Stability results for some nonlinear elliptic equations involving the <span class="mathjax-formula">$p$</span>-laplacian with critical Sobolev growth

URL:http://www.emath.fr/cocv/

STABILITY RESULTS FOR SOME NONLINEAR ELLIPTIC EQUATIONS INVOLVING THE p-LAPLACIAN WITH CRITICAL SOBOLEV GROWTH

Bruno Nazaret

Abstract. This article is devoted to the study of a perturbation with a viscosity term in an elliptic equation involving the p-Laplacian operator and related to the best contant problem in Sobolev in- equalities in the critical case. We prove first that this problem, together with the equation, is stable under this perturbation, assuming some conditions on the datas. In the next section, we show that the zero solution is strongly isolated in some sense, among the space of the solutions. Actually, we end the paper by giving some analoguous results in the case where the datas present symmetries.

R´esum´e. Dans cet article, l’auteur se propose d’´etudier une perturbation de type visqueux dans une ´equation elliptique contenant l’op´erateur dup-Laplacien, et provenant d’un probl`eme de meilleure constante pour les in´egalit´es de Sobolev dans le cas critique. Nous montrons ici que ce probl`eme, ainsi que l’´equation associ´ee est stable sous cette perturbation, moyennant quelques hypoth`eses sur les donn´ees. Dans la suite, nous prouvons que la solution identiquement nulle est, dans un certain sens, isol´ee dans l’espace des solutions. Enfin, nous terminons cette ´etude par des r´esultats analogues dans le cas o`u les donn´ees poss`edent des sym´etries.

AMS Subject Classification. 35J20, 35J60, 35J70, 49K20.

Received July 10, 1998. Revised December 4, 1998.

1. Introduction

In this paper, we are interested in the stability under a perturbation with a viscosity term of the following nonlinear elliptic PDE’s involving thep-Laplacian with critical Sobolev growth:

( −div

|∇u|^p−2∇u

+a(x)|u|^p−2u=f(x)|u|^p∗−2u

u∈W₀^1,p(Ω). (1)

Here, Ω denotes a bounded open set ofR^N,aandf are smooth on Ω,pis a real in (1, N), andp^∗=N p/(N−p) is the critical exponent for the Sobolev embedding ofW₀^1,p(Ω) inL^q(Ω).

Keywords and phrases: Variational problems, nonlinear elliptic PDE, optimal Sobolev inequalities, best constant problems, p-Laplacian operator.

1ENS Cachan, Antenne de Bretagne, Campus de Ker Lann, 35170 Bruz, France.

2Universit´e de Cergy-Pontoise, D´epartement de Math´ematiques, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise, France.

c EDP Sciences, SMAI 1999

Whenais constant andf ≡1, equation (1) is the Euler equation of the following minimisation problem:

λ(N, p,Ω) = inf

u∈W₀^1,p(Ω),u6=0

R

Ω(|∇u|^p+a|u|^p) R

Ω|u|^p∗_p∗^p · (2)

In other words,λ(N, p,Ω)^−1/pis the first best constant for the embedding ofW₀^1,p(Ω) inL^p∗(Ω) (see Hebey [11]).

In the case where Ω =R^N, the supremum defined by:

K(N, p) = sup

u∈C₀^∞(R^N)

R

R^N|u|^p∗p∗¹

R

R^N|∇u|^p1_p has been computed by Aubin [1] and Talenti [16], and has value:

K(N, p) = p−1 N−p

N−p N(p−1)

¹_p

 Γ(N + 1) Γ

N p

Γ

N+ 1−^Np

ωN−1





1 N

where ωN−1 denotes the volume of the unit sphere in R^N. Furthermore, this supremum is achieved on the following functions:

uµ(x) =

µ+r^p−1p 1−^Np

whereµis a parameter andrthe euclidean norm ofx.

