Multiple solutions of supercritical elliptic problems in perturbed domains

Multiple solutions of supercritical elliptic problems in perturbed domains

Multiplicité de solutions de problèmes elliptiques surcritiques en domaines perturbés

Riccardo Molle

, Donato Passaseo

aDipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica n. 1, 00133 Roma, Italy bDipartimento di Matematica “E. De Giorgi”, Università di Lecce, P.O. Box 193, 73100 Lecce, Italy

Received 23 July 2004; received in revised form 21 October 2004; accepted 20 May 2005 Available online 19 January 2006

Abstract

We are concerned with existence and multiplicity of nontrivial solutions for the Dirichlet problemu+ |u|^p−2u=0 inΩ, u=0 on∂Ω, whereΩis a bounded domain ofRⁿ,n3, andp >_n²ⁿ₋₂. We show that suitable perturbations of the domain, which modify its topological properties, give rise to a number of solutions which tends to infinity as the size of the perturbation tends to zero (some examples show that the perturbed domains may be even contractible). More precisely, we prove that for allk∈N, if the size of the perturbation is small enough (depending onk), there exist at leastkpairs of nontrivial solutions, which concentrate near the perturbation as the size of the perturbation tends to zero. The method we use, which is completely variational, gives also further informations on the qualitative properties of the solutions; in particular, these solutions (which may change sign) do not have more thanknodal regions and at least two solutions (which minimize the corresponding energy functional) have constant sign.

2005 Elsevier SAS. All rights reserved.

Résumé

Nous étudions l’existence et la multiplicité de solutions pour le problème de Dirichletu+ |u|^p−2u=0,u≡0 enΩ,u=0 sur∂Ω, oùΩ est un ouvert borné deRⁿ,n3, etp >_n²ⁿ₋₂. Nous démontrons que l’existence et le nombre de solutions sont liés à certaines perturbations du domaine, qui modifient ses propriétés topologiques. Chaque perturbation dépend d’un paramètreε (l’épaisseur de la perturbation) ; quandε→0, le nombre de solutions tend à l’infini et les solutions se concentrent près de la perturbation (des examples montrent que les domaines perturbés peuvent même être contractiles). Plus précisément, nous prouvons que pour toutk∈Nil existeε_k>0 tel que, pour toutε∈ ]0, εk[, le problème a au moinskpaires de solutions. La méthode que nous suivons, qui est complètement variationnelle, donne aussi des informations sur les propriétés qualitatives des solutions. En particulier, ces solutions (qui peuvent changer de signe) ne peuvent pas avoir plus quekrégions nodales ; de plus, au moins deux solutions (qui minimisent la fonctionnelle de l’énergie) ont signe constant.

2005 Elsevier SAS. All rights reserved.

✩ Work supported by the national research project “Metodi variazionali e topologici nello studio di fenomeni non lineari”.

* Corresponding author.

E-mail address:molle@mat.uniroma2.it (R. Molle).

0294-1449/$ – see front matter 2005 Elsevier SAS. All rights reserved.

doi:10.1016/j.anihpc.2005.05.003

MSC:35J60; 35J20; 35J25

Keywords:Supercritical problems; Changing sign solutions; Multiplicity of solutions; Number of nodal regions

1. Introduction

Let us consider the problem P (Ω)

u+ |u|^p−2u=0, u≡0 inΩ,

u=0 on∂Ω,

whereΩ is a bounded domain ofRⁿ,n3, andp >_n²ⁿ₋₂ is a fixed exponent (_n²ⁿ₋₂ is the critical Sobolev exponent).

In this paper we show that suitable perturbations of a given domain give rise to solutions of the problem in the perturbed domains; when the size of the perturbation tends to zero, the number of the solutions tends to infinity while the solutions concentrate as Dirac masses near the perturbation; this perturbation is realized by removing a thin subset of the domain in such a way to modify its topological properties (see Theorem 2.1 and Remark 2.2).

It is well known that, when 2< p <_n²ⁿ₋₂,P (Ω)has solutions in any domainΩ. On the contrary, whenp_n²ⁿ₋₂, the problem has no solution ifΩ is starshaped (see [21]) while it has infinitely many solutions ifΩ is, for example, an annulus (see [8]).

