On the existence of blowing-up solutions for a mean field equation

On the existence of blowing-up solutions for a mean field equation

Pierpaolo Esposito

, Massimo Grossi

, Angela Pistoia

aDipartimento di Matematica, Università degli Studi “Roma Tre”, Largo S. Leonardo Murialdo, 1, 00146 Roma, Italy bDipartimento di Matematica, Università di Roma “La Sapienza”, P.le A. Moro, 00166, Roma, Italy cDipartimento di Metodi e Modelli Matematici, Università di Roma “La Sapienza”, Via Scarpa, 16, 00166, Roma, Italy

Received 23 February 2004; received in revised form 6 June 2004

Abstract

In this paper we construct single and multiple blowing-up solutions to the mean field equation:

−u=λ^{V (x)eu}

ΩV (x)e^u inΩ, u=0 on∂Ω,

whereΩis a smooth bounded domain inR²,V is a smooth function positive somewhere inΩandλis a positive parameter.

2005 Elsevier SAS. All rights reserved.

Résumé

Dans ce papier nous construisons des solutions qui explosent pour l’équation de champ moyen : −u=λ^{V (x)eu}

ΩV (x)e^u inΩ, u=0 on∂Ω,

oùΩest un domaine borné dansR²,V est une fonction positive dansΩetλest un paramètre positif.

2005 Elsevier SAS. All rights reserved.

Keywords: Mean field equation; Peak solutions; Green’s function

✩ The first and second authors are supported by M.U.R.S.T., project “Variational methods and nonlinear differential equations”. The third author is supported by M.U.R.S.T., project “Metodi variazionali e topologici nello studio di fenomeni non lineari”.

* Corresponding author.

E-mail addresses: esposito@mat.uniroma3.it (P. Esposito), grossi@mat.uniroma1.it (M. Grossi), pistoia@dmmm.uniroma1.it (A. Pistoia).

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

doi:10.1016/j.anihpc.2004.12.001

1. Introduction

We consider the problem:

−u=λ^{V (x)eu}

ΩV (x)e^u inΩ,

u=0 on∂Ω,

(1.1) whereΩ is a smooth bounded domain in R²,V is a smooth function positive somewhere inΩ andλ∈Ris a positive parameter.

This equation occurs in various context such as: conformal geometry (cf. [2]), statistical mechanics (cf. [9,10]) and several other area of applied mathematics (cf. e.g. [6,13,23,34]). In statistical mechanics it is referred as the

“mean field equation”. In all those contexts, there is a definite interest to construct solutions which “blow-up” and

“concentrate” at a set of given points, whose location carries relevant information about the geometrical/physical properties of the problem under exam.

We are interested in finding solutionsu_λto (1.1) which blow-up atkdifferent pointsq₁, . . . , q_k inΩalong a sequenceλ→8π k, in the following sense:

λ V (x)e^uλ

ΩV (x)e^uλ 8π k i=1

δq_i in the sense of measures inΩ, (1.2)

whereδ_pdenotes the Dirac measure atp. HereΩ= {q∈Ω: V (q) >0}.

In order to state some new and old results it is useful to introduce some notation. LetG_Ω denote the Green’s function of−with Dirichlet boundary condition onΩ,namely for anyy∈Ωit holds

−_xG_Ω(x, y)=δ_y(x) ifx∈Ω, G_Ω(x, y)=0 ifx∈∂Ω,

and letH_Ω(x, y)=G_Ω(x, y)+_2π¹ log|x−y|be its regular part. We will refer toG_Ω,H_Ω simply asGandH respectively when the dependence inΩis not relevant.

First of all, let us point out that if inf_ΩV >0, it is known that if the sequenceu_λis a family of solutions to (1.1) which is not uniformly bounded from above forλbounded, thenuλblows-up atkdifferent pointsq₁, . . . , qkinΩ along a sequenceλ→8π kand(q₁, . . . , qk)is a critical point for the function:

F(ξ1, . . . , ξk)= k i=1

HΩ(ξi, ξi)+ k

i,j=1 i=j

GΩ(ξi, ξj)+ 1 4π

k i=1

logV (ξi) (1.3)

(cf. [8,22,28]; see also [35,38]). Secondly we note that, if λ∈(0,8π ) problem (1.1) admits a solution u_min,λ corresponding to a global minimum point for the functional

Jλ(u)=1 2

Ω

|∇u|²−λlog

Ω

V (x)e^u

, u∈H¹₀(Ω), which is coercive by the Moser–Trudinger inequality (cf. [32,40]).