When Ω is arbitrary, Hebey [10], Hebey-Vaugon [8] (in the case wherep= 2), and Demengel-Hebey (in the case wherep >1) study the existence of extremal functions (i.e. which realize the extremum) for problem (1) (see also Lions [13, 14]):

Letaandf beC^∞ functions on Ω. We assume that Ω is bounded and regular. Then, we define λ(Ω) = inf

u∈W₀^1,p(Ω)

R

Ω(|∇u|^p+a(x)|u|^p) R

Ωf(x)|u|^p∗_p∗^p ·

Note that, in this kind of problems, though the infimum value is in general not known, the caseR^N acts as reference, since the authors above show that, ifλ(Ω)< K(N, p)^−pkfk^(p/NL^{∞ )−1}, then this infimum is realized on a positive solution of the following equation:

−div

|∇u|^p−2∇u

+a(x)|u|^p−2u=λ(Ω)f(x)|u|^p∗−2u. (3) By regularity results, such as developped in Guedda-Veron [6, 7], u ∈ C^1,α Ω

, and by the Vazquez strict maximum principle [18],uis positive in Ω. They give also symmetry conditions on Ω and invariance conditions onaandf, which imply the existence’s conditionλ(Ω)< K(N, p)^−pkfk^p/NL^∞−1.

Our aim in this article is to show, forp <2 and under the condition above, that these solutions are stable under some viscosity perturbation. Furthermore, the proof of this result presents the advantage to give another proof of the existence theorem given by Demengel-Hebey.

Remark that, ifuis a solution of (3) and ifR

Ωf(x)|u|^p∗= 1, thenurealizes the infimumλ(Ω). Moreover, since the embedding of W₀^1,p(Ω) intoL^p∗(Ω) is not compact, we can not solve equation (3) by standard variational

arguments. In the perturbed equation, the viscosity term−∆u compensates the loss of compactness and give us a method in order to find a solution.

2. Notations and results

In this paper, Ω will denote aC¹ domain ofR^N, whereN ≥3. Letp∈(1, N) be a real, and leta,f be two C^∞ functions defined on Ω. We are interested in the following problem:

( −div

|∇u|^p−2∇u

+a(x)|u|^p−2u=f(x)|u|^p∗−2u u∈W₀^1,p(Ω)

(4) wherep^∗= _N^{N p}_−p is the critical Sobolev exponent for the embeddingsW^1,p(Ω),→L^q(Ω).

We make the following assumptions:

- the functionais such that the operator L(u) =−div

|∇u|^p−2∇u

+a(x)|u|^p−2u

is coercive, in the sense that there exists a positive constantC such that, for all functionsuinW₀^1,p(Ω), J(u) =

Z

Ω

(|∇u|^p+a(x)|u|^p)≥C Z

Ω

|u|^p.

- the functionf is positive somewhere in Ω (this assumption is necessary, sinceLis coercive).

Demengel and Hebey proved in [4] existence’s results in the case where the data Ω, a, and f present some symmetries (for a Riemannian manifold, this problem is treated by Druet in [3]). Since we are not interested here in finding concrete conditions for the existence of extremal functions, we only considere the general case.

(We make at the end of the paper a brief study of the presence of symmetries.) Then, the result of [4] can be written in the following simplified form: let us define the set

W^p∗=

v∈W₀^1,p(Ω);

Z

Ω

f(x)|v|^p∗= 1

·

Theorem 2.1. Let us suppose that, for allx∈Ωsuch that f(x)>0, we have:

K(N, p)^pλf(x)^1−p/N <1 where

λ= inf

v∈Wp∗J(v). (5)

Then, there exists a solutionuof (4) in W₀^1,p(Ω)∩C^1,α(Ω),α∈(0,1), which is positive. Moreover, urealizes the infimum in (5).

In the first section, we shall prove, in the case where p < 2, a stability result of the positive solutions of equation (4) under some perturbation by a viscous term. More precisely, we consider the variational problem:

λ=R inf

Ωf|v|^p∗=1

2

Z

Ω

|∇v|²+J(v)

(6)

whereis a positive real, which will tend to 0 later. Sincep <2,p^∗ is subcritical for the embedding ofH₀¹(Ω) in L^q(Ω), and then, by standard compactness arguments, this problem admits a non zero solutionu. In addition, u solves the following equation:

−

p∆u−div

|∇u|^p−2∇u

+a(x)|u|^p−2u=µf(x)|u|^p∗−2u (7) whereµ is some Lagrange multiplier.

Then, we prove the following result:

Theorem 2.2. We suppose that the assumptions in Theorem 2.1 hold. Let u be a solution of (6), which is positive. Then, up to a subsequence(u)converges strongly in W₀^1,p(Ω) to a solution uof (4) which belongs to W₀^1,p(Ω)∩C^1,α(Ω),α∈(0,1), is positive in Ω, and realizes the infimum in (5).