Forp=_n²ⁿ₋₂, a sufficient condition for the existence of a positive solution is thatΩ has nontrivial topology, in a suitable sense (see [1]); this condition is only sufficient but not necessary, as shown by some examples of contractible domains where the problem has positive solutions for p= _n²ⁿ₋₂ (see [5,7,16,18]). For p >_n²ⁿ₋₂, this nontriviality condition of the domain is neither a sufficient nor a necessary condition; in fact, nonexistence results hold for some p >_n²ⁿ₋₂ in some nontrivial domains (see [17,19]) while an arbitrarily large number of solutions can be obtained in some contractible domains for allp >_n²ⁿ₋₂(see [20]).

It is also worth pointing out that existence of solutions in the supercritical case has been proved even in some

“nearly starshaped” domains (see the definition introduced in [11]) forpsufficiently close to_n²ⁿ₋₂ (see [10,12,13]) or forplarge enough (see [14]); on the other hand, a different definition of “nearly starshaped” domain is used in [6] in order to extend Pohožaev’s nonexistence result to nonstarshaped domains whenpis large enough.

Let us observe that in [20] several perturbations have been used in order to obtain several solutions in contractible domains. On the contrary, the result proved in the present paper shows that a unique perturbation can produce an arbitrarily large number of solutions.

In order to overcome the difficulties related to the presence of the supercritical exponent, we proceed as follows: we modify the nonlinear term|u|^p−2uin such a way that the modified nonlinearityg(x, u)has subcritical growth outside a neighbourhood of the perturbation; then (exploiting also the symmetry of the domain with respect to an axis) we use topological methods of Calculus of Variations to find multiple solutions of the modified problem in the perturbed domain; we also show that, as the size of the perturbation tends to zero, these solutions concentrate as Dirac masses near the perturbation (where the nonlinear term has not been modified); finally, we prove that these solutions solve also the unmodified problem when the size of the perturbation is small enough.

2. Notations and statement of the main result

LetΩbe a bounded domain ofRⁿ, having radial symmetry with respect to thexn-axis, that is x=(x1, . . . , xn)∈Ω ⇐⇒

_n−1

i=1

x_i² 1/2

,0, . . . ,0, xn

∈Ω. (2.1)

Letx¹=(0, . . . ,0, x_n¹),x²=(0, . . . ,0, x_n²),x³=(0, . . . ,0, x_n³)be three points of thex_n-axis which satisfy

x_n¹< x_n²< x³_n, x¹∈ /Ω, x²∈Ω, x³∈ /Ω. (2.2)

For allε >0, set χ_ε=

x∈Rⁿ:

n−1

i=1

x_i²< ε², x_n¹< x_n< x_n³

(2.3) and

Ω_ε=Ω\ ¯χ_ε. (2.4)

Since the domainΩ_εhas radial symmetry with respect to thex_n-axis for allε >0, it is natural to look for solutions of problemP (Ω_ε)among the functions which have the same symmetry. Thus, we consider the subsets ofH₀^1,2(Ω_ε), H₀^1,2(Ω)which consist of the functions radially symmetric with respect to thex_n-axis and denote byH_S(Ω_ε),H_S(Ω) these subspaces. Moreover, we intend that every function of H₀^1,2(Ωε) is extended in all ofΩ by the value zero outsideΩε. In these spaces we shall use the norms

Du 2=

Ω

|Du|²dx

1/2

and

u _q=

Ω

|u|^qdx

1/q

∀u∈L^q(Ω), q1.

Theorem 2.1.Let p >_n²ⁿ₋₂,Ω be a bounded domain radially symmetric with respect to the x_n-axis,x¹, x², x³ be three points of thexn-axis satisfying condition(2.2)and, for allε >0,χεandΩεbe the domains defined in(2.3)and (2.4). Then, for all positive integerk, there existsε¯k>0such that, for allε∈ ]0,ε¯k[, problemP (Ωε)has at leastk pairs of solutions±u1,ε, . . . ,±uk,εinHS(Ωε)∩L^p(Ωε), which, for alli=1, . . . , k, satisfy:

(a) limε→0

Ωε|Du_i,ε|²dx=0, (b) limε→0

Ωε|u_i,ε|^pdx=0,

(c) limε→0sup{|u_i,ε(x)|: x∈Ω_ε\χ_ρ} =0 ∀ρ >0, (d) limε→0sup{|u_i,ε(x)|: x∈Ω_ε∩χ_ρ} = +∞ ∀ρ >0, (e) limε→0

Ω_ε∩χ_ρ|u_i,ε|^qdx= +∞ ∀q >ⁿ₂(p−1),∀ρ >0.

