Blow-up and nonexistence of sign changing solutions to the Brezis-Nirenberg problem in dimension three

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Blow-up and nonexistence of sign changing solutions to the Brezis–Nirenberg problem in dimension three

^✩

‘Blow-up’ et non-existence des solutions changeant de signe pour le problème de Brezis–Nirenberg en dimension trois

Mohamed Ben Ayed

^a

, Khalil El Mehdi

^b

, Filomena Pacella

^c,∗

aDépartement de Mathématiques, Faculté des Sciences de Sfax, Route Soukra, Sfax, Tunisia bFaculté des Sciences et Techniques, Université de Nouakchott, BP 5026, Nouakchott, Mauritania cDipartimento di Matematica, Universitá di Roma “La Sapienza”, P. le A. Moro 2, 00185, Roma, Italy

Received 28 February 2005; received in revised form 6 June 2005; accepted 7 July 2005 Available online 6 December 2005

Abstract

In this paper we study low energy sign changing solutions of the critical exponent problem(P_λ): −u=u⁵+λuinΩ,u=0 on∂Ω, whereΩis a smooth bounded domain inR³andλis a real positive parameter. We make a precise blow-up analysis of this kind of solutions and prove some comparison results among some limit values of the parameterλwhich are related to the existence of positive or of sign changing solutions.

Résumé

Dans cet article, nous étudions les solutions changeant de signe et à énergie minimale du problème avec exposant critique (P_λ):−u=u⁵+λudansΩ,u=0 sur∂Ω, oùΩest un domaine borné et régulier deR³etλest un paramètre réel strictement positif. Nous faisons une analyse précise du ‘blow-up’ de ce type de solutions et nous prouvons des résultats de comparaisons pour certaines valeurs limites du paramètreλqui sont liées à l’existence des solutions positives ou des solutions changeant de signe.

MSC:35J20; 35J60

Keywords:Blow-up analysis; Sign changing solutions; Nodal domains; Critical exponent

Mots-clés :Analyse du ‘blow-up’ ; Solutions changeant de signe ; Domaines nodaux ; Exposant critique

✩ M. Ben Ayed and F. Pacella are supported by MURST, Project “Metodi Variazionali ed Equazioni Differenziali non lineari”. K. El Mehdi is supported by Istituto Nazionale di Alta Matematica.

* Corresponding author.

E-mail addresses:Mohamed.Benayed@fss.rnu.tn (M. Ben Ayed), khalil@univ-nkc.mr (K. El Mehdi), pacella@mat.uniroma1.it (F. Pacella).

doi:10.1016/j.anihpc.2005.07.001

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1. Introduction

In this paper we study low energy sign changing solutions of the problem known as the Brezis–Nirenberg problem in smooth bounded domains inR³. We get two kind of results. First with simple arguments we prove some comparison results among some limit values of the parameterλwhich are connected with the existence of certain kind of positive or sign changing solutions. Second we make a precise blow-up analysis of sign changing solutions whose energy converges to the value 2S^3/2,Sbeing the best Sobolev constant for the embedding ofH₀¹(Ω)intoL⁶(Ω).

To be more precise we need some notations and recall previous results.

Let us consider the Brezis–Nirenberg problem which is the following elliptic problem with critical nonlinearity −u= |u|²^∗⁻²u+λu inΩ,

u=0 on∂Ω, (1.1)

whereΩ is a smooth bounded domain inRⁿ,n3,λis a real positive parameter and 2^∗=2n/(n−2)is the critical Sobolev exponent for the embedding ofH₀¹(Ω)intoL²^∗(Ω).

About twenty years ago Brezis and Nirenberg in the celebrated paper [8] proved that ifn4 there exists a positive solution of (1.1) for everyλ∈(0, λ1(Ω)),λ₁(Ω)being the first eigenvalue of−onΩ. Such a positive solution is a minimizer of the functional

J_λ(u)=

Ω|∇u|²−λ

Ω|u|² (

Ω|u|²^∗)^2/2^∗ (1.2)

in the spaceH₀¹(Ω). The three-dimensional case is quite different. In this case let us rewrite problem (1.1) as (Pλ)

−u=u⁵+λu inΩ,

u=0 on∂Ω.

Unlike the case of higher dimensions, whenΩ is a ballB inR³Brezis and Nirenberg [8] proved that a positive solution of(Pλ)exists if and only ifλ∈(^λ¹^(B)₄ , λ₁(B)). In view of [1] the positive solution is unique and is indeed the minimum of (1.2). For more general bounded domains they proved that ifΩis strictly starshaped about the origin then, defining

λ0(Ω)=inf

λ∈R|(1.1)has a positive solution

, (1.3)

it resultsλ₀(Ω) >0.