In the next section, we are interested in the weak continuity of the set of solutions for equation (4) and we prove the following theorem:

Theorem 2.3. Let(un)be a bounded sequence (in W₀^1,p(Ω)) of solutions of (4) which are non identically zero.

We assume that, at every point xin Ωwheref(x)>0,

K(N, p)^pλf(x)^1−p/N <1 (8)

and that

nlim→∞

Z

Ω

f(x)|un|^p∗=λ^N/p. Then,(un)converges strongly inW₀^1,p(Ω) to a nonzero solutionuof (4).

Finally, in the last section, we present a brief discussion concerning the case where the domain Ω is invariant under the action of a subgroupGof the orthogonal groupON(R^N) and give stability results for positive and nodal solutions (we say nodal for a solution wich changes sign).

3. Stability of positive solutions

Let us consider the following problem:

( −div

|∇u|^p−2∇u

+a(x)|u|^p−2u=f(x)|u|^p∗−2uin Ω

u= 0 on∂Ω, u >0 in Ω (9)

wherep <2.

This has been solved by Demengel-Hebey in [4] for everyp∈(1, N). Our goal here is to study its stability when the operatorL is perturbed by adding to it −∆ and when goes to zero. Since p < 2, the operator L−∆ is smoothingL. We now introduce some notations:

• H^p∗=

v∈H₀¹(Ω);R

Ωf(x)|v|^p∗ = 1 ·

• For≥0, andv∈H₀¹(Ω),

J(v) = 2

Z

Ω

|∇v|²+ Z

Ω

(|∇v|^p+a(x)|v|^p) (10) λ = inf

v∈Hp∗J(v). (11)

The functional J represents the energy functional for the operator−∆ +L and the real λ is the minimal energy under the condition R

Ωf|v|^p∗ = 1 (we shall see later that the real λis also the minimal energy of the initial problem).

3.1. The perturbed problem

Before studying the perturbed equation, let us note that:

Lemma 3.1. λ=λ0

Proof. We obviously haveλ≤λ0. To prove the reverse inequality, we first state that Hp^∗

W₀^1,p(Ω)

=Wp^∗. Indeed, letv∈W₀^1,p(Ω) such thatR

Ωf(x)|v|^p∗ = 1. Then, there exists a sequence (vn)n∈Nof functions inH₀¹(Ω) such that vn converges tov in W₀^1,p(Ω), almost everywhere in Ω and, due to the Sobolev embedding theorem, inL^p∗(Ω). From this, we deduce thatR

Ωf(x)|vn|^p∗−→R

Ωf(x)|v|^p∗= 1. Defining, fornlarge enough in order to have R

Ωf|u|^p∗1p∗

< ¹₂,

wn = vn

R

Ωf(x)|vn|^p∗_p∗¹ , one clearly hasJ0(wn)−→J0(v),i.e.

∀η >0,∃N ∈N, n > N=⇒ |J0(wn)−J0(v)|< η, and consequently

λ0≤J0(wn)< J0(v) +η.

This inequality being true for arbitraryv andη, the proof is completed. We now give the main result of this section.

Proposition 3.1. Let >0be given. Then, there existsu∈ H^p∗,u≥0a.e. inΩ, which is a solution of the minimization problem (11). Furthermore,u is a weak solution of the equation:

−

p∆u−div

|∇u|^p−2∇u

+a(x)|u|^p−2u=µf(x)|u|^p∗−2u (12) where

µ=λ+ 1

p−1 2

Z

Ω

|∇u|².

Proof. To prove the existence of u, let (vn)n∈N be a minimizing sequence forJ. Since |∇|vn|| =|∇vn|, one can assume that vn is nonnegative, for alln∈N. Using the coercivity ofL, one gets that (vn) is bounded in H0¹(Ω). From Banach-Aloaglu and Rellich-Kondrakov theorems, there exists a subsequence, still denoted (vn), and a function u inH0¹(Ω), such that:

• vn* u inH₀¹(Ω) (and then inW₀^1,p(Ω));

• vn−→u inL^p∗(Ω);

• vn−→u a.e. in Ω;

whereu∈H₀¹(Ω).

By the last two assertions, one gets that u ≥0 a.e. in Ω, and that R

Ωf|u|^p∗ = 1. By the lower semi- continuity of the norms inW₀^1,p(Ω) andH₀¹(Ω), one obtains

J(u)≤lim inf

n→∞J(vn) =λ(G)

and thenu solves (11). We now prove thatu is a weak solution of equation (12).