Let us consider the caseV (x)≡1. In [9,10] the authors study the asymptotic behavior ofu_min,λasλ→8π⁻, and show that eitheru_min,λconverges to a minimum ofJ_8πor admits a single blow-up pointq∈Ω,corresponding to a maximum point of the Robin’s functionH_Ω(·,·). They observe that both possibilities can occur. For instance, ifΩis a disk or a simply connected domain sufficiently close to a disk, then concentration does occur asλ→8π⁻. On the contrary, there are simply connected regions (for instance, rectangles with large ratio between the sides) for which concentration cannot occur, asλ→8π⁻.We also mention that Suzuki in [39] proves that the solution of (1.1) is unique whenλ∈(0,8π )andΩ is a simply connected domain. Ifλ8π the situation becomes more

complex. As well known (see [4]), ifΩ is the unit disk andλ8π, then there are no solutions to (1.1). In [41]

the author constructs a sequence of solutions on simply connected domains “blowing-up” at a critical point of the Robin’s function (see also [29,30] and [31] for similar results in a more general setting). Non-simply connected domains are considered in [19].

The general case concerning the existence of solutions with multiple blow-up points so far has been treated only by Baraket–Pacard in [5]. They prove that any nondegenerate critical point(q₁, . . . , q_k)of the functionFdefined in (1.3) generates a family of solutionsu_λ which blow-up atq₁, . . . , q_k asλ→8π k.In [21] the author extends the previous result by allowing a general weight functionV (x). We are aware of only another successful approach to handle multiple peak blow-up solutions which has been introduced by Chen and Lin in [16] for the annulus.

A general degree formula for the corresponding Fredholm map can be found in [15].

Perturbative problems with exponential nonlinearities in dimension two seem to be much more difficult to han- dle (beside [5], see also [11,12] and [36]) in contrast to similar problems in higher dimensions. Baraket–Pacard’s method is too demanding in terms of assumptions (the non-degeneracy of the critical point) and functional frame- work, and pays in return with a very accurate control on the asymptotics of the solutions. Instead, in the spirit of some perturbation methods available in higher dimension (see e.g. [3] and [37]), we propose an alternative approach to the existence of blowing-up solutions by introducing for our problem a perturbation setting in the spaceH₀¹(Ω). This more flexible approach allows to replace the non-degeneracy assumption of Baraket–Pacard, by showing that “stable” (in a suitable sense) critical points of the functionFin (1.3) generate solutions for (1.1) which blow-up at those points. It is important to point out that this “weaker” stability of critical points enables us to construct some domains where a large number of blowing-up solutions exists.

Let us state now the main results of the paper. RecallΩ= {q∈Ω: V (q) >0}and let∆= {(q₁, . . . , qk)∈ Ω^k:q_i=q_jfor somei=j}.

Theorem 1.1. Assume thatK is a stable critical set forFin(Ω)^k\∆(see (1.3) and Definition 5.1). Then there exists a family of solutions of (1.1) which blow-up at pointsq₁, . . . , q_kwith the property(q₁, . . . , q_k)∈K,along a sequenceλ→8π k, in the sense that (1.2) holds.

In Section 5 we takeV (x)≡1 and we exhibit some simply connected domains where many solutions to (1.1) blowing-up at one or more points exist: see Theorems 5.4, 5.5 and Corollary 5.6. This result is in striking contrast with Suzuki’s uniqueness result [39] for the caseλ <8π.

As a consequence of Theorem 1.1, changing sign weight functionsV (x)generate many solutions as we show in the following result:

Theorem 1.2. LetΩ₁, . . . , Ω_ν be the connected components ofΩ inΩ.Then there exist at leastν families of solutions of (1.1) which blow-up at a maximum pointξiof the functionFinΩ_ifori=1, . . . , ν,along a sequence λ→8π, in the sense that (1.2) holds.

As far as it concerns the existence of solutions blowing-up at a point whenV (x)≡1, we would like also to quote the following existence result:

Theorem 1.3. LetV (x)≡1. Assume thatcis a stable critical value for the Robin’s functionH_Ω(·,·)(see De- finition 6.1). Then there exists a family of solutions of (1.1) which blow-up at a point q with the properties H_Ω(q, q)=cand∇H_Ω(q, q)=0,along a sequenceλ→8π, in the sense that (1.2) holds.

