Concentration phenomena for solutions of superlinear elliptic problems

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Concentration phenomena for solutions of superlinear elliptic problems

Phénomènes de concentration pour les solutions de problèmes elliptiques surlinéaires

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 29 July 2004; received in revised form 13 January 2005; accepted 2 February 2005 Available online 20 April 2005

Abstract

In this paper we look for positive solutions of the problem−u+λu=u^p−1inΩ,u=0 on∂Ω, whereΩis a bounded domain inRⁿ,n3,p >2 andλis a positive parameter. We describe new concentration phenomena, which occur asλ→ +∞, and exploit them to construct (forλlarge enough) positive solutions that concentrate near spheres of codimension 2 asλ→ +∞; these spheres approach the boundary ofΩasλ→ +∞. Notice that the existence and multiplicity results we obtain hold also in contractible domains arbitrarily close to starshaped domains (no solution can exist ifp_n²ⁿ₋₂andΩis starshaped, because of Pohožaev’s identity). The method we use is completely variational and based on a blow up analysis in the equivariant setting.

In order to avoid concentration phenomena near points and to overcome some difficulties related to the lack of compactness, we first modify the nonlinear term in a suitable region, then we solve the modified problem by minimizing the related energy functional on a suitable infinite dimensional manifold and, finally, we show that the solutions of the modified problem solve also our problem, forλlarge enough, because they are localized in the prescribed region where the nonlinear term has not been modified.

2005 Elsevier SAS. All rights reserved.

Résumé

Nous démontrons l’existence de solutions positives pour le problème−u+λu=u^p−1enΩ,u=0 sur∂Ω, oùΩest un domaine borné deRⁿ,n3,p >2 etλ >0. Nous décrivons de nouveaux phénomènes de concentration qui apparaissent quandλ→ +∞. Grâce à ceux-ci nous construisons des solutions positives pourλassez grand donc qui se concentrent près

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

* Corresponding author.

E-mail address: [email protected] (R. Molle).

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

doi:10.1016/j.anihpc.2005.02.002

des sphères de codimention 2 quandλ→ +∞; ces sphères approchent du bord de omega quandλ→ +∞. Il faut remarquer que l’existence de solutions est prouvée pour des domaines qui peuvent être arbitrairement proches de domaines étoilés (quand p_n²ⁿ₋₂etΩest étoilé il n’y a pas de solutions, ce qui se déduit de l’identité de Pohožaev). La méthode que nous suivons pour la démonstration est complètement variationnelle. Pour surmonter les difficultés liées à la présence d’opérateurs non compacts, d’abord nous modifions le terme non linéaire ; ensuite nous trouvons des solutions du problème modifié en minimisant la fonctionnelle de l’énergie sur une varieté de dimension infinie ; enfin nous démontrons que les solutions du problème modifié sont aussi solutions du problème original, parce-qu’elles sont localisées, dans la région où le terme non linéaire n’a pas été modifié.

2005 Elsevier SAS. All rights reserved.

MSC: 35J20; 35J60; 35J65

Keywords: Variational methods; Lack of compactness; Concentration phenomena; Blow up analysis; Nearly starshaped domains

1. Introduction

Let us consider the following problem





−u+λu=u^p−1 inΩ,

u >0 inΩ,

u=0 on∂Ω,

(1.1) whereΩis a bounded domain ofRⁿ,n3,p >2 andλis a positive parameter.

In [13] and [15] this problem has been studied in the casep=_n²ⁿ₋₂(the critical Sobolev exponent). In this case some concentration phenomena occur whenλ→0: the solutions tend to concentrate near points of the domain.

This fact is used in [13,15] to obtain, forλ >0 small enough, an arbitrarily large number of positive solutions, having an arbitrarily large number of spikes, under suitable assumptions on the domainΩ.

In the present paper we point out other concentration phenomena, which occur, for allp >2, asλ→ +∞and exploit them to construct positive solutions, forλ >0 large enough, in the same domains considered in [13,15].

Notice that, while the solutions obtained in [13,15] concentrate and blow-up as λ→0 near a finite number of points, the solutions we construct in the present paper concentrate near spheres of codimension 2. In [13,15] we proved that for all positive integerkthere exist, forλ >0 small enough, solutions blowing-up asλ→0 at exactly kpoints, which approach the boundary ofΩ ask→ ∞; the solutions we obtain in this paper, forλlarge enough, concentrate near spheres which approach the boundary ofΩ asλ→ +∞and the rate of concentration is greater than the rate of approaching the boundary. In [13,15] the basic tool is a finite dimensional reduction method of Lyapunov–Schmidt type; here we use a completely variational method. Thus, it is clear that the concentration phenomena and the methods we used in [13,15] are deeply different in nature with respect to the ones we use in the present paper; however, they allow us to state existence and multiplicity results in the same domains (see Examples 2.2 and Remark 2.3) which may be also contractible and even arbitrarily close to starshaped domains in the sense we described in [13] (it is well known that the problem cannot have solutions ifΩis starshaped andp_n²ⁿ₋₂, as a consequence of the Pohožaev’s identity: see [22,4,5]).