Letv∈ D(Ω) be given. Then, for every realt small enough, J



 u+tv R

Ωf|u+tv|^p∗p∗¹



≥J(u).

By expanding the left hand side in powers oftto the first order, one gets:

t

p Z

Ω∇u∇v + Z

Ω|∇u|^p−2∇u∇v+ Z

Ω

a(x)|u|^p−2uv

≥t Z

Ω

f(x)|u|^p∗−2uv

×

p Z

Ω|∇u|²+ Z

Ω

(|∇u|^p+a(x)|u|^p)

+O(t²).

It follows that for every functionv inD(Ω),

p Z

Ω

∇u∇v+ Z

Ω

|∇u|^p−2∇u∇v+ Z

Ω

a(x)|u|^p−2uv=µ

Z

Ω

f(x)|u|^p∗−2uv (13) withµ=λe+

1 p−¹2

R

Ω|∇u|². This completes the proof.

In order to prove the stability of problem (4) under the perturbation defined in (7), we need some further results on the behaviour of the sequences (u) and (µ).

Proposition 3.2. The sequence(λ)tends to λasgoes to0.

Proof. Let > η > 0 be given. Then, ∀v ∈ H^p∗, Jη(v) < J(v). Thus, λη ≤ λ. We derive from this that (λ) has a limit asgoes to 0. Moreover, by Lemma 3.1, this limit is greater thanλ. Let us prove the reverse inequality.

Letδ >0 be given. As we remarked in the proof of Lemma 3.1, there existsvδ ∈ H^p∗ such that, λ≤J0(vδ)≤λ+δ.

Hence,

λ≤J(vδ)< λ+δ+ 2

Z

Ω

|∇vδ|². δbeing arbitrary, we letgo to 0, and obtain

lim→0λ≤λ+δ which ends the proof.

Now, we give a strong convergence result concerning the perturbation term.

Proposition 3.3. Let(u)be a sequence of solutions given by Proposition 3.1. Then,(√

u)converges strongly to0in H₀¹(Ω).

Proof. For > η >0 being given, one has

λ−λη≥J(u)−Jη(u)≥−η 2

Z

Ω

|∇u|².

By two repeated applications of Proposition 3.2 above, letting first η, and next go to 0, one obtains the result.

Corollary 3.4. The sequence(µ)tends to λasgoes to0.

3.2. Convergence of the perturbed problem

We shall get a solution of (4) by extracting subsequences from the initial sequence (u)>0. By the way, the coercivity of the operatorL, together with the convergence of (λ), imply that the sequence (u) is bounded in W₀^1,p(Ω). Then, up to a subsequence, it converges weakly in this space. The main difficulty here is to prove that the limit is not identically zero. This will be done in the next section.

Proposition 3.5. Let us suppose that there exists a subsequence of (u) which converges weakly in W₀^1,p(Ω) to some function u6≡0. Then, ubelongs to W₀^1,p(Ω)∩C^1,α(Ω), for all α∈(0,1), and u is a solution of (4).

Furthermore,urealizes the infimum (5).

Proof. Up to a subsequence, one can assume that:

• u* uweakly inW₀^1,p(Ω);

• u−→uinL^k(Ω),∀k < p^∗;

• u−→ua.e. in Ω.

It turns out thatu≥0 a.e. in Ω.

In addition, the sequence (|∇u|) is bounded inL^p(Ω), so

|∇u|^p−2∇u

is bounded inL^p0(Ω), wherep⁰ is the H¨older conjugate of p. Hence, there exists Σ∈L^p0(Ω), such that, up to a subsequence,

• |∇u|^p−2∇u*Σ, weakly inL^p0(Ω).

Since (√

u) converges strongly to 0 inH₀¹(Ω), passing to limit in (12), one obtains

−div (Σ) +a(x)|u|^p−2u=λf(x)|u|^p∗−2u.

Moreover, from (12), div

∇u+|∇u|^p−2∇u

is bounded inL

p∗−1p∗ (Ω), and then inL¹(Ω). Using Lemma 3.2 below, one obtains that Σ =|∇u|^p−2∇u, and thatuis a solution of

−div

|∇u|^p−2∇u

+a(x)|u|^p−2u=λf(x)|u|^p∗−2u.