Another important number connected with the existence of positive solutions is the following λ^∗(Ω)=inf

λ∈R| a minimizer for(1.2)exists

. (1.4)

Note that, by Remark 2.3 in Section 2, we have λ^∗(Ω)=inf

λ∈R|(Pλ)has a positive solutionuλwithuλ²λS^3/2 , whereSdenotes the Sobolev constant, that is,S=inf_u_∈_H1

0(Ω), u≡0(u²/u²_L6(Ω)), withu²=

Ω|∇u|²and where u_λ²_λ=

Ω

|∇u|²−λ

Ω

u².

As recalled before, by [8], we have thatλ^∗(B)=λ0(B)=λ1(B)/4 in the case of the ball.

Obviously for general domainsλ0(Ω)λ^∗(Ω)and, as far as we know, is not yet clear in which casesλ0(Ω)= λ^∗(Ω).

A recent interesting result which states the equivalence between the existence of a minimizer for (1.2) and the fact that the infimum is strictly smaller thanSand as well relates the existence of a minimizer to the fact that the regular part of the Green function of the operator(−−λ)becomes negative at some point of the domainΩ is contained in [11].

Concerning the existence of sign changing solutions of (1.1), several results have been obtained if n4. As expected, in this case one can get sign changing solutions for everyλ∈(0, λ1(Ω))or even forλ > λ₁(Ω). For details one can see the papers by Atkinson, Brezis and Peletier [2,3], Clapp and Weth [10] and the references therein.

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Obviously the casen=3 presents the same difficulties enlightened in [8] for positive solutions.

In the first part of this paper, with a simple argument, we prove that there cannot exist sign changing solutions with low energy in correspondence to a value of the parameterλsmaller thanλ^∗(Ω).

More precisely, let us define λ(Ω)¯ =inf

λ∈R|(Pλ)has a sign changing solutionuλwithuλ²λ2S^3/2

. (1.5)

Note that sign changing solutions with energy smaller than 2S^3/2always exist ifΩ is a ball as it can be easily seen by considering the positive solution in the half ball which minimizes (1.2) and extending it by oddness to the whole ball.

It should also be pointed out that solutions to (1.1) are smooth as a consequence of a result due to Brezis and Kato [7] (see also [6]).

Now, we state our comparison results:

Theorem 1.1.Letλbe such that there exists a nontrivial solutionu_λof(Pλ)satisfying:

there exists a connected componentΩ₁ofΩ\ {x∈Ω|u_λ(x)=0}such that

Ω1|∇u_λ|²−λ

Ω1|u_λ|² (

Ω1|u_λ|²^∗)^2/2^∗ S. (1.6)

Then we have λλ^∗(Ω).

In the case of sign changing solutions this theorem essentially claims that ifλ < λ^∗(Ω)there cannot exist a sign changing solutionu_λwith a nodal regionΩ₁in which the energy ofu_λ_|_Ω₁ is smaller than or equal toS^3/2.

Corollary 1.2.We have that λ(Ω)¯ λ^∗(Ω).

Corollary 1.3.Ifλ(Ω)¯ is achieved, then we have an alternative:

either

λ(Ω) > λ¯ ^∗(Ω) or

λ(Ω)¯ =λ^∗(Ω) and henceλ^∗(Ω)is achieved.

In particular, ifΩ is a ballBthen λ(B) > λ¯ ₁(B)/4.

Another result that is obtained with same proof as for Theorem 1.1 is the following

Corollary 1.4.Assume thatΩ is symmetric with respect to the planeT = {x=(x1, x2, x3)∈R³|x1=0}and letD1

be the setD1= {x∈Ω|x1<0}. Then we have λ(Ω)¯ λ^∗(D₁) and λ^∗(D₁)λ^∗(Ω).

Though it applies only to symmetric domains, the result of Corollary 1.4 indicates a kind of monotonicity of the parameterλ^∗(Ω)with respect to the domainΩas it happens for the first Dirichlet eigenvalue of−onΩ. In view of the result of [11] it also gives a relation between the sign of the regular part of the Green function of the operator (−−λ)inΩ and inD₁.

Theorem 1.1 and Corollaries 1.2 and 1.3 are also related to the following question raised by H. Brezis IfΩ is a ballBinR³, could exist sign changing solutions of(Pλ)whenλis smaller thanλ^∗(B)=λ₁(B)/4?

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Obviously the above results give a very partial answer to the above question since they only concern the case of sign changing solutionsu_λwith a nodal regionΩ₁in which the energyu_λ²_λis not bigger thanS^3/2.

Note that ifλ(Ω)¯ was achieved by a sign changing solutionu_λthen either the energy ofu_λ should be less than 2S^3/2 andu_λ should be degenerate otherwise it could be continued to a solutionu_λ,λ<λ(Ω), contradicting the¯ definition ofλ(Ω), or the energy of¯ u_λshould be equal to 2S^3/2.