The proof is reported in Section 3, where also other properties of the solutions are specified (see Propositions 3.8 and 3.9).

Remark 2.2. A result analogous to Theorem 2.1 holds in some more general perturbed domains. In fact, letΩ, x¹, x², x³be as in Theorem 2.1; consider a sequence of subdomainsΩ_jofΩ, satisfying the radial symmetry condition (2.1) for allj∈N; now, set

cap(Ω\Ω_j)=inf

Rⁿ

|Du|²dx: u∈C₀^∞(Rⁿ), u(x)1∀x∈Ω\Ω_j

(2.5) and assume that

(a) limj→∞cap(Ω\Ωj)=0,

(b) for allj∈Nthere existsρj>0 such thatχρ_j∩Ωj= ∅.

Then, arguing as in the proof of Theorem 2.1, one can prove that, for all p > _n²ⁿ₋₂, problem P (Ωj), forj large enough, has at leastkpairs of solutions±u1,j, . . . ,±uk,j inHS(Ωj)∩L^p(Ωj), which, asj → ∞, present the same asymptotic behaviour that the solutions given by Theorem 2.1 present asε→0 (see (a)–(e) in Theorem 2.1 as well as the properties described in Propositions 3.8 and 3.9).

Finally, let us remark that the methods we use in this paper can be easily adapted to deal with more general nonlinear terms having supercritical growth.

Examples 2.3.Theorem 2.1 allows us to obtain an arbitrarily large number of solutions even in some contractible domains. Consider, for example, the domainA\ ¯χ_ε(a pierced annulus) whereAis the annulus

A=

x∈Rⁿ: 0< r1<|x|< r2

andχ_ε is the cylinder defined as in (2.3) with x_n¹=0 andx_n³> r₂. It is clear that, because of this choice ofx¹ andx³,A\ ¯χε is a contractible domain for allε >0; moreover, Theorem 2.1 applies whenε >0 is small enough and guarantees the existence of an arbitrarily large number of solutions. Asε→0, the number of solutions tends to infinity and the solutions concentrate near points of thex_n-axis. Notice that these solutions do not converge asε→0 to solutions of the problem in the limit domain (i.e. the annulusAwhere, on the other hand, it is easy to find infinitely many radial solutions).

Let us remark that, in order to have an arbitrarily large number of solutions in contractible domains, in [20] some domainsΩ_ε^h of the following form have been considered: Ω_ε^h=Ω^h\h

j=1χ¯ε^j whereχε^j, forj =1, . . . , h, is the cylinder of the form (2.3) withx¹_n=j andx_n³=j+1 andΩ^h=T^h\_h

j=1B(c_j, r_j), where

T^h=

x=(x1, . . . , xn)∈Rⁿ:

n−1

i=1

x_i²<1, 0< xn< h+1

(2.6) andB(c_j, r_j) is the ball of radius r_j ∈ ]0,1/2[ and centre c_j =(0, . . . ,0, j ) ∈Rⁿ. In [20] it is proved that for allj =1, . . . , hthere exists a solution ofP (Ω_ε^h)whose positive or negative part concentrates as ε→0 near the cylinderχ_ε^j. Now, it is clear that Theorem 2.1 can be applied in the domainΩ^h withh separate terns of points x¹, x², x³satisfying condition (2.2); so, for eachj=1, . . . , h, forεsmall enough, we obtain many solutions localized nearχ_ε^j, whose number tends to infinity asε→0.

Notice that, taking into account Remark 2.2, similar multiplicity results can be obtained also in contractible do- mainsΩ_r^s of the form

Ω_r^s=

x=(x₁, . . . , x_n)∈Rⁿ: 1<|x|< r, _n−1

i=1

x_i² 1/2

> sx_n

(2.7) for allr >1, whens >0 is small enough.

