Kamal OULD BOUH Department of Mathematics,College of Sciences,Taibah University,P.O.Box 30097, Almadinah Almunawwarah,KSA E-mail:kamal bouh@yahoo.fr;hbouh@taibahu.edu.sa

EXISTENCE AND NONEXISTENCE OF SOLUTIONS FOR A HARMONIC EQUATION WITH CRITICAL NONLINEARITY∗

Kamal OULD BOUH Department of Mathematics,College of Sciences,Taibah University,P.O.Box 30097, Almadinah Almunawwarah,KSA E-mail:kamal bouh@yahoo.fr;hbouh@taibahu.edu.sa

This paper is concerned with the harmonic equation(P∓ε):Δu=0,u>0 inwhere Bnis the unit ball in ℝn,n≥4 with Euclidean metric go,∂Bn=Sn_1is its boundary,K is a function on Sn_1and ε is a small positive parameter.We construct solutions of the subcritical equation(P_ε)which blow up at one critical point of K.We give also a sufficient condition on the function K to ensure the nonexistence of solutions for(P_ε)which blow up at one point.Finally,we prove a nonexistence result of single peaked solutions for the supercritical equation(P+ε).

variational problem;critical points;harmonic equation;mean curvature;critical exponent

2010 MR Subject Classification35J20;35J60

1　Introduction

Let us consider the following nonlinear boundary value problem on n-dimensional Riemannian manifold with boundary(M,g),with n≥3:

whereM˚=M∂M denotes the interior of M,Rgis the scalar curvature of M,Kgis the mean curvature of∂M,v is the outward unit vector with respect to the metric g,and c is a constant whose sign is uniquely determined by the conformal structure.Indeed,ifthen the metric¯g has zero scalar curvature and the boundary has a constant mean curvature with respect to¯g.

Escobar studied this problem in[11].He showed that most compact manifolds with boundary admit such conformally related metrics.

In view of the above equation,it is natural to consider the problem of prescribing boundary mean curvature with zero scalar curvature,that is:given a function K:∂M−→ℝ,does thereexists a metric g′conformally equivalent to g such that Rg′≡o and Kg′≡K?From equation (1.1),the problem is equivalent to finding a smooth positive solution v to the following equation,

In this article,we are interested in the case where a non compact group of conformal transformations acts on the equation so that Kazdan-Warner type conditions give rise to obstructions as in the Nirenberg problem(see[15]).The simplest situation is the following one.

Let Bnbe the unit ball in ℝnwith Euclidean metric go.Its boundary will be denoted by∂Bn=Sn−1and will be endowed by the standard metric go.Let K be a function on Sn−1.In this case,our problem becomes

Previously,Cherrier[8]studied the regularity question for this equation.He showed that solution of(P)which are of class H1are also smooth.In[12],Escobar studied this problem(P) on manifolds which are not equivalent to the standard ball.On the ball,sufficient conditions on K in dimensions 3 and 4 were given in[14],and[9],and a perturbative results were obtained in[7].In[1],the authors developed a Morse theoretical approach to this problem in the 4-dimensional case providing some multiplicity results under generic conditions on the function K.

where ε is a small positive parameter.

Our aim,in this paper,is to give sufficient conditions on K such that problem(P∓ε)admits a positive solution.It is easy to see that a necessary condition for solving the problem is that K has to be positive somewhere.Note that some related problems of type(P∓ε),in case of bounded domains,were studied in[4–6,1o,16,17]and the references therein.

Our first result deals with construction of single peaked solutions for the subcritical harmonic problem(P−ε)with ε>o.More precisely,we have

Theorem 1.1Let n≥4 and y be a nondegenerate critical point of K with−ΔK(y)>o. Then,there exists εo>o such that for each ε∈(o,εo),problem(P−ε)has a solution(uε)of the form

with vε∈E(xε,λε)and as ε→o,

The method of this type of theorem was done firstly by Bahri,Li and Rey[3]when they studied an approximation problem of the Yamabe type problem on domains.Many authors used this idea to construct some solutions to other problems.The method becomes standard. Here we will follow the idea of[3]and take account the new estimates since we have another equation than studied in[3].

In the second result,we give a sufficient condition on the function K to ensure the nonexistence of single peaked solutions of(P−ε)with ε>o.

Theorem 1.2Let n≥4 and y be a nondegenerate critical point of K with ΔK(y)>o. Then,there exists εo>o such that for each ε∈(o,εo),problem(P−ε)has no solution(uε)of form(1.3)satisfying(1.4).

Concerning the supercritical case,problem(P+ε)becomes more delicate since we loose the Sobolev embedding which is an important point to overcome.

In contrast with the subcritical case,we have the following nonexistence result for the supercritical problem.