In the case whenλ(Ω)¯ is not achieved it is possible to prove (see Lemma 4.2 below) that there exists a family of solutionsu_λsuch thatu_λ²→2S^3/2andu_λ0 inH₀¹(Ω), asλ→ ¯λ(Ω), i.e. the solutionsu_λconcentrate. Hence it is an interesting question to analyze the concentration phenomenon of these solutions.

Indeed the second part of the paper is devoted to analyze the behavior of sign changing solutions of(Pλ)which converge weakly to zero and whose energy converges to 2S^3/2 as λ→ ¯λ(Ω). More precisely we prove that these solutions blow-up at two pointsa¯₁anda¯₂which are the limit of the concentration pointsa_λ,1anda_λ,2of the positive and negative part of the solutions. Moreover the distance betweena_λ,1anda_λ,2is bounded from below by a positive constant depending only onΩ and the speeds of concentrations of the positive and negative part are comparable.

We think that these results, whose precise statements are contained in Theorems 3.1 and 4.1 below, are interesting in theirselves and important to face the study of sign changing solutions of problems with critical Sobolev exponent.

They were also difficult to get since we could not always exploit the same arguments used in the study of positive solutions blowing up at two points.

It is also interesting that, once the blow-up analysis is carried out, we can give an alternative proof of Corollary 1.2, in the case whenλ(Ω)¯ is not achieved, which relies on Pohozaev’s identity and on the sign of the regular part of the Green function of(−− ¯λ(Ω))as deduced by the result of [11].

A final comment is that one expects that results analogous to those of Theorems 3.1 and 4.1 below should also hold in higher dimensions and with similar or even simpler proof. Surprisingly our proof only works in dimension 3 because in applying Pohozaev’s identity and getting the convergence of certain integrals the role of the dimension is crucial. We think that other arguments could be used forn4 and a further investigation in this direction is in progress.

The outline of the paper is the following. In Section 2 we prove Theorem 1.1 and its corollaries and we also show a qualitative result which gives the connection between the energy of a solution and the number of its nodal regions.

Section 3 is devoted to state and prove Theorem 3.1. In Section 4 we state and prove Theorem 4.1 and we then give another proof of Corollary 1.2 in the case whenλ(Ω)¯ is not achieved. Finally, Appendix A is devoted to the local blow-up analysis needed in Sections 3 and 4.

2. Proof of Theorem 1.1 and its corollaries

We start by proving Theorem 1.1.

Proof of Theorem 1.1. First, ifu_λdoes not change sign we deduce thatΩ₁=Ω. In this case, it is easy to see that J_λ(u_λ)S.

Assume that inf

u∈H₀¹(Ω), u=0

J_λ(u)=S,

thenu_λis a minimizer forJ_λ which contradicts the result of [11] according to which the infimum is achieved if and only if it is smaller thanS. Thus

inf

u∈H₀¹(Ω), u=0

Jλ(u) < S

and hence again by the result of [11],J_λhas a minimizerv_λ. Thusλλ^∗(Ω).

Now, we analyze the case whenuλchanges sign. In this case, we definewλas w_λ=u_λ in Ω₁, w_λ=0 inΩ\Ω₁.

Without loss of generality, we can assume thatw_λ0. Sinceu_λ=0 on∂Ω₁, we havew_λ∈H₀¹(Ω). Now, multiplying (Pλ)byw_λand integrating onΩ, we obtain

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w_λ²λ:=

Ω

|∇w_λ|²−λ

Ω

w²_λ=

Ω

w_λ⁶. (2.1)

Observe that, because of the assumption (1.6) J_λ(w_λ):= wλ²_λ

(

Ωw_λ⁶)^1/3S. (2.2)

Assume that inf

u∈H₀¹(Ω), u=0

Jλ(u)=S,

thenw_λis a minimizer forJ_λ, which is a contradiction since, by [11], the infimum can only be achieved if it is strictly smaller thanS. Thus

inf

u∈H₀¹(Ω), u=0

J_λ(u) < S

and hence by the result of [11],J_λhas a minimizerv_λ. Thusλλ^∗(Ω). Theorem 1.1 is thereby proved. 2 Next, we are going to give the proofs of Corollaries 1.2–1.4.