Domains of this type have been also considered in [10,12–14]. Whenp >_n²ⁿ₋₂is a fixed exponent, the concentration phenomena, which allow us to obtain the multiplicity result proved in the present paper, occur as s→0; on the contrary, other different concentration phenomena, which arise asp→ _n²ⁿ₋₂(see [10,12,13]) orp→ +∞(see [14]), have been used to find solutions for allr >1 ands >0, not necessarily small, whenpis close to _n²ⁿ₋₂ or it is large enough.

In the casep_n²ⁿ₋₂, the solutions ofP (Ω)are obtained as critical points of the functional f (u)˜ =1

2

Ω

|Du|²dx− 1 p

Ω

|u|^pdx.

In the casep >_n²ⁿ₋₂, there are some difficulties in dealing with this functional. Therefore, we introduce the following modified functionalf_ε:H_S(Ω_ε)→R, defined by

f_ε(u)=1 2

Ω_ε

|Du|²dx−

Ω_ε

G(x, u)dx, (2.8)

where G(x, u)=_u

0 g(x, t )dt with g(x, t ) defined as follows. For all x ∈Ω ∩ ¯χ₁ we set g(x, t )= |t|^p−2t. If Ω\ ¯χ₁= ∅, in order to defineg(x, t )forx∈Ω\ ¯χ₁, we first remark that

inf

Ω\ ¯χ₁

|Du|²dx: u∈H₀^1,2(Ω),

Ω\ ¯χ₁

u²dx=1

>0; (2.9)

then we chooset₀>0 small enough such that (p−1)t₀^p−2<inf

Ω\ ¯χ1

|Du|²dx: u∈H₀^1,2(Ω),

Ω\ ¯χ1

u²dx=1

; (2.10)

hence, for allx∈Ω\ ¯χ₁, we set

g(x, t )=







|t|^p−2t if|t|t0, t₀^p−1+(p−1)t₀^p−2(t−t₀) iftt₀,

−t₀^p−1+(p−1)t₀^p−2(t+t0) ift−t0.

(2.11)

Let us remark that this choice oft₀implies the existence of a constantc >˜ 0 such that

Ω\ ¯χ1

|Du|²−g(x, u)u dxc˜

Ω\ ¯χ1

|Du|²dx ∀u∈H₀^1,2(Ω), (2.12)

as one can easily verify.

Notice that all nontrivial critical points offεbelong to the set Mε=

u∈HS(Ωε): u≡0, f_ε(u)[u] =0

. (2.13)

The solutions of problemP (Ωε)will be obtained, forε >0 small enough, as critical points forfεconstrained onMε. 3. Proof of Theorem 2.1 and behaviour of the solutions

Lemma 3.1.For all ε >0, the functionalfε (see(2.8))is well defined and belongs to the classC². Moreover, the following properties hold.

(a) Letu∈HS(Ωε),u≡0;then eitherf_ε(t u)[u]>0∀t >0 (what happens, for example, ifu≡0inχ1)or there exists a uniquet >¯ 0such thatt u¯ ∈Mε;in the second case,f_ε(t u)[u]>0∀t∈ ]0,t¯[andf_ε(t u)[u]<0∀t >t¯ (the second case occurs, for example, ifu≡0inΩε\χ1).

(b) For allε >0such thatMε= ∅, we haveinfM_εfε>0.

(c) There existsε >¯ 0such thatMε= ∅ ∀ε∈ ]0,ε¯[.

(d) limε→0infM_εfε=0.

(e) IfM_ε= ∅, thenM_ε is aC¹-manifold of codimension1.

(f) Every critical point forf_εconstrained onM_εis a critical point forf_ε. Proof. First, let us remark that the functionals

u−→

Ω_ε\ ¯χ₁

G(x, u)dx and u−→

Ω_ε∩χ₁

G(x, u)dx, (3.1)

defined respectively inL²(Ω_ε\ ¯χ₁)and inL^p(Ω_ε∩χ₁), areC²-functionals. Moreover, notice that

Ωε\ ¯χ₁

u²dxc¯

Ωε

|Du|²dx ∀u∈H₀^1,2(Ωε) (3.2)

for a suitable constantc >¯ 0 and, for allε >0, there existsc¯_ε>0 such that

Ωε∩χ1

|u|^pdx

2/p

c¯_ε

Ωε

|Du|²dx ∀u∈H_S(Ω_ε). (3.3)

Hence, by standard arguments, it is easy to verify thatfε is a well definedC²-functional inHS(Ωε).