Notice that, on the contrary, Dancer and Zhang (see [7]) obtained nonexistence results for supercritical problems in some domains which are nearly starshaped in a different sense with respect to [13] (see also [19,20,25] for other nonexistence results concerning supercritical problems in nonstarshaped domains).

It is worth pointing out that also forλ=0 some concentration phenomena occur, related to the exponentp.

Whenp→_n²ⁿ₋₂, we have concentration near points and this fact has been used to describe the effect of the domain’s shape on the existence and the multiplicity of solutions (see [1,2,6,23]) and to construct multispike solutions when pis sufficiently close to _n²ⁿ₋₂ (see, for example, [12,14] and the references therein). Whenpis not close to _n²ⁿ₋₂, concentration may occur not only near points but also near some more complex sets; for example, in [17] we prove that, forλ=0 andplarge enough, there exist solutions which concentrate near spheres asp→ +∞.

The domains we consider in this paper have radial symmetry with respect to thex_n-axis (see condition (2.1)) and we look for solutions having the same symmetry. The method we use is completely variational and the problem reduces to minimizing the corresponding functional constrained on a suitable manifold. However, let us remark that we do not apply this method directly to problem (1.1): we first modify in a suitable region the nonlinear term, then we solve the modified problem and, finally, we show that the solutions of the modified problem solve also problem (1.1) forλlarge enough, because they are localized in the prescribed region where the nonlinear term has not been modified. This modification of the nonlinear term is due to several reasons, specially whenΩ meets thexn-axis.

In fact (see Remark 3.12) if we try to apply the minimizing arguments without having modified the nonlinear term, we see that the minimum is not achieved ifp_n²ⁿ₋₂(because of concentration phenomena near points of the x_n-axis) while, forp <_n²ⁿ₋₂, the minimum is achieved but the minimum points give solutions which concentrate, asλ→ +∞, near points of thexn-axis, not near spheres (solutions of this type would not be interesting because already well known).

Finally, let us remark that (unlike the finite dimensional reduction methods of Lyapunov–Schmidt type) our construction does not need to know the limit profile of the solution u_λ; however, the method we use suggests that, suitably rescaled, the functionw_λ(ρ, τ )=u_λ(ρ,0, . . . ,0, τ )converges asλ→ +∞to the unique solution of problem

−v+v=v^p−1, v >0 inR², v∈H^1,2(R²), v(0)=max

R² v (1.2)

(see Remark 3.11 for more details).

2. Statement of the main results and examples

Let us consider a bounded domainΩ ofRⁿ, satisfying the following symmetry condition x=(x₁, . . . , x_n)∈Ω ⇐⇒

ρ(x),0, . . . ,0, x_n

∈Ω, (2.1)

whereρ(x)=(n−1

i=1x_i²)^1/2. We say that a functionudefined inΩhas radial symmetry with respect to thexn-axis ifu(x)=u(ρ(x),0, . . . ,0, xn)for allx∈Ω.

Let us set Σ (Ω)=

(ρ, τ )∈R²: ρ >0, (ρ,0, . . . ,0, τ )∈Ω (2.2)

and denote byH_S(Ω)the subspace ofH₀^1,2(Ω)consisting of the functions having radial symmetry with respect to thex_n-axis.

Theorem 2.1. LetΩ be a bounded domain ofRⁿ,n3, satisfying condition (2.1) and assume that there exists an open subsetAofR²such that

0<inf

ρ(x): x∈Ω,

ρ(x), x_n

∈A <inf

ρ(x): x∈Ω,

ρ(x), x_n

∈∂A . (2.3)

Also assumep >2. Then, there existsλ >¯ 0 such that, for allλλ, problem (1.1) has a solution¯ u_λ∈H_S(Ω).

For allλλ,¯ uλ is a bounded smooth function and there existscλ=(ρλ, τλ)∈Σ (Ω)such that, for allη >0, we have

lim

λ→+∞

1 λ^p−2sup

u_λ(x): x∈Ω,

ρ(x), x_n

∈/B(c_λ, η) =0, (2.4)

while sup

Ω

u_λλ^p−2 ∀λλ.¯ (2.5)

Furthermore,c_λ∈Aforλ >0 large enough and

λ→+∞lim ρλ=inf

ρ(x): x∈Ω, ρ(x), xn

∈A . (2.6)

Moreover, lim sup

λ→+∞λ^−1/(p−2)u_λH^1,2

0 (Ω)<+∞, (2.7)

lim sup

λ→+∞λ^{(p−4)/(p−2)}u_λ²_L2(Ω)<+∞, (2.8)

lim sup

λ→+∞λ^−2/(p−2)u_λ^p_Lp(Ω)<+∞. (2.9)

The proof is reported in Section 3.