Proof of Corollary 1.2. Letλλ(Ω)¯ be such that there exists a sign changing solution u_λ of(Pλ)withu_λ²_λ 2S^3/2. Sinceuλ²_λ= u⁺_λ²_λ+ u⁻_λ²_λ, we have thatu⁺_λ²_λS^3/2oru⁻_λ²_λS^3/2. Then the assumptions of Theo- rem 1.1 are satisfied and thereforeλλ^∗(Ω). Hence the corollary follows. 2

Proof of Corollary 1.3. By Theorem 1.1, we know thatλ(Ω)¯ λ^∗(Ω). Ifλ(Ω)is achieved then as in the proof of Corollary 1.2, we obtainu⁺_¯

λ(Ω)²_λ(Ω)_¯ S^3/2oru⁻_¯

λ(Ω)²_λ(Ω)_¯ S^3/2. So, following the proof of Theorem 1.1, we derive thatJ_λ(Ω)_¯ has a minimizer. Now, if we assume thatλ(Ω)¯ =λ^∗(Ω)we derive thatJλ^∗(Ω)has a minimizer and thereforeλ^∗(Ω)is achieved.

Since whenΩ is a ballBwe know by [8] thatλ^∗(B)=λ₁(B)/4 is not achieved we getλ(Ω) > λ¯ ₁(B)/4. 2 Proof of Corollary 1.4. Letλ > λ^∗(D1)such that there exists a positive solutionu_λof the problem(Pλ)onD1with

D₁|∇uλ|²−λ

D₁u²_λS^3/2. Extendinguλby oddness with respect toT, we can construct a sign changing solution vλof(Pλ)which satisfies

Ω

|∇vλ|²−λ

Ω

v²_λ=2

D₁

|∇uλ|²−λ

D₁

u²_λ

2S^3/2.

Thusλλ(Ω)¯ and therefore

λ^∗(D₁)λ(Ω).¯ (2.3)

Corollary 1.4 follows immediately from (2.3) and Corollary 1.2. 2

We now state and prove a general result on the relation between the energy of a sign changing solution of(Pλ)and the number of its nodal regions.

Proposition 2.1.Assume thatμ < λ₁(Ω)and letu_λbe a sign changing solution of(Pλ)such that u_λ0 asλ→μ.

Then, we have

Ω

|∇u_λ|²kS^3/2 1+o(1) ,

wherekis the number of the connected components ofΩ\Z_λ, withZ_λ= {x∈Ω|u_λ(x)=0}.

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Proof. LetΩ₁be a connected component ofΩ\Z_λ. We observe that −u_λ=u⁵_λ+λu_λ inΩ₁,

u_λ=0 on∂Ω₁. (2.4)

Multiplying (2.4) byu_λand integrating onΩ₁, we obtain

Ω1

|∇u_λ|²=

Ω1

u⁶_λ+λ

Ω1

u²_λ 1 S³

Ω1

|∇u_λ|² 3

+ λ λ1(Ω1)

Ω1

|∇u_λ|², (2.5)

whereλ₁(Ω₁)is the first Dirichlet eigenvalue of−onΩ₁. Notice thatλ1(Ω1)λ1(Ω)and therefore (2.5) implies that

1− λ

λ₁(Ω)

Ω₁

|∇uλ|² 1 S³

Ω₁

|∇uλ|² 3

. Hence

Ω1

|∇u_λ|²c >0 whenλis close toμ, (2.6)

wherecis a positive constant which depends only onμandλ₁(Ω).

On the other hand, sinceuλ0 inΩ asλ→μ, we have that

Ω₁u²_λ→0 and hence 1+o(1)

Ω₁

|∇uλ|² 1 S³

Ω₁

|∇uλ|² 3

. Thus, by (2.6)

Ω₁

|∇u_λ|²S^3/2 1+o(1) . Therefore

Ω

|∇u_λ|²kS^3/2 1+o(1) ,

wherekis the number of the connected components ofΩ\Z_λ. Therefore our proposition is established. 2 Clearly, Proposition 2.1 implies the following:

Corollary 2.2.Assume thatλ(Ω) < λ¯ ₁(Ω)and letu_λbe a sign changing solution of(Pλ)such that

Ω

|∇u_λ|²→2S^3/2 and u_λ0 asλ→ ¯λ(Ω).

Then the setΩ\ {x∈Ω|u_λ(x)=0}has exactly two connected components.

Before ending this section, let us mention the following remark:

Remark 2.3.Let λ(Ω)=inf

λ∈R|(Pλ)has a positive solutionu_λwithu_λ²λS^3/2 , withu_λ²_λ=

Ω|∇u_λ|²−λ

Ωu²_λ. Then we have λ(Ω)=λ^∗(Ω),

whereλ^∗(Ω)is defined by (1.4).

Proof. It follows from the same proof as for the case of Theorem 1.1 whenu_λis a positive solution. 2

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3. Blow-up analysis

This section is devoted to the proof of the following result:

Theorem 3.1.Assume thatλ(Ω) < λ¯ ₁(Ω)and let(u_λ)be a family of sign changing solutions of(Pλ)which satisfies u_λ²:=

Ω

|∇u_λ|²→2S^3/2 and u_λ0 asλ→ ¯λ(Ω).