By regularity results, such as developped in Guedda-Veron [6], Tolksdorf [17], and by the Vazquez strict max- imum principle [18], one gets thatubelongs to C^1,α(Ω), for all α∈(0,1) and thatu >0 in Ω. Furthermore, multiplying the equation byuand integrating over Ω, one can see thatλandR

Ωf|u|^p∗are positive. Multiplying ubyλ^N−pp2 , one obtains a solution of (4).

Lemma 3.2. LetΩbe a bounded domain of R^N,pa real strictly larger than1, and (u)a sequence ofH₀¹(Ω).

We assume that:

1. √

u−→0strongly inH₀¹(Ω).

2. (u)_>0 is bounded inW₀^1,p(Ω).

3.

div

p∇u+|∇u|^p−2∇u

>0is bounded in L¹(Ω).

Then, there existsu∈W₀^1,p(Ω)such that, up to a subsequence,(u)>0converges a.e. to u,(∇u)_>0 converges a.e. to∇u,

|∇u|^p−2∇u

>0 converges a.e. and weakly in L^p0(Ω) to|∇u|^p−2∇u(In fact, (∇u) converges strongly to∇uinL^q(Ω), for all q < p).

Proof. The proof is based on Evans ([9] Th. 3, Chap. 4) and Courilleau-Demengel’s arguments ([5] Prop. 3.1).

From the boundedness in L^p0(Ω) of

|∇u|^p−2∇u

, one obtains that, up to a subsequence,

• √

∇u−→0 a.e. in Ω;

• u−→ua.e. in Ω;

• Σ=|∇u|^p−2∇u*Σ weakly inL^p0(Ω);

• u* uweakly inW₀^1,p(Ω).

Let δ > 0 be given. By Egoroff’s theorem, there exists a universally measurable set Eδ ⊂⊂ Ω such that meas (Ω\Eδ) < δ, and u (respectively √

∇u) tends to u (respectively 0) uniformly in Eδ. This implies in particular that√

∇u−→0 strongly inL^q(Eδ), for allq≤ ∞.

Now, letη >0 be given. By the uniform convergence inEδ, there exists0>0 such that

< 0 =⇒ ∀x∈Eδ,|u(x)−u(x)|< η 2

·

Let us consider the following cut-off functionβη: βη(t) =

t if |t| ≤η η_|^t_t| if |t|> η

and defineΣ =e |∇u|^p−2∇u(eΣ∈L^p0(Ω)). Sinceβη is piecewiseC¹ and continuous, one has thatβη◦(u−u)

∈W^1,p(Ω). Furthermore, one can easily see that

Σ−Σe

· ∇(βη◦(u−u))≥0 dans Ω, since∇(βη◦(u−u)) =∇(u−u) for < 0 in Eδ. Now consider the integral

Z

E_δ

Σ−Σe

· ∇(u−u)

(x)dx= Z

E_δ

p∇u+ Σ−Σe

· ∇(u−u)

(x)dx−Z

E_δ

p(∇u· ∇(u−u)) (x)dx.

Writing the second integral on the right hand side as

− p

Z

E_δ

(∇u· ∇(u−u)) (x)dx=1 p

−k√

uk²L²(Eδ)+ Z

E_δ

(∇u· ∇u) (x)dx

one sees that it goes to 0 when goes to 0, since √

u converges to 0 inL^p0(Eδ) and in L²(Eδ). Now, let us treat the first integral as follows:

Z

E_δ

p∇u+ Σ−Σe

· ∇(u−u)

(x)dx= Z

Ω

p∇u+ Σ−Σe

· ∇(βη◦(u−u))

(x)dx

=− Z

Ω

div

p∇u+ Σ

(βη◦(u−u))

(x)dx

−Z

Ω

Σe· ∇(βη◦(u−u)) (x)dx.

On the one hand,βη◦(u−u) converges weakly inW₀^1,p(Ω), hence

lim→0

Z

Ω

Σe· ∇(βη◦(u−u))

(x)dx= 0.

On the other hand, by point 3, there existsC >0 such that, for all >0, −Z

Ω

div

p∇u+ Σ

(βη◦(u−u))

(x)dx ≤η

Z

Ω

div

p∇u+ Σ

(x)

dx≤Cη.