Examples 2.2. It is clear that, because of the behaviour of the solution uλ as λ→ +∞, for λ large enough Theorem 2.1 guarantees the existence ofkdistinct solutions if the domainΩ satisfies condition (2.3) with respect tokopen subsetsA₁, . . . , Akpairwise disjoint. For example, in domainsΩsuch that

Ω=

(x₁, . . . , x_n)∈Rⁿ: ax_nb, ρ₁(x_n)ρ(x)ρ₂(x_n) (2.10) for suitablea, binR,a < b, andρ₁, ρ₂nonnegative functions in[a, b], the number of distinct solutions, forλlarge enough, is related to the number of local (strict) minimum points of the functionρ₁, with positive minimum values.

This fact allows us to construct examples of domains with trivial topology (even contractible) where the number of solutions is arbitrarily large. For example, let us consider, for allk∈N,r >1 ands >0, the domain

Ω_r,k^s =

x∈Rⁿ: 1<|x|< r, ρ(x) > sx_n,dist(x, C_m^s) >1 form=1, . . . , k−1 , (2.11) where

C_m^s =

x∈Rⁿ: ρ(x)=3m, x_n=3m s

.

Ifr > ^3k_s (1+s²)^1/2, Theorem 2.1 applies with kpairwise disjoint open subsetsA₁, . . . , A_k and guarantees, for λ large enough, the existence of k distinct solutions which concentrate, as λ→ +∞, neark distinct(n−2)- dimensional spheres of the boundary of Ω_r,k^s . Notice that similar domains have been also considered in [12–14, 16,17,21], where concentration phenomena of different type have been used to obtain existence and multiplicity results.

Remark 2.3. The example of the domainΩ_r,k^s (see (2.11)) is particularly significant whenp_n²ⁿ₋₂. In fact, in this case problem (1.1) has no solution forλ0 ifΩ is a starshaped domain, as a consequence of the well known Pohožaev’s identity. The domains of the formΩ_r,k^s allow us to show that problem (1.1), forλ >0 large enough, can have an arbitrarily large number of solutions in domains arbitrarily close to starshaped domains in a sense (introduced in [13]) we describe below (notice that a different definition of nearly starshaped domain has been given in [7] in order to extend Pohožaev nonexistence result).

For each smooth bounded domainΩ inRⁿ, let us set σ (Ω)= sup

x₀∈Ω

inf

ν(x)· x−x0

|x−x0|: x∈∂Ω

, (2.12)

where ν(x) denotes the outward normal to ∂Ω in x. We say that Ω is nearly starshaped if σ (Ω)⁻ = max{0,−σ (Ω)}is small (δ-nearly-starshaped ifσ (Ω)⁻δ).

SinceΩ_r,k^s is not a smooth domain, let us consider its neighbourhood Nη(Ω_r,k^s )=

x∈Rⁿ: dist(x, Ω_r,k^s ) < η .

Then, ifη∈ ]0,1[,Nη(Ω_r,k^s )is a smooth domain forr andslarge enough, limr,s→+∞σ (Nη(Ω_r,k^s ))=0 and Theorem 2.1 guarantees the existence ofkdistinct solutions forλlarge enough.

3. Preliminary results and proof of the main theorem

Notice that, for allλ >0, a functionu∈H₀^1,2(Ω)solves the equation−u+λu=u^p−1 if and only if the functionλ^−1/(p−2)usolves the equation−λ⁻¹u+u=u^p−1. Therefore, if we setε=λ^−1/2, finding solutions of problem (1.1) for largeλ >0 is equivalent to finding solutions, for smallε, of the following problem

−ε²u+u=u^p−1 inΩ,

u >0 inΩ,

u=0 on∂Ω.

(3.1) Let us chooseδ >0 small enough such that, if we set

A_δ=

(ρ, τ )∈A: dist

(ρ, τ ),R²\A

> δ , (3.2)

thenΣ (Ω)∩A_δ= ∅and inf

ρ(x): x∈Ω,

ρ(x), x_n

∈A_δ =inf

ρ(x): x∈Ω,

ρ(x), x_n

∈A

<inf

ρ(x): x∈Ω,

ρ(x), x_n

∈∂A_δ . (3.3)

Then consider a smooth functionθ:R²→ [0,1]such thatθ (ρ, τ )=1 ∀(ρ, τ )∈Aδ andθ (ρ, τ )=0 ∀(ρ, τ )∈ R²\A.

Letg₁:R→R⁺ be the function defined by g₁(t )=0 ∀t 0, g₁(t )=t^p−1 ∀t0 and consider a smooth convex functiong₀:R→R⁺such that lim_t→+∞g₀(t ) <1,g₀(t )g₁(t )∀t∈Randg₀(t )=g₁(t )∀tt₀for a suitablet₀>0.