Then, there exist two local extremum pointsa_λ,1,a_λ,2ofu_λand a positive constantα, which depends only onΩ, such that

u_λ−P δ_(a_λ,1_,μ_λ,1₎+P δ_(a_λ,2_,μ_λ,2₎ →0, asλ→ ¯λ(Ω), μ_λ,i:=3⁻^1/2u_λ(a_λ,i)²→ +∞, asλ→ ¯λ(Ω), fori∈ {1,2},

d(a_λ,i, ∂Ω)α, fori∈ {1,2} and |a_λ,1−a_λ,2|α forλclose toλ(Ω),¯ whereP δ_(a,μ)denotes the projection ofδ_(a,μ)onH₀¹(Ω), that is,

P δ_(a,μ)=δ_(a,μ) inΩ, P δ_(a,μ)=0 on∂Ω, and δ_(a,μ)(x)= 3^1/4μ^1/2 (1+μ²|x−a|²)^1/2.

To prove Theorem 3.1, we need some preliminary results. In the sequel we denote by(uλ)the family of solutions which satisfies the assumption of Theorem 3.1.

Lemma 3.2.We have that (i)

Ω

∇u⁺_λ²→S^3/2,

Ω

∇u⁻_λ²→S³/2 asλ→ ¯λ(Ω), (ii)

Ω

u⁺_λ6

→S^3/2,

Ω

u⁻_λ6

→S^3/2 asλ→ ¯λ(Ω),

whereu⁺_λ =max(uλ,0)andu⁻_λ =max(0,−u_λ).

Proof. Multiplying(Pλ)byu^±_λ and integrating onΩ, we obtain

Ω

∇u⁺_λ²=

Ω

u⁺_λ6

+λ

Ω

u⁺_λ2

and

Ω

∇u⁻_λ²=

Ω

u⁻_λ6

+λ

Ω

u⁻_λ2

. (3.1)

On the other hand, sinceu_λ0 inΩasλ→λ(Ω), we have

Ω

u⁺_λ2

→0 and

Ω

u⁻_λ2

→0 asλ→ ¯λ(Ω). (3.2)

Therefore, arguing as in the proof of Proposition 2.1, we find u⁺_λ²S^3/2 1+o(1)

and u⁻_λ²S^3/2 1+o(1) . Clearly

u_λ²=u⁺

λ²+u⁻

λ²=2S^3/2 1+o(1) .

Therefore claim (i) of Lemma 3.2 follows. Claim (ii) follows from (3.1), (3.2) and claim (i). 2 Lemma 3.3.We have that

M_λ,₊:=max

Ω u⁺_λ → +∞, M_λ,₋:=max

Ω u⁻_λ → +∞ asλ→ ¯λ(Ω).

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Proof. Arguing by contradiction, we assume thatM_λ,₊casλ→λ(Ω). Thereforeu⁺_λ ∈L^∞(Ω)andu⁺_λ →0 a.e.

Thus(u⁺_λ)⁶→0 inL¹(Ω)which contradicts Lemma 3.2. HenceM_λ,₊→ ∞asλ→ ¯λ(Ω).

In the same way, we prove thatM_λ,₋→ +∞asλ→ ¯λ(Ω). 2 Without loss of generality, we can assume in the sequel that

M_λ,₊M_λ,₋. (3.3)

LetΩλ:=M_λ,²₊(Ω−aλ,1), whereaλ,1∈Ω such thatMλ,+=uλ(aλ,1), and we denote byvλ the function defined onΩ_λby

v_λ(y)=M_λ,⁻¹₊u_λ aλ,1+M_λ,⁻²₊y

. (3.4)

It is easy to see thatv_λsatisfies (Qλ)

⎧⎨

⎩

−v_λ=v⁵_λ+λM_λ,⁻⁴₊v_λ inΩ_λ,

v_λ=0 on∂Ω_λ,

v_λ(0)=1, |v_λ|1 inΩ_λ. (Recall that, we have assumedMλ,+Mλ,−.)

Using the assumptions of Theorem 3.1, we see that

Ωλ

|∇v_λ|²=

Ω

|∇u_λ|²→2S^3/2,

Ωλ

|v_λ|⁶=

Ω

|u_λ|⁶→2S^3/2 asλ→ ¯λ(Ω).

Let us prove the following lemma.

Lemma 3.4.We have that

M_λ,²₊d(aλ,1, ∂Ω)→ +∞ asλ→ ¯λ(Ω).

Proof. As in the proof of Lemma 2.3 of [5], we can show that l:= lim

λ→¯λ(Ω)

M_λ,²₊d(aλ,1, ∂Ω) >0.