(a) Notice that, for allt >0, f_ε(t u)[u] =t

Ω_ε

|Du|²dx−

Ω_ε

g(x, t u) t udx

,

where (by the definition ofg) the functiont→

Ωε

g(x,t u)

t udx is strictly increasing in]0,+∞[for allu≡0. Then the assertion (a) follows, taking also into account thatg(x, t u)u=t^p−1|u|^p ifu≡0 inΩ_ε\χ₁whileg(x, t u)u (p−1)t₀^p−2t u²ifu≡0 inχ₁.

(b) From property (a) we infer that, for allu∈M_ε,

f_ε(u)f_ε(t u) ∀t0. (3.4)

On the other hand, we havef_ε(0)=0 and f_ε(0)[u, u] =

Ω_ε|Du|²dx. Therefore there exists r >0 andα >0 such that

inf

f_ε(u): u∈H_S(Ω_ε),

Ωε

|Du|²dx=r²

α.

It follows that infMε

f_εα >0.

(c) Sincex²∈Ω, there existsρ₂>0 such thatB(x², ρ₂)⊂Ω. Chooseϕ¯∈H_S(B(x², ρ₂)\ ¯χ_ρ2/2)and, for all ε >0, set

¯

ϕε(x)= ¯ϕ

x²+ρ₂ 2ε

x−x² (3.5)

(we intend that ϕ¯ is extended by zero outside B(x², ρ₂)\ ¯χ_ρ2/2). Then, it is easy to verify thatϕ¯_ε∈H_S(Ω_ε)for ε∈ ]0, ρ2/2[andϕ¯_ε≡0 inΩ_ε\χ₁forε∈ ]0,1/2[; thus, for 0< ε <min{1/2, ρ2/2}, there existst¯_ε>0 such that

¯

t_εϕ¯_ε∈M_ε.

(d) For the proof of (d), it suffices to observe that, sincep > _n²ⁿ₋₂,

εlim→0f_ε(¯t_εϕ¯_ε)=0 (3.6)

as one can easily verify by a direct computation.

(e) Let us set

F_ε(u)=f_ε(u)[u] (3.7)

(notice thatF_ε is aC¹-functional inH_S(Ω_ε)). We shall prove that F_ε(u)=0 for allu∈M_ε, so the assertion will follow by the implicit function theorem. In fact, ifu∈M_ε, thenu≡0 andF_ε(u)=0, that is

Ωε

|Du|²dx−

Ωε

g(x, u)udx=0.

Therefore, we have F_ε(u)[u] =2

Ω_ε

|Du|²dx−

Ω_ε

g(x, u)u²dx−

Ω_ε

g(x, u)udx

=

Ωε

g(x, u)u−g(x, u)u² dx

(hereg(x, t )denotes the derivative ofg(x, t )with respect tot).

Sinceg(x, t )t−g(x, t )t²<0∀t=0 and sinceu≡0, we infer thatF_ε(u)[u]<0 and soF_ε(u)=0.

(f) Ifu∈Mεis a critical point forfεconstrained onMε, there exists a Lagrange multiplierµsuch that

f_ε(u)+µF_ε(u)=0. (3.8)

In particular, we have

f_ε(u)[u] +µF_ε(u)[u] =0, (3.9)

which impliesµ=0 sincef_ε(u)[u] =0 (becauseu∈M_ε) andF_ε(u)[u] =0. Therefore,f_ε(u)=0. 2

Thus, finding nontrivial critical points forfε is equivalent to finding critical points forfεconstrained onMε. The compactness property proved in the following lemma plays an important role to find critical points forfεconstrained onM_ε.

Lemma 3.2.Letε >0be such thatM_ε= ∅and(u_i)_i be a sequence inM_ε. The following properties hold.

(a) Ifsup_i∈Nf_ε(u_i) <+∞, then the sequence(u_i)_i is bounded inH₀^1,2(Ω_ε).