Then, let us consider the smooth functiong:Ω×R→Rdefined by g(x, t )=θ

ρ(x), x_n

g₁(t )+ 1−θ

ρ(x), x_n

g₀(t ) (3.4)

and the functionalf_ε:H_S(Ω)→Rdefined by f_ε(u)=1

2

Ω

ε²|Du|²+u² dx−

Ω

G(x, u)dx, (3.5)

whereG(x, t )=t

0g(x, τ )dτ.

Throughout this paper, for everyE⊆R², we denote byEthe set defined as follows E=

x=(x₁, . . . , xn)∈Rⁿ: ρ(x), xn

∈E . (3.6)

Notice that, since inf{ρ(x): x∈Ω∩ ˜A}>0 because of condition (2.3), one can verify thatfε is well defined in HS(Ω)and is aC²-functional whose critical points (by the maximum principle) solve the problem

−ε²u+u=g(x, u) inΩ,

u >0 inΩ,

u=0 on∂Ω.

(3.7) It is clear that all nontrivial critical points forf_εbelong to the set

M_ε=

u∈H_S(Ω): u≡0, f_ε(u)[u] =0 . (3.8)

The following lemma describes some properties off_εandM_ε and, in particular, shows that looking for nontrivial critical points forf_εis equivalent to searching critical points forf_εconstrained onM_ε.

Lemma 3.1. For allp >2 andε >0, the following properties hold for the functionalf_ε.

(a) Assumeu∈HS(Ω),u≡0; then, eitherf_ε(t u)[u]>0∀t >0 (what happens, for example, ifu⁺≡0 inΩ∩ ˜A) or there exists a uniquetε>0 such thattεu∈Mε; in this case,f_ε(t u)[u]>0∀t∈ ]0, tε[andf_ε(t u)[u]<0

∀t > t_ε (this case occurs, for example, ifu⁺≡0 inΩ∩ ˜A_δ).

(b) M_ε= ∅and infMεf_ε>0.

(c) M_ε is aC¹-manifold of codimension 1; moreover, every critical point forf_ε constrained onM_ε is a critical point forf_ε.

Proof. (a) First observe that, because of the definition ofg, we have f_ε(t u)[u] =t

Ω

ε²|Du|²+u²

dx >0 ∀t >0 (3.9)

for allusuch thatu⁺≡0. Ifu⁺≡0, the functiont→¹_t

Ωg(x, t u)udx is strictly increasing in]0,+∞[. More- over, for allt >0, we haveg(x, t u)u=t^p−1(u⁺)^p ifx∈Ω∩ ˜A_δ while 0g(x, t u)u[lim_s→+∞g₀(s)]t u²if x∈Ω\ ˜A.

Therefore, property (a) follows easily taking into account that f_ε(t u)[u] =t

Ω

ε²|Du|²+u² dx−1

t

Ω

g(x, t u)udx

∀t >0, (3.10)

where lim

t→0

1 t

Ω

g(x, t u)udx=0, (3.11)

1 t

Ω∩ ˜A_δ

g(x, t u)udx=t^p−2

Ω∩ ˜A_δ

(u⁺)^pdx ∀t >0 (3.12)

and, since limt→+∞g₀(t ) <1, 01

t

Ω\ ˜A

g(x, t u)udx

Ω\ ˜A

u²dx ∀t >0. (3.13)

(b) In order to prove thatM_ε= ∅for allε >0 andp >2, it suffices to chooseϕ¯∈H_S(Ω),ϕ¯0, such that

¯

ϕ≡0 inΩ∩ ˜A_δ(notice thatΩ∩ ˜A_δ= ∅because of the choice ofδ). Then, from property (a) we infer that there existst¯_ε>0 such thatt¯_εϕ¯∈M_ε. Moreover, property (a) implies that, for allu∈M_ε,

fε(u)fε(t u) ∀t0. (3.14)

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

Ω[ε²|Du|²+u²]dx.

Therefore, there existsr_ε>0 andα_ε>0 such that inf

f_ε(u): u∈H_S(Ω),

Ω

|Du|²dx=r_ε²

α_ε. (3.15)

It follows that inf

Mε

f_εα_ε>0, (3.16)

which completes the proof of (b).

(c) The fact thatMε is aC¹ manifold of codimension 1 follows from the implicit function theorem. In fact, consider the functionalΦε:HS(Ω)→Rdefined by

Φε(u)=f_ε(u)[u] ∀u∈HS(Ω). (3.17)

Taking into account condition (2.3), one can verify thatΦεis aC¹functional. Moreover,Φ_ε(u)=0 for allu∈Mε. In fact, ifu∈M_ε, thenu⁺≡0 andΦ_ε(u)=0 that is

ε²

Ω

|Du|²dx+

Ω

u²dx−

Ω

g(x, u)udx=0. (3.18)

Therefore, ifg(x, t )denotes the derivative ofg(x, t )with respect tot, we have Φ_ε(u)[u] =2ε²

Ω

|Du|²dx+2

Ω

u²dx−

Ω

g(x, u)u²dx−

Ω

g(x, u)udx

=

Ω

g(x, u)u−g(x, u)u²

dx. (3.19)

On the other hand, a direct computation shows thatg(x, t )t−g(x, t )t²<0 ∀t >0. Therefore, sinceu⁺≡0, it follows thatΦ_ε(u)[u]<0. SoΦ_ε(u)=0 and the implicit function theorem applies.