Theorem 1.3Let n≥4 and y be a nondegenerate critical point of K with ΔK(y)o such that for each ε∈(o,εo),problem(P+ε)has no solution(uε)of form(1.3)satisfying(1.4)andis bounded.

Remark 1.4The assumption thatis bounded allows us to recover the Sobolev embedding.It is used to obtain useful estimates in the supercritical case.Note that this assumption is always satisfied in the subcritical case(P−ε).We think that it is still satisfied in the supercritical case but it is too technical to discuss this problem in this paper.

The remainder of this paper is organized as follows.In Section 2,we recall some preliminaries.In Section 3,we give some careful expansions of gradient of the associated variational functional Iεfor(ε>o).Wile Sections 4,5 and 6 are devoted to the proofs of Theorem 1.1, Theorem 1.2 and Theorem 1.3,respectively.

2　Preliminary Results

In this section,we recall the functional setting and the variational problem and its main features.Let p+1=for ε>o,problem(P−ε)occurs as the Euler equation of the variational functional

defined on H1(Bn)equipped with the norm

and〈·,·〉denotes the scalar product defined on H1(Bn)by,

where dvgoand dσgodenote the Riemannian measures on Bnand Sn−1induced by the metric go.

Note that if u is a positive critical point of Iε,then u is a solution of(P−ε),and inversely.

Since the Sobolev embedding H1(Bn)→Lp+1(Sn−1)being not compact,the functional Iεdoes not satisfy the Palais-Smale condition.For this reason,standard variational methods cannot be applied to find critical points of Iε.In order to characterize the sequences failing the Palais-Smale condition,we need to introduce some notations.

We will use the notation x for the variables belonging to the unit ball Bnor to the half spaceWe will also use the notation x=(x′,xn)for

It will be convenient to perform some stereographic projection in order to reduce the abovedenote the completion of,with respect to the Dirichlet norm.The stereographic projection πqthrough an appropriate point q∈Sn−1induces an isometry i:according to the following formula

where x′=(x1,···,xn−1).In particular,we can check that the following relations hold true for every u∈H1(Bn),

In the sequel,we will identify the function K and its composition with the stereographic projection πq.We will also identify a point x of Bnand its image by πq.These facts will be assumed as understood in the sequel.and λ>o,we define the function:

where vois a small positive constant and here,xjdenotes the j-th component of x.Let us define the function

For sake of simplicity,we will writeeδ instead ofeδ(x,λ)and therefore,for m=(α,λ,x,v)∈Mεwe can write u=αeδ+v.

3　Expansions of the Gradient of the Functional Iε

In this Section,we collect some expansions of the gradient of the functional Iεassociated to the problem(P−ε)for ε>o which are needed in Section 4.We start by giving the following remark which is firstly given by Rey[18]when he studied a similar problem in dimension 3.

Remark 3.1Assume that εlogλ small enough.Then for ε>o andeδ(a,λ)defined in+O(ε2log2λ)in Bn.

Now,explicit computations,using Remark 3.1 and the properties of Mε,yield the following propositions

Proposition 3.2For u=αeδ(x,λ)+v,with(α,λ,x,v)∈Mε,we have:

A computation similar to the one performed in[2]shows that

Combining(3.1),(3.2)and(3.3),we easily derive our proposition.□

Proposition 3.3For u=α˜δ(x,λ)+v,with(α,λ,x,v)∈Mε,we have the following expansion:

ProofA computation similar to the one performed in[2],shows that,

Combining(3.1),(3.4)and(3.5),we easily derive our proposition.

Proposition 3.4For u=α˜δ(x,λ)+v,with(α,λ,x,v)∈Mε,we have

ProofA computation similar to the one performed in[2],shows that,

Using(3.1),(3.6)and(3.7),our proposition follows.

4　Proof of Theorem 1.1

Using the Euler-Lagrange’s coefficients,it is easy to get the following proposition.

Proposition 4.1Let m=(α,λ,x,v)∈Mε.m is a critical point of Ψεif and only if u=α˜δ(x,λ)+v is a critical point of Iε,i.e.,if and only if there exists?A,B,C?∈ℝ×ℝ×ℝnsuch that the following holds

The results of Theorem 1.1 will be obtained through a careful analysis of(4.1)–(4.4)on Mε.As usual in this type of problems,we first deal with the v-part of u,in order to show that it is negligible with respect to the concentration phenomenon.The study of(Ev)yields.

Proposition 4.2There exists a smooth map which to any(ε,α,λ,x)such that(α,λ,x,o) in Mεassociates v∈E(x,λ)such that‖v‖

ProofExpanding Iεwith respect to v∈E(x,λ),we obtain

Q(·,·)is a quadratic form positive definite(for more details,see[2]),f(·)is a linear form and R(v)satisfies R(v)=o(‖v‖2),R′(v)=o(‖v‖)and R′′(v)=o(1).