Consequently, one has that

lim→0sup Z

Eδ

Σ−Σe

· ∇(u−u)

(x)dx≤Cη.

Since this is true for allη, one gets that

Σ−Σe

· ∇(u−u) converges a.e. to 0 on Eδ. Using Lemma 3.3 below, one has that (∇u) converges a.e. to ∇uin Eδ, for all δ >0, and δ being arbitrary,∇u tends to ∇u almost everywhere in Ω. This implies in particular that Σ converges toΣ a.e., and, since (Σe ) is bounded in L^p0(Ω), weakly inL^p0(Ω). Finally, Σ =Σ.e

It could be easily derived from Egoroff’s theorem that the convergence of (∇u) to ∇uholds also in every L^q(Ω) spaces, for q < p.

To complete the proof, we give Lemma 3.3, which may be found in [4].

Lemma 3.3. Letpbe in (1,∞), and let(Xk)be a sequence of R^N andX ∈R^N, such that

klim→∞

|Xk|^p−2Xk− |X|^p−2X

(Xk−X) = 0.

Then, limk→∞Xk=X. 3.3. Localisation method

We have proved in Section 3.2 that if (u) converges weakly inW₀^1,p(Ω) to a functionuwhich is non identically zero, then this limit is a positive solution of (4). We denote by (H1) this assumption.

Let (H2) be the following condition:

K(N, p)^pλ|f(x)|^1−p/N <1 at every pointxin Ω, wheref(x)>0.

We prove in this section that (H1) follows from (H2). For that aim, we adapt the isometry-concentration method used by Hebey [10] and Demengel-Hebey [4] (see also [3] and [12]).

In what follows, we assume that (u) converges weakly to 0 inW₀^1,p(Ω).

LetP ∈Ω,δ >0, andη∈C₀^∞(R^N), 0≤η≤1, such that:

η(x) =

1 if x∈BP(δ/2) 0 if x6∈BP(δ).

Let us multiply equation (12) byη^pu and integrate over Ω. One obtains

p Z

Ω

∇u· ∇(η^pu) + Z

Ω

|∇u|^p−2∇u· ∇(η^pu) +a(x)|u|^pη^p

=µ

Z

Ω

f(x)|u|^p∗η^p. (14) On the one hand,

p

Z

Ω∇u· ∇(η^pu) = p

Z

Ω

η^p|∇u|²+ Z

Ω

η^p−1u∇u· ∇η=o(1) (→0).

On the other hand,

Z

Ω

a(x)|u|^pη^p=o(1) (→0)

sinceu−→0 inL^p(Ω). Let us now treat the second term in the left-hand side of (14):

Z

Ω|∇u|^p−2∇u· ∇(η^pu) = Z

Ω

η^p|∇u|^p+p Z

Ω

η^p−1|∇u|^p−2u∇u· ∇η. (15) First, one has

p Z

Ω

η^p−1u|∇u|^p−2∇u· ∇η

≤ kηk^p_∞⁻¹k∇ηk∞

Z

Ω

u|∇u|^p−1

≤ kηk^p_∞⁻¹k∇ηk∞

Z

Ω

|u|^p

1/pZ

Ω

|∇u|^p 1−1/p

=o(1) (→0). (16) (Here, K denotes a generic positive constant.)

In addition,

Z

Ω

|∇(ηu)|^p = Z

Ω

|η∇u+u∇η|^p. (17)

TakingX =u∇η andY =η∇u in the following inequality, valid for vectors in R^N:

||X+Y|^p− |Y|^p| ≤p |X|^p−1+|Y|^p−1

|X| one gets

p Z

Ω

|X|^p = p Z

Ω

|u|^p|∇η|^p=o(1) (→0) (18) p

Z

Ω

|Y|^p−1|X| = p Z

Ω

η^p−1u|∇η| |∇u|^p−1=o(1) (→0) (19) by (16). Using (15–18), and (19), one finally gets

Z

Ω

|∇(ηu)|^p+o(1) = Z

Ω

µf(x)|u|^p∗η^p.

Assume first thatf(P)<0. Then, by choosingδsmall enough, one has that f is negative onBP(δ). Hence

lim→0

Z

Ω

|∇(ηu)|^p= 0.