Arguing by contradiction, we suppose thatl <∞. Then it follows from(Qλ)and standard elliptic theories that there exists some functionv, such thatv_λ→vinC_loc² (Π ), whereΠ is a half space ofR³, andvsatisfies

⎧⎪

⎪⎨

⎪⎪

⎩

−v=v⁵, |v|1 inΠ,

v=0 on∂Π,

v(0)=1, ∇v(0)=0, v²2S^3/2, vL⁶c.

But ifΠis a half space ofR³, by Pohozaev Identity,vhas to vanish identically. Thus, we derive a contradiction and our lemma follows. 2

From Lemma 3.4, we derive that there exists some functionv, such that,vλ→vinC_loc² (R³), andvsatisfies

⎧⎨

⎩

−v=v⁵, |v|1 inR³, v(0)=1, ∇v(0)=0,

v²2S^3/2, vL⁶c.

(3.5) But, we know that ifv is a sign changing solution, thenv²>2S^3/2(see p. 170 of [17]). Thenvhas to be positive and therefore, it follows from [9] that

v(y)=δ_(0,α₀₎(y), withα₀=1/√ 3.

Thus

M_λ,⁻¹₊u_λ a_λ,1+M_λ,⁻²₊y

−δ_(0,α₀₎(y)→0 inC²_loc R³

asλ→ ¯λ(Ω).

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Observe that

M_λ,⁻¹₊u_λ a_λ,1+M_λ,⁻²₊y

−δ_(0,α₀₎(y)=M_λ,⁻¹₊ u_λ(x)−δ_(a_λ,1_,μ_λ,1₎(x) , whereμλ,1=3⁻^1/2M_λ,²₊.

Lemma 3.5.Letu_λ,1(x)=u_λ(x)−P δ_(a_λ,1_,μ_λ,1₎(x). Then we have (i)

Ω

|∇u_λ,1|²=

Ω

|∇u_λ|²−S^3/2+o(1) asλ→ ¯λ(Ω),

(ii)

Ω

|u_λ,1|⁶=

Ω

|u_λ|⁶−S^3/2+o(1) asλ→ ¯λ(Ω).

Proof. We have

Ω

|∇u_λ,1|²=

Ω

|∇u_λ|²+

Ω

|∇P δ_(a_λ,1_,μ_λ,1₎|²−2

Ω

∇u_λ∇P δ_(a_λ,1_,μ_λ,1₎. (3.6)

According to Bahri [4] (see also Rey [14]), we have thanks to Lemma 3.4

Ω

|∇P δ_(a_λ,1_,μ_λ,1₎|²=S^3/2+o(1) asλ→ ¯λ(Ω). (3.7)

We also have

Ω

∇u_λ∇P δ_(a_λ,1_,μ_λ,1₎=

Ω

u_λδ_(a⁵

λ,1,μ_λ,1)=

Ω

(u_λ−δ_(a_λ,1_,μ_λ,1₎)δ⁵_(a

λ,1,μ_λ,1)+

Ω

δ⁶_(a

λ,1,μ_λ,1)

=

Ωλ

(v_λ−δ_(0,α₀₎)δ⁵_(0,α

0)+

Ω

δ⁶_(a

λ,1,μ_λ,1). Using Lemma 3.4 and Bahri [4] (see also Rey [14]), we have

Ω

δ_(a⁶

λ,1,μ_λ,1)=S^3/2+o(1) asλ→ ¯λ(Ω).

We now notice that

R³\B(0,R)

(vλ−δ(0,α₀))δ⁵_(0,α

0)=o(1), forRlarge enough,

and

B(0,R)

(v_λ−δ_(0,α₀₎)δ⁵_(0,α

0)=o(1), because ofv_λ→δ_(0,α₀₎inC_loc² R³ . Then

Ω

∇u_λ∇P δ_(a_λ,1_,μ_λ,1₎=S^3/2+o(1). (3.8)

Therefore claim (i) follows from (3.6), (3.7) and (3.8). The proof of claim (ii) is similar to the proof of claim (i), so we will omit it. 2

Now, let us introduce the following notations:

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h_λ:=max

x∈Ω |x−a_λ,1|^1/2u_λ(x), (3.9)

Zλ:=

x∈Ω|uλ(x)=0

, (3.10)

Ωλ,+:= the connected component ofΩ\Zλwhich containsaλ,1, (3.11)

dλ,1:=d(aλ,1, ∂Ω_λ,₊). (3.12)

Proposition 3.6.We have that h_λ:=max

x∈Ω |x−a_λ,1|^1/2u_λ(x)→ +∞ asλ→ ¯λ(Ω).