Since Q(v,v)is positive definite,we derive that the following problem

is achieved by a unique functionwhich satisfies‖v‖≤c‖f‖.Using the ideas of[2]we get estimate(4.5).Sinceis orthogonal to the functions{˜δ,∂˜δ/∂λ,∂˜δ/∂xj,j≤n},there exist A, B and C such that

The proposition follows.

Proof of Theorem 1.1Once v is defined by Proposition 4.2,we estimate the corresponding numbers A,B,C by taking the scalar product in H1(Bn)of(Ev)with˜δ,∂˜δ/∂λ and∂˜δ/∂x respectively.Thus we get a quasi-diagonal system whose coefficients are given by

with Γ1,Γ2are positive constants.

The other hand side is given by

Using Proposition 3.2,some computations yield

In the same way,using Proposition 3.3,we get

where Vλis a smooth function satisfying

Lastly,using Proposition 3.4,we have

where Vxis a smooth function such that

Notice that these estimates imply

The solution of the system in A,B and C shows that

This allows us to evaluate the right hand side in equations(Eλ)and(Ex),namely,

where we have used the following estimates

Now,we consider a point y∈Bnsuch that y is nondegenerate critical point of K.We set

where ζ∈ℝ,ξ∈ℝnare assumed to be small and Λ=Λ(y)verifies

With these changes of variables and using(4.7),(Eα)is equivalent to

Now,using(4.9),an easy computation shows that

This implies that(Eλ)is equivalent,while using(4.1o)and(4.14),to

Lastly,using(4.11),(4.12)and(4.15),(Ex)is equivalent to We remark that Vα,Vλand Vxare smooth functions.

This system may also be written

where L is a fixed linear operator on ℝn+1defined by(4.17)and(4.18)and V,W are smooth functions satisfying

Moreover,a simple computation shows that the determinant of L is not equal to zero.Hence L is invertible,and Brouwer’s fixed point theorem shows that(4.19)has a solution(βε,ζε,ξε) for ε small enough,such that

Hence,we have constructed mε=(αε,λε,xε)such thatsatisfies(4.1)– (4.5).Therefore,by Proposition 4.1,uεis a critical point of Iε,i.e.,uεsatisfies

Hence,the proof of Theorem 1.1 is thereby completed.

5　Proof of Theorem 1.2

Arguing by contradiction,suppose that(P−ε)has a solution of form(1.3)and satisfying

(1.4).We start by showing thatIndeed,multiplying(P−ε)by˜δ(xε,λε)and integrating over Bn,we obtain

Consequently by(3.2)and(3.3),we get

where o(1)→o as ε→o.Since αε→K(y)p−1and xε→y as ε→o,we deduce from(5.2)

Next,we estimate vε.Multiplying(P−ε)by vεand integrating over Bn,we obtain

It follows from[2],that there exists a ρ>o,such that

In other hand,expanding the function K around xεand using the Holder’s inequality,we obtain

Combining(5.3)and(5.4),we get

Now,multiplying(P−ε)by λε∂˜δ/∂λεand integrating over Bn,we derive

Arguing as in the proof of Proposition 3.3,we easily arrive at

where we have used the previous estimate of vε,and the fact that

which is a contradiction with the assumption of Theorem 1.2.

6　Proof of Theorem 1.3

Arguing by contradiction,suppose that(P+ε)has a solution as stated in Theorem 1.3.We

As in(3.3),we have

Consequently by(3.2)and(6.2),we obtain

where o(1)→o as ε→o.Since αε→K(y)p−1and xε→y as ε→o,we deduce from(6.3)

Remark 6.1We remark that

(ii)We also point out that it follows from the assumption thatis bounded and→1 as ε→o thatis bounded–a fact which is used in the proof of Lemma 6.2.

Next,we are going to estimate the vε-part of uεin order to show that it is negligible with respect to the concentration phenomenon.Namely,we have the following estimate.

Lemma 6.2The function vεdefined in(1.3),satisfies the following estimate

ProofMultiplying(P+ε)by vεand integrating over Bn,we obtain

Since αε1 as ε→o,it follows from[2],that there exists a positive constant ρ>o independent of ε,such that

Combining(6.5),(6.7)and(6.8),we get estimate(6.4).

Now,we turn to the proof of Theorem 1.3.Multiplying(P+ε)by λε∂˜δ/∂λεand integrating over Bn,we derive

Arguing as in the proof of Proposition 3.3,we easily derive

where we have used the previous estimate of vε,and the fact that

which is a contradiction with the assumption of Theorem 1.3.

AcknowledgementsThe author gratefully acknowledges the Deanship of Scientific Research at Taibah University on material and moral support in the financing of this research project.

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∗June 23,2015;revised November 17,2015.