(b) Let us setF_ε(u)=f_ε(u)[u];ifu_i uweakly inH₀^1,2(Ω_ε)and there exists a sequence(µ_i)_i inRsuch that

f_ε(u_i)+µ_iF_ε(u_i)−→0 inH⁻¹(Ω_ε), (3.10)

thenlimi→∞µi=0,u∈Mεandui→uinH₀^1,2(Ωε).

(c) The functionalf_εconstrained onM_ε satisfies the Palais–Smale condition at any levelc∈R, i.e. every sequence (u_i)_i inM_ε, such that

ilim→∞f_ε(u_i)=c and grad_Mεf_ε(u_i)→0 inH⁻¹(Ω_ε), (3.11) is relatively compact inH₀^1,2(Ω_ε).

Proof. (a) Sinceui∈Mε, taking into account the definition ofG(x, t ), we have f_ε(u_i)=1

2

Ωε

|Du_i|²dx−

Ωε

G(x, u_i)dx

1

2

Ωε

|Du_i|²dx−1 2

Ωε\ ¯χ1

g(x, u_i)u_idx−1 p

Ωε∩χ1

|u_i|^pdx

= 1

2−1 p

Ω_ε

|Du_i|²dx− 1

2−1 p

Ωε\ ¯χ1

g(x, u_i)u_idx.

It follows that sup

i∈N Ωε

|Du_i|²dx−

Ωε\ ¯χ1

g(x, u_i)u_idx

<+∞. (3.12)

Hence (3.12) and (2.12) imply sup_i∈N

Ω_ε|Dui|²dx <+∞. (b) First, let us prove thatu≡0. Notice that

ilim→∞

Ωε\ ¯χ1

|u_i−u|²dx=0; (3.13)

moreover, sinceε >0 andui∈HS(Ωε),

ilim→∞

Ωε∩χ₁

|ui−u|^pdx=0. (3.14)

Notice thatui ∈Mεimplies

Ωε\ ¯χ₁

|Du_i|²−g(x, u_i)u_i dx+

Ωε∩χ1

|Du_i|²− |u_i|^p

dx=0. (3.15)

Let us observe that we must have

Ω_ε∩χ₁

|ui|^pdx >0 ∀i∈N, (3.16)

otherwise, sinceu_i≡0 inΩ_ε, we should have

Ω_ε\ ¯χ₁

|Dui|²−g(x, ui)ui

dx >0,

because of the choice oft₀(see (2.10)), which contradicts (3.15).

Since

Ω_ε\ ¯χ₁

|Du|²−g(x, u)u

dx0 ∀u∈H₀^1,2(Ωε), (3.17)

we infer from (3.15)

Ωε∩χ1

|u_i|^pdx

Ωε∩χ1

|Du_i|²dx.

On the other hand, asε >0 andui∈HS(Ωε), there exists a constantc >¯ 0 such that

Ωε∩χ1

|Du|²dxc¯

Ωε∩χ1

|u|^pdx

2/p

∀u∈H_S(Ω_ε). (3.18)

Hence we get

Ω_ε∩χ₁

|ui|^pdx

Ω_ε∩χ₁

|Dui|²dxc¯

Ω_ε∩χ₁

|ui|^pdx

2/p

∀i∈N,

which, because of (3.16), implies

iinf∈N

Ω_ε∩χ₁

|ui|^pdx >0.

Taking into account (3.14), it follows that

Ω_ε∩χ₁

|u|^pdx >0. (3.19)

Notice that, asu_i∈M_ε, (3.10) implies

ilim→∞µiF_ε(ui)[ui] =0. (3.20)

Moreover,u_i∈M_εimplies F_ε(u_i)[u_i] =2

Ωε

|Du_i|²dx−

Ωε

g(x, u_i)u²_i dx−

Ωε

g(x, u_i)u_idx=

Ωε

g(x, u_i)u_i−g(x, u_i)u²_i

dx. (3.21) Hence, from (3.13) and (3.14) we infer that

ilim→∞F_ε(u_i)[u_i] =

Ωε

g(x, u)u−g(x, u)u²

dx. (3.22)

Notice that

Ωε

g(x, u)u−g(x, u)u² dx <0

becauseg(x, t )t−g(x, t )t²<0∀t=0 andu≡0 inΩ_ε(as (3.19) holds). Taking into account (3.20), it follows that

ilim→∞µ_i=0. (3.23)

Using again (3.10), we obtain

ilim→∞

f_ε(u_i)[u−u_i] +µ_iF_ε(u_i)[u−u_i]

=0, (3.24)

which, taking into account (3.13), (3.14), (3.20), (3.23) and the weak convergence ofu_i, implies

ilim→∞

Ωε

|Du_i|²dx=

Ωε

|Du|²dx.