Assume now thatf(P)≥0. Writing that µ

Z

Ω

f(x)u^p∗η^p ≤ µ sup

B_P(δ)|f|^p/p∗Z

Ω

|ηu|^p|f(x)|^1−p/p∗|u|^p∗−p

≤ µ sup

B_P(δ)|f|^p/p∗ Z

Ω

|ηu|^p∗

p/p^∗ Z

BP(δ)

|f(x)||u|^p∗

!1−p/p^∗

and using the definition ofK(N, p), one has that Z

Ω

|ηu|^p∗

≤K(N, p)^p Z

Ω

|∇(ηu)|^p.

Thus,

µ

Z

Ω

f(x)|u|^p∗η^p≤µ sup

B_P(δ)|f|^p/p∗K(N, p)^p Z

B_P(δ)|f(x)||u|^p∗

!1−p/p^∗Z

Ω|∇(ηu)|^p. Now, iff(P) = 0, by choosingδsmall enough we have

µ sup

B_P(δ)|f|^p/p∗K(N, p)^p Z

BP(δ)

|f(x)||u|^p∗

!1−p/p^∗

<1.

Hence,

elim→0

Z

Ω

|∇(ηu)|^p = 0.

In the same manner, iff(P)>0, one can chooseδsmall enough so thatf is positive inBP(δ). Now, assuming that

λK(N, p)^pf(P)^1−N/plim

→0

Z

BP(δ)

|f(x)||u|^p∗

!1−p/p^∗

<1 once more we get

lim→0

Z

Ω

|∇(ηu)|^p= 0.

We have obtained the following result

Lemma 3.4. Assume that (H1) does not hold and that, for every pointP inΩsuch thatf(P)>0, there exists some realδP >0satisfying

λK(N, p)^pf(P)^1−p/Nlim

→0

Z

BP(δP)

|f(x)||u|^p∗

!1−p/p^∗

<1.

Then,

lim→0

Z

Ω

|∇(ηPu)|^p= 0

whereηP denotes the cut-off function defined at the beginning of the section.

We can now prove that (H1) follows from (H2). According to the assumptions of Lemma 3.4, for allx∈Ω, there existsδx>0, and a cut-off functionηx, verifying

lim→0

Z

Ω

|∇(ηxu)|^p= 0.

By the compactness of Ω, one can find a finite number of points (xi)1≤i≤k and reals (δx_i)1≤i≤k such that Ω =

[k i=1

Bxi(δxi).

By convexity, one gets

Z

Ω|∇u|^p≤k^p−1 Xk

i=1

Z

Ω|∇(ηxiu)|^p thus,

lim→0

Z

Ω

|∇u|^p= 0 which contradictsu∈ H^p∗, sinceR

Ωf|u|^p∗= 1,∀ >0.

Consequently, there exists P∈Ω, such thatf(P)>0, which verifies

∀δ >0, K(N, p)^pλf(P)^p/p∗lim

→0

Z

B_P(δ)|f(x)||u|^p∗

!1−p/p^∗

≥1.

Furthermore, one can chooseδsmall enough in order to have 1≥

Z

B_P(δ)|f(x)||u|^p∗+ Z

f≤0

f(x)|u|^p∗.

The previous computations proves that (u) converges strongly in L^p∗ in the neighbourhood of every point wheref is nonnegative, and then, one obtains that

lim→0

Z

BP(δ)

|f(x)||u|^p∗≤1.

This contradicts (H2), and then Lemma 3.4 implies that the limit ucannot be identically zero. To complete the proof of Theorem 2.2, it remains to show that (a subsequence of) (u) converges strongly to a solutionu.

Proposition 3.6. Up to a subsequence,(u)converges strongly touinW₀^1,p(Ω)as tends to0.

Proof. We have proved before that every subsequence of (u) which converges weakly inW₀^1,p(Ω) has a limit u which is positive in Ω. Let us show that, in fact, the convergence is strong. For that aim, it is sufficient to show that the L^p-norm of the gradient (∇u) converge tok∇uk^Lp. According to Proposition 3.5,uis a solution of

−div

|∇u|^p−2∇u

+a(x)|u|^p−2u=λf(x)|u|^p∗−2u then, multiplying byuand using Green’s formula, one gets

Z

Ω

(|∇u|^p+a(x)|u|^p) =λ Z

Ω

f(x)|u|^p∗. First, let us prove thatλ=J0(u).