It allows to find solutions blowing-up at critical points of the Robin’s function of “saddle-type”, which a priori are not stable according to Definition 5.1. In Section 6 we exhibit some domains where this kind of solutions appears: see Theorems 6.2 and 6.3.

Let us sketch the main ideas involved in the proof. A crucial role in the construction of our solution is played by the problem:

−U=ρ²e^U inR²,

R² e^U<+∞, (1.4)

whereρ=0 is a fixed parameter. In [14] (see also the Liouville representation formula in [27]) it is shown that all the solutions to (1.4) take the form:

Uτ,ξ(x)=log 8τ²

(τ²ρ²+ |x−ξ|²)², x∈R², (1.5)

forτ >0 andξ∈R². Following Bahri’s idea (see [3] and also [37]) we look for solutions to (1.1) in the form u(x)=

k i=1

P U_{τλ,i,ξλ,i}(x)+φ^λ(x)

for suitable positive parametersτ_λ=(τ_λ,1, . . . , τ_λ,k)and pointsξ_λ=(ξ_λ,1, . . . , ξ_λ,k).Here, the remainder termφ^λ goes to zero in H¹₀(Ω)as λgoes to 8π k andP U_τ,ξ denotes the projection ofU_τ,ξ into H¹₀(Ω), in other words, P U_τ,ξ is uniquely defined as satisfying:

−P U_τ,ξ= −U_τ,ξ =ρ²e^Uτ,ξ inΩ,

P U_τ,ξ =0 on∂Ω. (1.6)

In order to determine τ_λ andξ_λ in Section 4 we reduce the problem to a finite dimensional one. Although this problem has many similarities to equations with critical growth in smooth domains ofR^N,N3, our proof displays important differences. First of all, we point out that the parameterτλwill be a priori prescribed in terms of the pointξ_λ(see formula (2.6)). An important consequence is that the functionφ^λ will be found in a space of codimension 2k.We recall that the standard procedure in elliptic problems involving the critical Sobolev exponent takes place in spaces of codimension(N+1)k. This different choice of the space will require delicate computations in order to establish some invertibility property of the corresponding linearized operator, which plays a crucial role in the finite dimensional reduction. This analysis will be carried out in Section 3. In Section 4 we study the reduced problem and give the expansion of the functional associated to the problem (1.1). We have collected some technical computations in Appendix A, B, C, D.

We learnt that related results to this paper have been proved at the same time, independently and using different arguments by Del Pino, Kowalczyk and Musso (see [17]).

2. Setting of the problem

In order to prove the existence of blow-up solutions to (1.1), we will consider the problem:

−u=ρ²V (x)e^u inΩ,

u=0 on∂Ω, (2.1)

and seek solutions such that ρ²V (x)e^u8π

k i=1

δ_qi in the sense of measures inΩ, asρ→0, for some pointq=(q₁, . . . , q_k)∈(Ω)^k\∆. Via the transformationρ²=λ/(

ΩV (x)e^u)it is possible to show that for concentrating solutions problem (2.1) is equivalent to (1.1).

It will be useful to rewrite problem (2.1) in a more convenient setting. For this purpose, let us introduce the following definition.

Definition 2.1. Forp >1, leti_p^∗: L^p →H¹₀(Ω)be the adjoint operator relative to the immersion i: H¹₀(Ω) → L

p

p−1(Ω)and leti^∗:

p>1L^p→H¹₀(Ω)be defined by the propertyi^∗|L^p=i_p^∗for anyp >1.

By definition,v=i^∗(u)holds if and only if (v, ϕ)=

Ω

uϕdx, ∀ϕ∈H¹₀(Ω), and the following estimate holds:

Lemma 2.2. For anyp >1 there existsc_p>0 such that i^∗(u) c_pu^p_L, ∀u∈H¹₀(Ω).

H¹₀(Ω)is an Hilbert space equipped with the usual inner product (u, v)=

Ω

∇u∇vdx and induced norm

u =

Ω

|∇u|²dx 1/2

.

Next, we recall the Moser–Trudinger inequality (cf. [32,40]):

Lemma 2.3. There exists a constantc >0 such that for any smooth bounded domainΩ⊂R²we have:

Ω

e

4π u²/u²

H10c|Ω|, ∀u∈H¹₀(Ω).