Thus,u_i→ustrongly inH₀^1,2(Ω_ε)andu∈M_ε.

(c) Property (a) implies that every Palais–Smale sequence (u_i)_i is bounded in H₀^1,2(Ω_ε); hence, up to a subse- quence, it converges weakly in H₀^1,2(Ω_ε); then, property (b) guarantees the strong convergence in H₀^1,2(Ω_ε)to a functionu∈Mε. 2

Lemma 3.3.Let (ε_i)_i be a sequence of positive numbers and(u_i)_i a sequence of functions inH_S(Ω_εi)such that f_ε

i(u_i)=0andu_i≡0inΩ_εifor alli∈N.

Iflimi→∞f_εi(u_i)=0, then (a) limi→∞

Ω_εi|Du_i|²dx=0, (b) limi→∞ε_i=0,

(c) limi→∞sup{|u_i(x)|: x∈Ω_εi\χ_ρ} =0 ∀ρ >0.

Proof. (a) Clearly, it suffices to prove that there exists a constantc >¯ 0 for which f_ε(u)c¯

Ω_ε

|Du|²dx (3.25)

for allε >0 and for allu∈H_S(Ω_ε)such thatf_ε(u)=0.

Iff_ε(u)=0, arguing as in the proof of Lemma 3.2, we have f_ε(u)1

2

Ω_ε

|Du|²dx−1 2

Ω_ε\ ¯χ₁

g(x, u)udx− 1 p

Ω_ε∩χ₁

|u|^pdx

= 1

2−1 p

Ω_ε\ ¯χ₁

|Du|²−g(x, u)u dx+

Ωε∩χ₁

|Du|²dx

. (3.26)

Hence, the existence of a constantc >¯ 0, such that (3.25) holds, follows easily from (2.12).

(b) Arguing by contradiction, assume that, up to a subsequence, infi∈Nε_i>0 and chooseε¯∈ ]0,infi∈Nε_i[. Thus, u_i∈H_S(Ω_ε¯)for alli∈N. Sinceε >¯ 0, there exists a constantc¯_¯ε>0 such that

Ωε_¯∩χ1

|Du|²dxc¯_ε¯

Ω_¯ε∩χ1

|u|^pdx

2/p

∀u∈H_S(Ω_ε¯). (3.27)

Hence, property (a) implies

ilim→∞

Ω_¯ε∩χ₁

|ui|^pdx=0. (3.28)

On the other hand, we havef_ε¯(ui)[ui] =0, that is

Ωε_¯∩χ1

|Du_i|²− |u_i|^p dx+

Ωε_¯\ ¯χ₁

|Du_i|²−g(x, u_i)u_i

dx=0. (3.29)

First, let us remark that (3.29) implies

Ω_¯ε∩χ₁

|ui|^pdx >0 ∀i∈N (3.30)

otherwise, asui≡0 inΩε_¯, we should have (because of (2.12))

Ω_¯ε\ ¯χ1

|Du_i|²−g(x, u_i)u_i dx >0 and so (3.29) cannot hold.

Moreover, from (3.29) and (2.12) we infer that

Ω_¯ε∩χ1

|u_i|^pdx

Ωε_¯∩χ1

|Du_i|²dx ∀i∈N. (3.31)

Therefore, taking into account (3.27), (3.30) and (3.31), we get

iinf∈N

Ωε_¯∩χ1

|u_i|^pdx >0, (3.32)

which contradicts (3.28).

(c) For allρ >0, let us consider the cylinder Cρ=

x=(x1, . . . , xn)∈Rⁿ:

n−1

i=1

x_i²ρ²

.