Proof. First, we recall that the functionv_λ, defined in (3.4), converges toδ_(0,α₀₎inC_loc² (R³). Therefored_λ,1M_λ,²₊→ +∞asλ→ ¯λ(Ω). Secondly, we are going to prove the following crucial claim:

h_λc⇒d_λ,10 asλ→ ¯λ(Ω), (3.13)

wherecis a positive constant independent ofλ, forλclose toλ(Ω).¯

Arguing by contradiction, we assume thath_λcandd_λ,1→0 asλ→ ¯λ(Ω). We set

w_λ(y)=d_λ,1^1/2u_λ(a_λ,1+d_λ,1y), y∈Ω_λ,₊:=d_λ,1⁻¹(Ω_λ,₊−a_λ,1). (3.14) Observe thatB(0,1)⊂Ω_λ,₊, and we have

|y|^1/2w_λ(y)c, for ally∈B(0,1),

w_λ(0)=d_λ,1^1/2u_λ(aλ,1)=d_λ,1^1/2M_λ,₊→ +∞ asλ→ ¯λ(Ω). (3.15) Therefore 0 is an isolated blow-up point ofwλ(see Appendix A for definition). Notice thatwλsatisfies

−wλ=w⁵_λ+λd_λ,1² wλ, wλ>0 inΩλ,+,

w_λ=0 on∂Ω_λ,₊.

Then it follows from Proposition A.9 in Appendix A that 0 is an isolated simple blow-up point inB(0,1). By Propo- sitions A.6 and A.7 of Appendix A, we know that there exist positive constantsc₁andc₂such that

c₁w_λ(0)1+α²₀w⁴_λ(0)|y|²₋1/2

w_λ(y)c₂w_λ⁻¹(0)|y|⁻¹ fory∈B(0,1/2)\ {0}. (3.16) Observe that

w_λ+V_λw_λ=0, withV_λ=w⁴_λ+λd_λ,1² . Notice that, sinceh_λis bounded, we have

V_λ(y)c for ally∈Ω_λ,₊\B(0,1/4).

By Harnack Inequality (see Corollary 8.21 of [12]), we deduce that, for any compact setKofΩλ,+\ {0}, we have

wλ(y)cKwλ(0)⁻¹ for ally∈K. (3.17)

Now, we set

w_λ(y)=w_λ(0)wλ(y).

It is easy to see thatw_λsatisfies

−w_λ=w_λ(0)⁻⁴w_λ⁵+λd_λ,1² w_λ, w_λ>0 inΩ_λ,₊,

wλ=0 on∂Ωλ,+.

By (3.17),w_λis bounded in any compact set ofΩ_λ,₊\ {0}. It follows from standard elliptic theories thatw_λconverges inC_loc² (Π\ {0})to some positive functionw∈C²(Π\ {0}), whereΠ is the limit domain ofΩ_λ,₊whenλ→ ¯λ(Ω).

Sinced_λ,1→0, we see thatwsatisfies

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−w=0, w0 inΠ\ {0},

w=0 on∂Π.

It follows from (3.16) that 0 is a nonremovable singularity ofw. Therefore

wλ→αGΠ(0,·) inC_loc² (Π\ {0}),

whereG_Π(0,·)is the Green function of the Laplacian operator with Dirichlet boundary condition defined on the limit domain andαis a positive constant. Such a Green function can be written as

G_Π(0, y)= |y|⁻¹−H (0, y),

where by the maximum principleH (0, y) >0.

Applying Pohozaev Identity in the form of Theorem 1.1 of [13], we derive that, for 0< r <1/2 λd_λ,1²

B_r

w²_λ−r 6

∂B_r

w⁶_λ−r 2λd_λ,1²

∂B_r

w²_λ=

∂B_r

B(r, x, wλ,∇wλ)dx, (3.18)

whereB_r=B(0, r)and where B(r, x, u,∇u)=1

2u∂u

∂ν −r

2|∇u|²+r ∂u

∂ν 2

. (3.19)

Multiplying (3.18) byw_λ²(0)and using the homogeneity of the operatorB, we obtain λd_λ,1²

Br

w²_λ−r

6w⁻_λ⁴(0)

∂Br

w_λ⁶−r

2λd_λ,1²

∂Br

w_λ²=

∂Br

B(r, x,w_λ,∇w_λ)dx. (3.20) Using (3.16), we derive that

rw_λ⁻⁴(0)

∂B_r

w⁶_λcw_λ⁻⁴(0)r⁻³→0 asλ→ ¯λ(Ω),

λd_λ,1²

B_r

w²_λcλd_λ,1²

B_r

dy

|y|² cλd_λ,1² →0 asλ→ ¯λ(Ω), λd_λ,1²

∂Br

w²_λcλd_λ,1² →0 asλ→ ¯λ(Ω).

Therefore the left-hand side of (3.20) tends to zero asλ→λ(Ω).