Now, letu∈Mε be a critical point forfεconstrained onMε. Then, there exists a Lagrange multiplierµsuch that

f_ε(u)+µΦ_ε(u)=0. (3.20)

It follows, in particular, that

f_ε(u)[u] +µΦ_ε(u)[u] =0, (3.21)

which impliesµ=0 becausef_ε(u)[u] =0 andΦ_ε(u)[u] =0. Thus,uis a critical point forfε. 2

Lemma 3.2. Under the assumptions of Theorem 2.1, the following properties hold for the functionalf_ε and the manifoldM_ε:

(a) f_ε(u) 1

2−1 p

ε²

Ω

|Du|²dx+

1−g₀(∞)

Ω

u²dx

∀u∈M_ε, whereg₀(∞)=limt→+∞g₀(t );

(b) inf

Ω∩ ˜A

(u⁺)^pdx: u∈M_ε

>0.

Proof. In order to prove (a), let us observe that, for allu∈Mε, 0=

Ω

ε²|Du|²+u² dx−

Ω

g(x, u)udx

=

Ω

ε²|Du|²+u² dx−

Ω

˜

θ (x)(u⁺)^pdx−

Ω

1− ˜θ (x)

g₀(u)udx, (3.22)

whereθ (x)˜ =θ (ρ(x), x_n)∀x∈Rⁿ. Now, setG₀(t )=_t

0g₀(τ )dτ and observe that G₀(t )1

2g₀(t )t ∀t∈R (3.23)

becauseg₀is a convex function. As a consequence, we obtain f_ε(u)=1

2

Ω

ε²|Du|²+u² dx−

Ω

G(x, u)dx

1

2

Ω

ε²|Du|²+u² dx−1

p

Ω

θ (x)(u˜ ⁺)^pdx−1 2

Ω

1− ˜θ (x)

g₀(u)udx. (3.24)

It follows that, for allu∈M_ε, f_ε(u)

1 2 −1

p

Ω

ε²|Du|²+u² dx−

Ω

1− ˜θ (x)

g₀(u)udx

1

2 −1 p

Ω

ε²|Du|²+u² dx−

Ω

g₀(u)udx

. (3.25)

Now observe that, sinceg0is convex, we haveg0(t )tg₀(∞)t²∀t∈R, which implies

Ω

g₀(u)udxg₀(∞)

Ω

u²dx. (3.26)

Thus, (a) follows easily from (3.25) and (3.26).

For the proof of (b), let us first prove that

Ω∩ ˜A

(u⁺)^pdx >0 ∀u∈Mε. (3.27)

For allu∈M_εwe have

Ω∩ ˜A

(u⁺)^pdx

Ω∩ ˜A

g(x, u)udx=

Ω

ε²|Du|²+u² dx−

Ω\ ˜A

g(x, u)udx

ε²

Ω

|Du|²dx+

1−g₀(∞)

Ω

u²dx, (3.28)

which implies (3.27) (becauseg₀(∞) <1 andu≡0∀u∈M_ε).

Now observe that, since inf{ρ(x): x∈Ω∩ ˜A}>0 (see condition (2.3)), the subspace ofH^1,2(Ω∩ ˜A)consisting of the radial functions, is compactly embedded inL^p; in particular, for a suitable positive constantc¯_p, we have

Ω∩ ˜A

|Du|²dxc¯_p

Ω∩ ˜A

|u|^pdx 2/p

∀u∈H_S(Ω). (3.29)

Therefore, taking into account (3.28), we obtain

Ω∩ ˜A

(u⁺)^pdxε²

Ω

|Du|²dxε²

Ω∩ ˜A

|Du⁺|²dxε²c¯_p

Ω∩ ˜A

(u⁺)^pdx 2/p

∀u∈M_ε. (3.30)

Since

Ω∩ ˜A(u⁺)^pdx=0 (see (3.27)), it follows

Ω∩ ˜A

(u⁺)^pdx(ε²c¯_p)^p/(p−2) ∀u∈M_ε, (3.31)

which implies property (b). 2

Lemma 3.3. The minimum minM_εfε is achieved for all ε >0. Moreover, every minimizing function uε solves problem (3.7).