Since the subspace of the radial functions in H¹(Ω\C_ρ)is embedded in L^q(Ω\C_ρ)for all ρ >0 andq 1, property (a) implies

ilim→∞

Ω_εi\C_ρ

|ui|^qdx=0 ∀q1, ∀ρ >0. (3.33)

Now, let us introduce the functionw_idefined by w_i(x)= 1

n(n−2)ωn

Ω_εi

|g(y, u_i(y))|

|x−y|ⁿ⁻² dy,

whereω_nis the measure of the unit sphere ofRⁿ. It is well known thatw_i solves the equation wi(x)+g

x, ui(x)=0;

moreover, we havew_i0 on∂Ω_εi, which implies

wi(x)ui(x) ∀x∈Ωε_i. (3.34)

Now, let us prove that

ilim→∞sup

w_i(x): x∈Ω\C_ρ

=0 ∀ρ >0. (3.35)

For allx∈Ω\Cρ, we writewi as w_i(x)= 1

n(n−2)ωn

Ω_εi\B(x,ρ/2)

|g(y, u_i(y))|

|x−y|ⁿ⁻² dy+

Ω_εi∩B(x,ρ/2)

|g(y, u_i(y))|

|x−y|ⁿ⁻² dy

. (3.36)

The first integral in (3.36) can be estimated as follows:

Ω_εi\B(x,ρ/2)

|g(y, u_i(y))|

|x−y|ⁿ⁻² dy 2

ρ

n−2

Ω_εi

|g(y, u_i(y))|dy. (3.37)

Asf_ε

i(u_i)=0, property (a) implies

ilim→∞

Ω_εi

g(x, ui)uidx=0. (3.38)

Since

g(x, t )g(x, t )t+g(x,1) ∀x∈Ω, ∀t∈R, (3.39) taking into account the definition ofg(x, t ), we easily infer that

ilim→∞

Ω_εi

g

y, u_i(y)dy=0,

which, by (3.37), implies

ilim→∞

Ω_εi\B(x,ρ/2)

|g(y, u_i(y))|

|x−y|ⁿ⁻² dy=0 (3.40)

uniformly with respect tox∈Ω\Cρ.

In order to deal with the second integral in (3.36), let us remark that, sincex∈Ω\Cρ,

ilim→∞

Ω_εi∩B(x,ρ/2)

g

y, u_i(y)^qdy=0 ∀q1, ∀ρ >0

because of (3.33). SinceΩ_εi∩B(x, ρ/2)⊂Ω\C_ρ/2for allx∈Ω\C_ρ, forq >ⁿ₂ we obtain

Ω_εi∩B(x,ρ/2)

|g(y, u_i(y))|

|x−y|ⁿ⁻² dyc(p, q)

Ω_εi\C_ρ/2

g

y, u_i(y)^qdy

1/q

∀x∈Ω\C_ρ,

wherec(p, q)is a suitable constant depending only onpandq. Taking into account (3.33), it follows that

ilim→∞

Ω_εi∩B(x,ρ/2)

|g(y, u_i(y))|

|x−y|ⁿ⁻² dy=0 (3.41)

uniformly with respect tox ∈Ω\Cρ. Hence, (3.35) follows from (3.36), (3.40) and (3.41). Notice that, since x¹ andx²do not belong toΩ, (3.34) and (3.35) imply, in particular, that ui→0 uniformly on the boundary ofΩ\ ¯χ1. Therefore, taking into account the definition of g(x, t )forx∈Ω\ ¯χ1, it follows thatui→0 uniformly inΩ\ ¯χ1. Thus, property (c) is completely proved. 2

Corollary 3.4.There exists a positive constantc(Ω, p)¯ (depending only onΩ andp)such that, ifu∈HS(Ωε)is a critical point forfεandfε(u)c(Ω, p), then¯

(a) |u(x)|t0∀x∈Ωε\ ¯χ1,

(b) usolves problemP (Ωε)(providedu≡0inΩε).

The proof follows easily from property (c) of Lemma 3.3.

Proposition 3.5.For allε >0, the minimum of the functionalf_εconstrained onM_εis achieved. Moreover, (a) the minimizing functions have constant sign;

(b) there existsε₁>0such that, for allε∈ ]0, ε1[, the minimizing functions solve problemP (Ω_ε).