Setv= R

f(x)|u|^p∗−1/p^∗

u. The equation above writes Z

Ω

f(x)|u|^p∗ p/p^∗Z

Ω

(|∇v|^p+a(x)|v|^p) =λ Z

Ω

f(x)|u|^p∗.

Sincev∈ W^p∗,J0(v)≥λ, and then

Z

Ω

f(x)|u|^p∗≥1.

Furthermore, sinceu−→uin W₀^1,p(Ω), one can write λ

Z

Ω

f(x)|u|^p∗= Z

Ω

(|∇u|^p+a(x)|u|^p)≤lim inf

→0 Z

Ω

|∇u|²+ Z

Ω

(|∇u|^p+a(x)|u|^p)≤lim inf

→0λ ≤ λ which gives the result. Actually, J(u) tends toJ0(u), then

lim→0

Z

Ω

|∇u|^p= Z

Ω

|∇u|^p

and the proof of Theorem 2.2 is completed.

4. A continuity result

In this section, we are interested in the continuity of the set of the solutions for equation (4). Here, we prove thatu≡0 is isolated among the solutions of (4), in some sense which will be precised later. First, we give a result concerning the isolation of 0 in a strong sense.

Lemma 4.1. Letu∈W₀^1,p(Ω) be a solution of (4).

•If R

Ωf(x)|u|^p∗= 0, thenu≡0.

•If this integral is not zero, then we have R

Ωf(x)|u|^p∗≥λ^N/p. Proof. Suppose thatR

Ωf(x)|u|^p∗= 0, multiply byuand integrate over Ω, to obtain J(u) =

Z

Ω

(|∇u|^p+a(x)|u|^p) = 0 thusu≡0 by coercivity ofL.

Suppose now thatu6≡0, so that R

Ωf(x)|u|^p∗ >0. Defining w=

Z

Ω

f(x)|u|^p∗ −1/p^∗

u one has, by construction, thatw∈ W^p∗(G), hence

λ≤ Z

Ω

f(x)|u|^p∗ −p/p^∗

J(u)≤ Z

Ω

f(x)|u|^p∗

1−p/p^∗

and actually

Z

Ω

f(x)|u|^p∗≥λ^N/p.

Remark. Here, one can see thatλrepresents a minimal energy level for non zero solutions of (4).

Now, we prove a stronger result which prevents, under some assumption on the energy level of considered solutions, from concentration phenomena that could happen if the sequence converges weakly but not strongly inW₀^1,p(Ω).

In what follows, we assume that the existence’s condition of Theorem 2.2 holds, that is on every pointx∈Ω where f(x)>0,

λf(x)^1−p/NK(N, p)^p<1.

Theorem 4.1. Let(un)be a bounded sequence (in W₀^1,p(Ω)) of solutions of (4) which are non identically zero.

We assume that

nlim→∞

Z

Ω

f(x)|un|^p∗=λ^N/p.

Then, if(un)converges weakly tou,uis a non identically zero solution of (4) (In fact, (un)converges strongly touin W^1,p(Ω).)

Remark. This result can be extended to positive and nodal solutions, when the data Ω,aand f present some symmetries.

Proof. The fact thatuis a solution of (4) immediately follows from Section 3.2. It is sufficient to show thatu is not the trivial function.

For that aim, we use the localisation method in Section 3.3. Let P ∈ Ω, δ be a positive real, and η be the cut-off function defined in Section 3.3. Following the proof of Lemma 3.4, and assuming that (un)*0 in W₀^1,p(Ω), one obtains that

Z

Ω|∇(ηun)|^p+o(1) = Z

Ω

f(x)|un|^p∗η^p (n→ ∞).

- Iff(P)<0, one has, forδ small enough, thatR

Ω|∇(ηun)|^p tends to 0 asntends to∞. - Iff(P)≥0, one has the following estimate:

Z

Ω

f(x)|un|^p∗η^p ≤ sup

BP(δ)|f|^p/p∗K(N, p)^p Z

B_P(δ)|f(x)||un|^p∗

!1−p/p^∗Z

Ω|∇ |ηun)|^p. (20) Hence,

∗forf(P) = 0, the conclusion is the same;

∗forf(P)>0, one assumes that

K(N, p)^pf(P)^1−p/N lim

n→∞

Z

B_P(δ)|f(x)||un|^p∗

!1−p/p^∗

<1

which yields that

nlim→∞

Z

Ω

|∇(ηun)|^p = 0.