Proof. Let(ui)i be a minimizing sequence forfε onMε, that isui ∈Mε ∀i∈Nand limi→∞fε(ui)=infM_εfε. From (a) of Lemma 3.2 we infer that the sequence(u_i)_iis bounded inH₀^1,2(Ω). It follows that, up to a subsequence, it converges to a functionu∈H_S(Ω)weakly inH₀^1,2(Ω)and inL²(Ω). Taking into account that the subspace of H^1,2(Ω∩ ˜A) consisting of the radial functions is compactly embedded inL^p (because of condition (2.3)), we obtain

ilim→∞

Ω∩ ˜A

(u⁺_i )^pdx=

Ω∩ ˜A

(u⁺)^pdx, (3.32)

lim

i→∞

Ω

g(x, u_i)u_idx=

Ω

g(x, u)udx (3.33)

and lim

i→∞

Ω

G(x, u_i)dx=

Ω

G(x, u)dx. (3.34)

In particular, (3.34) implies

ilim→∞fε(ui)fε(u). (3.35)

Therefore, if we show thatu∈Mε, we can conclude thatuis a minimizing function forfεconstrained onMε. Notice thatu≡0 because

Ω∩ ˜A(u⁺)^pdx >0, as we can infer from (3.32) taking into account (b) of Lemma 3.2.

Thus, it remains to prove thatf_ε(u)[u] =0, which (because of (3.33)) is equivalent to showing that

ilim→∞

Ω

|Du_i|²dx=

Ω

|Du|²dx. (3.36)

In order to prove (3.36), we argue by contradiction and assume that (up to a subsequence)

Ω

|Du|²dx < lim

i→∞

Ω

|Du_i|²dx, (3.37)

which (because of (3.33)) impliesf_ε(u)[u]<0. As a consequence, taking into account (a) of Lemma 3.1, there existst∈ ]0,1[such thatt u∈Mε. It follows that

f_ε(t u)=

Ω

1

2g(x, t u)t u−G(x, t u)

dx <

Ω

1

2g(x, u)u−G(x, u)

dx

= lim

i→∞

Ω

1

2g(x, u_i)u_i−G(x, u_i)

dx= lim

i→∞f_ε(u_i)=inf

Mε

f_ε, (3.38)

which is a contradiction (notice that here the strict inequality holds becauseu⁺≡0).

Then, we can conclude thatu∈M_ε andf_ε(u)=min_Mεf_ε.

Finally, notice that every minimizing function forf_εonM_ε is, in particular, a critical point forf_ε constrained onM_ε; hence, the conclusion follows taking into account (c) of Lemma 3.1. 2

For each domainΣinR², let us set m_ε(Σ )=inf

Σ

ε²|Dv|²+v²

dξ: v∈H₀^1,2(Σ ),

Σ

|v|^pdξ=1

. (3.39)

It is well known (see [3,8,9,26]) that, forΣ=R², the minimummε(R²)is achieved by a positive function which is unique modulo translations, radially symmetric with respect to the origin, decreasing when the radial coordinate increases and decaying exponentially at infinity together with its derivatives.

Lemma 3.4. For each domainΣofR², the following properties hold for allε >0 andp >2:

(a) m_ε(Σ )=ε^2(p−2)/pm₁(¹_εΣ ); (b) limε→0ε^{−2(p−2)/p}m_ε(Σ )=m₁(R²).

Proof. For each functionv∈H₀^1,2(Σ ), consider the functionv_εinH₀^1,2(¹_εΣ )defined by v_ε(ξ )=v(εξ ) ∀ξ∈ 1

εΣ. (3.40)

A direct computation shows that

Σ

ε²|Dv|²+v² dξ=ε²

1 εΣ

|Dv_ε|²+v²_ε dξ

(3.41)

and

Σ

|v|^pdξ=ε²

1 εΣ

|v_ε|^pdξ. (3.42)

Ifv≡0, setv¯=v/vL^p(Σ )andv¯_ε=v_ε/v_εLp(¹_εΣ ). Then, the above computations show that

Σ

ε²|Dv¯|²+ ¯v²

dξ=ε^2(p−2)/2

1 εΣ

|Dv¯ε|²+ ¯v_ε²

dξ, (3.43)

which clearly implies (a).

Taking into account (a), the proof of (b) is equivalent to showing that lim

ε→0m₁ 1

εΣ

=m₁(R²). (3.44)

It is clear that m₁(¹_εΣ )m₁(R²) for all ε >0 because each function of H₀^1,2(¹_εΣ ) can be extended by the value zero inR²\ ¹_εΣ. Then, it is sufficient to show that for allε >0 there exists a functionw_ε∈H₀^1,2(¹_εΣ ),

w_εL^p₍¹

εΣ )=1,such that lim

ε→0

1 εΣ

|Dw_ε|²+ w²_ε

dξ=m₁(R²). (3.45)

Letw∈H^1,2(R²)be the positive function (see [3,8,9,26]) such that sup

R²

w= w(0),

R²

| w|^pdξ=1,

R²

|Dw|²+ w²

dξ=m₁(R²). (3.46)

Let us fix ξ₀∈Σ and consider a cut-off function ϕ∈C₀^∞(Σ ), such that 0ϕ(ξ )1 ∀ξ ∈Σ and ϕ(ξ )=1

∀ξ∈B(ξ₀, r₀)for a suitabler₀>0.

