, Fractional conformal Laplacians and fractional Yamabe problems, preprint, arXiv:1012.0579.[30] C.R. Graham, R. Jenne, L.J. Mason, G.A.J. Sparling, Conformally invariant powers of the Laplacian I. Existence,

J. Lond. Math. Soc. 46 (1992) 557–565.[31] C.R. Graham, M. Zworski, Scattering matrix in conformal geometry, Invent. Math. 152 (2003) 89–118.[32] T. Jin, J. Xiong, A fractional Yamabe flow and some applications, J. Reine Angew. Math. (2013), http://dx.doi.org/

10.1515/crelle-2012-0110, in press; arXiv:1110.5664v1.[33] Y. Li, Prescribing scalar curvature on Sn and related topics, Part I, J. Differential Equations 120 (1995) 319–410.[34] Y.Y. Li, M. Zhu, Uniqueness theorems through the method of moving spheres, Duke Math. J. 80 (1995) 383–417.[35] J. Qing, D. Raske, On positive solutions to semilinear conformally invariant equations on locally conformally flat

manifolds, Int. Math. Res. Not. 2006 (2006), Art. ID 94172, 20 pp.[36] R. Schoen, D. Zhang, Prescribed scalar curvature on the n-sphere, Calc. Var. Partial Differential Equations 4 (1996)

1–25.[37] E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Math. Ser., vol. 30, Princeton

University Press, Princeton, NJ, 1970.[38] P.R. Stinga, J.L. Torrea, Extension problem and Harnack’s inequality for some fractional operators, Comm. Partial

Differential Equations 35 (11) (2010) 2092–2122.[39] M. Stuwe, A global compactness result for elliptic boundary value problem involving limiting nonlinearities,

Math. Z. 187 (1984) 511–517.[40] J. Wei, X. Xu, Prescribing Q-curvature problem on Sn , J. Funct. Anal. 257 (2009) 1995–2023.



http://refhub.elsevier.com/S0022-1236(13)00310-8/bib63677A31s1


http://refhub.elsevier.com/S0022-1236(13)00310-8/bib63734Cs1


http://refhub.elsevier.com/S0022-1236(13)00310-8/bib436874696F7569s1






http://refhub.elsevier.com/S0022-1236(13)00310-8/bib444D4F33s1




http://refhub.elsevier.com/S0022-1236(13)00310-8/bib676D7331s1

http://refhub.elsevier.com/S0022-1236(13)00310-8/bib676D7331s1


http://refhub.elsevier.com/S0022-1236(13)00310-8/bib676A6D7331s1

http://refhub.elsevier.com/S0022-1236(13)00310-8/bib676A6D7331s1


http://dx.doi.org/10.1515/crelle-2012-0110





http://refhub.elsevier.com/S0022-1236(13)00310-8/bib535As1

http://refhub.elsevier.com/S0022-1236(13)00310-8/bib535As1

http://refhub.elsevier.com/S0022-1236(13)00310-8/bib537465696Es1

http://refhub.elsevier.com/S0022-1236(13)00310-8/bib537465696Es1






http://dx.doi.org/10.1515/crelle-2012-0110

on a nirenberg-type problem involving the square root of the laplacian

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