Now, since

w_λ→αG_Π(0,·) inC_loc² (∂B_r)for 0< r <1/2 and forrsmall

G_Π(0, x)= |x|⁻¹−H (0,0)+o |x|

, with|x| =r, we deduce that

lim

λ→¯λ(Ω), r→0

∂B_r

B(r, x,wλ,∇wλ)dx=1

2α²ω3H (0,0) >0,

whereω3is the area of the unit sphere inR³. This leads a contradiction and therefore claim (3.13) follows.

Now, we are going to prove Proposition 3.6. Arguing by contradiction, we suppose that h_λc, withcis a positive constant independent ofλ.

By claim (3.13), we haved_λ,1c >0 asλ→λ(Ω). Observe that|a_λ,1−x_λ,₋|d_λ,1, wherex_λ,₋∈Ω such that

|u_λ(x_λ,₋)| =maxu⁻_λ. Thus, using Lemma 3.3, we obtain

|a_λ,1−x_λ,₋|^1/2u_λ(x_λ,₋)→ +∞ asλ→ ¯λ(Ω).

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This implies thath_λ→ +∞asλ→ ¯λ(Ω). Therefore we obtain a contradiction and our proposition follows. 2 Now, letx_λ,2∈ΩandM_λ,2>0 such that

h_λ= |a_λ,1−x_λ,2|^1/2M_λ,2, whereh_λis defined by (3.9). We set

˜

uλ(y)=M_λ,2⁻¹uλ xλ,2+M_λ,2⁻²y

fory∈Ωλ,2:=M_λ,2² (Ω−xλ,2).

Recall that, for anyx∈Ω, we have

|x−a_λ,1|^1/2u_λ(x)|x_λ,2−a_λ,1|^1/2u_λ(x_λ,2).

Notice that, forx∈B(x_λ,2,|x_λ,2−a_λ,1|/2), we have|x−a_λ,1||x_λ,2−a_λ,1|/2. Hence M_λ,2⁻¹u_λ(x)√

2 for anyx∈B x_λ,2,|x_λ,2−a_λ,1|/2

∩Ω.

Thus, we obtain u˜_λ(y)√

2 for anyy∈B 0, M_λ,2² |x_λ,2−a_λ,1|/2

∩Ω_λ,2. As in Lemma 3.4, we can prove that

M_λ,2² d(x_λ,2, ∂Ω)→ +∞ asλ→ ¯λ(Ω),

and thereforeu˜_λconverges inC_loc² (R³)to some functionϕsuch that

⎧⎪

⎨

⎪⎩

−ϕ=ϕ⁵ inR³, ϕ(0)=1, ϕ(x)√

2, ϕ²2S^3/2.

Thusϕdoes not change sign and therefore two cases may occur:

Case1:u_λ(x_λ,2) >0. In this caseϕ >0 and therefore there existb∈R³andμ >0 such thatϕ=δ_(b,μ). Using Proposition 3.6, we see that

u⁺

λ²2S^3/2+o(1) asλ→ ¯λ(Ω)

which contradicts Lemma 3.2. Thus this case cannot happen.

Case2:u_λ(x_λ,2) <0. Thusϕ <0 and therefore there existb∈R³andμ >0 such thatϕ= −δ_(b,μ). Sincebis a nondegenerate critical point of δ_(b,μ), there existsb_λ→b as λ→ ¯λ(Ω), such that∇u_λ(b_λ)=0 and u˜_λ(b_λ)→ 3^1/4μ^1/2. We see that

˜

uλ+δ_(b_λ_,3−1/2u˜λ(bλ)²)→0 inC_loc² R³

asλ→ ¯λ(Ω).

Thus we have found a second blow-up pointaλ,2ofu_λwith the concentrationμλ,2defined by aλ,2=xλ,2+M_λ,2⁻¹bλ and μλ,2=3⁻^1/2uλ(aλ,2)².

Clearly, we have

∇u_λ(a_λ,2)=0 and

Ω

∇P δ_(a_λ,1_,μ_λ,1₎· ∇P δ_(a_λ,2_,μ_λ,2₎→0 asλ→ ¯λ(Ω).

Therefore, as in Lemma 3.5, we obtain

u_λ−P δ_(a_λ,1_,μ_λ,1₎+P δ_(a_λ,2_,μ_λ,2₎ →0 asλ→ ¯λ(Ω). (3.21) Now, we are in position to prove Theorem 3.1.

Proof of Theorem 3.1. We have already proved the existence of two local extremum pointsa_λ,1,a_λ,2ofu_λsatisfying (3.21) and

μ_λ,i:=3⁻^1/2u_λ(a_λ,i)²→ +∞, μ_λ,id(a_λ,i, ∂Ω)→ +∞ asλ→ ¯λ(Ω), fori∈ {1,2}.