Now setw_ε(ξ )=ϕ(εξ )w(ξ−ξ₀/ε)∀ξ∈¹_εΣandw_ε=w_ε/w_εL^p₍¹

εΣ ).

Then, taking into account thatwand its derivatives decay exponentially at infinity, a direct computation shows that (3.45) holds for the functionw_εdefined as above. 2

Lemma 3.5. For allp >2, we have lim sup

ε→0

ε⁻²min

M_ε f_ε<+∞. (3.47)

Proof. Clearly, it suffices to show that for allε >0 there existsu˜_ε∈M_εsuch that lim sup

ε→0

ε⁻²f_ε(u˜_ε) <+∞. (3.48)

Choose ξ₀∈Σ (Ω)∩A_δ and r₀>0 such that B(ξ₀, r₀)⊂Σ (Ω)∩A_δ. Since B(ξ₀, r₀)is a bounded domain of R², for all p >2 there exists a minimizing function vε∈H₀^1,2(B(ξ₀, r₀)), vε0 in B(ξ₀, r₀), such that

B(ξ₀,r₀)|vε|^pdξ=1 and

B(ξ0,r0)

ε²|Dv_ε|²+v_ε²

dξ=m_ε

B(ξ₀, r₀)

. (3.49)

Letuε∈H₀^1,2(Ω)be the function defined byuε(x)=vε(ρ(x), xn)if(ρ(x), xn)∈B(ξ0, r0),uε(x)=0 otherwise.

A simple computation shows that there exist two constantsc1andc2, depending only onξ0andr0, such that

Ω

ε²|Duε|²+u²_ε dxc₁

B(ξ0,r0)

ε²|Dvε|²+v²_ε

dξ (3.50)

and

Ω

|u_ε|^pdxc₂

B(ξ0,r0)

|v_ε|^pdξ =c₂; (3.51)

moreover, because of condition (2.3), we can choosec₂>0.

It follows that, if we setu¯ε=uε/uεL^p(Ω), we have

Ω

ε²|Du¯_ε|²+ ¯u²_ε

dxc₃m_ε

B(ξ₀, r₀)

(3.52) for a suitable constantc3depending only onξ0, r0andp.

Taking into account thatg(x, tu¯_ε(x))= |tu¯_ε(x)|^p−1 ∀t >0, we infer that there existst¯_ε>0 such thatu˜_ε=

¯

t_εu¯_ε∈M_ε.

Moreover, a direct computation shows that f_ε(u˜_ε)=

1 2 −1

p

Ω

ε²|Du¯_ε|²+ ¯u²_ε dx

p/(p−2)

. (3.53)

It follows that, for a suitable constantc₄(depending only onξ₀, r₀, p) inf

Mε

f_εc₄ m_ε

B(ξ₀, r₀)p/(p−2)

. (3.54)

Hence (3.47) follows easily, taking into account (b) of Lemma 3.4. 2

Corollary 3.6. For allε >0, letu_εbe a minimizing function forf_εconstrained onM_ε. Then lim sup

ε→0

Ω

|Du_ε|²dx <+∞ and lim sup

ε→0

ε⁻²

Ω

u²_εdx <+∞ (3.55)

(which, in particular, impliesu_ε→0 inL²(Ω)asε→0).

The proof is a direct consequence of Lemma 3.5 and (a) of Lemma 3.2 (taking into account thatg₀(∞) <1).

Lemma 3.7. For allε >0, letu_εbe a critical point forf_ε constrained onM_ε. Then sup

Ω

u_ε1. (3.56)

Proof. Arguing by contradiction, assume that, for someε >0, sup

Ω

uε<1. (3.57)

Then, we have 0< g

x, u_ε(x)

< u_ε(x) ∀x∈Ω (3.58)

because of the maximum principle and the definition ofg.

Lete1be a function inH₀^1,2(Ω)such thate1>0 inΩande1+λ1e1=0, whereλ1denotes the first eigenvalue of the Laplace operator−inH₀^1,2(Ω). Sinceu_εsolves problem (3.7), a direct computation shows that

Ω

g(x, u_ε)−u_ε−ε²λ₁u_ε

e₁dx=0, (3.59)

which is a contradiction because of (3.58). 2

Lemma 3.8. For allε >0, letu_εbe a minimizing function forf_εconstrained onM_ε. Thenu_εis bounded inΩand it is a smooth solution of problem (3.7).

Proof. First observe that, since

Ω|Du_ε|²dx <+∞and inf{ρ(x): x∈Ω∩ ˜A}>0,the functionu_ε (which has radial symmetry with respect to thexn-axis) belongs toL^q(Ω∩ ˜A)for allq1. In particular, forq > (p−1)ⁿ₂, it follows that the functionwεdefined by

wε(x)= 1 n(n−2)ω_nε²

Ω∩ ˜A

u^p_ε⁻¹(y)

|x−y|ⁿ⁻²dy,

whereωn denotes the measure of the unit ball ofRⁿ, is a bounded function inHS(Ω). Taking into account that g0(t )t ∀t0, we have(wε−uε)0 (in weak sense) inΩ; sincewε0 on∂Ω, it follows thatuεwε

inΩ. Therefore,uεis bounded too; thus, sinceg(x, t )is a smooth function, it follows thatuεis a smooth solution of (3.7). 2

Lemma 3.9. For allα∈ ]0,1[andε >0, letu_εbe a minimizing function forf_ε constrained onM_εand consider the set

Σ_ε^α=

(ρ, τ )∈Σ (Ω): u_ε(ρ,0, . . . ,0, τ ) > α . (3.60)

Then

(a) meas(Σ_ε^α) >0,∀ε >0, ∀α∈ ]0,1[;

(b) limε→0meas(Σ_ε^α)=0, ∀α∈ ]0,1[;

(c) for allε >0 andα∈ ]0,1[, there existc_α,ε=(ρ_α,ε, τ_α,ε)∈Σ (Ω)andr_α,ε>0 such that Σ_ε^α⊆B(cα,ε, rα,ε) and lim

ε→0rα,ε=0 ∀α∈ ]0,1[. (3.61)

Moreover, we have

εlim→0dist(cα,ε, A)=0. (3.62)

Proof. Property (a) is a direct consequence of Lemma 3.7, while (b) follows from Corollary 3.6 (becauseuε→0 inL²(Ω)asε→0).

Property (c) is a consequence of the fact thatf_ε(u_ε)=minM_εf_ε, which implies the existence ofc_α,ε∈Σ (Ω) andr_α,ε >0 such that (3.61) holds. In fact, let us argue by contradiction and assume that, for someα∈ ]0,1[, (3.61) does not hold for any choice ofc_α,εandr_α,ε. Taking into account the definition ofg(x, t ), we can fixα >¯ 0, small enough, such thatg(x, α)−α <0∀α∈ ]0,α¯[,∀x∈Ω. SinceΣ_α,ε⊆Σ_α,ε forα> α, it is clear that, in order to prove our assertion, it suffices to consider only the caseα∈ ]0,α¯[.

Then, fixα∈ ]0,α¯[, consider the setA^θ= {(ρ, τ )∈R²: θ (ρ, τ ) >0}and observe thatΣα,ε∩A^θ= ∅ ∀ε >0;

in fact, otherwise, we should haveg(x, uε(x))−uε(x) <0 a.e. inΩ (sinceg0(t ) < t∀t >0) which is impossible (by the maximum principle) becauseu_ε>0.

If the diameter of Σ_α,ε does not tend to zero as ε→0 (as we are assuming by contradiction), then Σ_α,ε (see (3.6)) has at least two connected components forε >0 small enough.

In fact, assume that (up to a subsequence)

εlim→0diam(Σ_α,ε) >0 (3.63)

and, arguing by contradiction, assume also that (up to a subsequence)Σ_α,ε is connected for allε >0.

From (3.63) and condition (2.3) we infer that there existsd >0 such that d <min

1 2 lim

ε→0diam(Σ_α,ε), inf

ρ: (ρ, τ )∈Σ (Ω)∩A . (3.64)

SinceΣ_α,ε∩A^θ = ∅ ∀ε >0 andA^θ⊂A, we can choose a pointp_ε inΣ_α,ε∩A∀ε >0; then, forε >0 small enough, there existp_ε ∈Σ_α,ε, such that dist(p_ε, p_ε)=d, and a continuous path inΣ_α,ε∩B(p_ε, d), joiningp_ε andp_ε(in factΣ_α,εis open connected, just asΣ_α,ε, and cannot be contained inB(p_ε, d)as diam(Σ_α,ε) >2d).

Now observe that, because of (3.64), we can chooseρ >¯ 0 such that 2ρ¯+d <inf

ρ: (ρ, τ )∈Σ (Ω)∩A . (3.65)

Then, consider a smooth functionζ:R→ [0,1]such thatζ (t )=0∀tρ,¯ ζ (t )=1∀t2ρ¯and setϕε(ρ, τ )= ζ (ρ)uε(ρ,0, . . . ,0, τ ) ∀(ρ, τ )∈Σ (Ω). It is clear that ϕε is a smooth function in H₀¹(Σ (Ω)) andϕε(ρ, τ )= u_ε(ρ,0, . . . ,0, τ )∀(ρ, τ )∈Σ (Ω)∩B(p_ε, d).

Moreover, sinceρ >¯ 0, Corollary 3.6 implies that lim sup

ε→0

Σ (Ω)

|Dϕ_ε|²dρdτ <+∞ and lim sup

ε→0

1 ε²

Σ (Ω)

ϕ_ε²dρdτ <+∞. (3.66)