In particular, by Lemma 2.3 we deduce the following useful estimate:

Remark 2.4. There exists a constantc >0 such that for anyη >0,

Ω

e^ηuc|Ω|e

η2 16πu²

H10, ∀u∈H¹₀(Ω).

In particular, the map:

H¹₀(Ω)→L^p(Ω), u→e^u

is continuous for everyp >1 and we can rewrite problem (2.1) in the following equivalent form:

u=i^∗(ρ²V (x)e^u) inΩ,

u∈H¹₀(Ω). (2.2)

LetP Uτ,ξ be the projection onto H¹₀(Ω)ofUτ,ξ (see (1.5), (1.6)). Thus, by Definition 2.1 we have P Uτ,ξ:=i^∗(ρ²e^Uτ,ξ).

We are looking for solutions to (2.2) of the form u(x)=

k i=1

P U_{τρ,i,ξρ,i}(x)+φ^ρ(x)

for suitable positive parametersτ_ρ=(τ_ρ,1, . . . , τ_ρ,1) and pointsξ_ρ =(ξ_ρ,1, . . . , ξ_ρ,k). The remainder term φ^ρ belongs to a suitable space which will be defined as follows. Let

ψ_τ,ξ⁰ (x)=∂Uτ,ξ

∂τ (x)=2 τ

|x−ξ|²−τ²ρ²

|x−ξ|²+τ²ρ², x∈Ω, (2.3)

and forj =1,2 let ψ_τ,ξ^j (x)=∂U_τ,ξ

∂ξ_j (x)=4 (x−ξ )_j

τ²ρ²+ |x−ξ|², x∈Ω. (2.4)

Forj=0,1,2, the functionψ_τ,ξ^j is a solution for the equation−ψ=ρ²e^Uτ,ξψinR². LetP ψ_τ,ξ^j be the projection into H¹₀(Ω)ofψ_τ,ξ^j , i.e.

−P ψ_τ,ξ^j =ρ²e^Uτ,ξψ_τ,ξ^j inΩ,

P ψ_τ,ξ^j =0 on∂Ω. (2.5)

For anyξ=(ξ1, . . . , ξk)∈(Ω)^k\∆, set τ_i(ξ )=

V (ξi)

8 e^{4π(HΩ(ξi,ξi)+}

j=iGΩ(ξi,ξj)). (2.6)

SetU_i=U_{τi(ξ ),ξi} andψ_i^j:=ψ_τ^j

i(ξ ),ξi forj=0,1,2,i=1, . . . , kandξ∈(Ω)^k\∆. We consider the subspace of H¹₀(Ω):

K_ξ=span{P ψ_i^j, j=1,2, i=1, . . . , k}, and its complement

K_ξ^⊥=

φ∈H¹₀(Ω)|(φ, P ψ_i^j)=0, j=1,2, i=1, . . . , k , and consider the corresponding orthogonal projections:

Πξ: H¹₀(Ω)→Kξ and Π_ξ^⊥: H¹₀(Ω)→K_ξ^⊥.

3. The linear problem: a key lemma

Let us introduce the linear operatorL^ρ_ξ:K_ξ^⊥→K_ξ^⊥defined as follows:

L^ρ_ξ(φ)=Π_ξ^⊥ φ−i^∗

ρ²V (x)e^{ki=1P Ui}φ .

In order to solve Eq. (4.1), a crucial ingredient is given by the following result, which is similar in spirit to an invertibility property established in [11]:

Proposition 3.1. Letξ∈(Ω)^k\∆. There existsρ₀>0 and a constantc >0 such that for anyρ∈(0, ρ₀)we have L^ρ_ξ(φ) c

|logρ|φ, ∀φ∈K_ξ^⊥.

In particular, the operatorL^ρ_ξ is invertible and(L^ρ_ξ)⁻¹|logρ|/c.

Moreover, the estimates are uniform in compact sets of(Ω)^k\∆.

Proof. We need only to establish the validity of the following estimate:

L^ρ_ξ(φ) c

|logρ|φ ∀φ∈K_ξ^⊥, (3.1)

for some uniform constantc >0. Indeed, by (3.1) we deduce thatL^ρ_ξ is an injective operator with closed image.

SinceL^ρ_ξ is selfadjoint, we also conclude that it is onto and its inverse(L^ρ_ξ)⁻¹satisfies(L^ρ_ξ)⁻¹|logρ|/c.

To prove (3.1), we argue by contradiction. Suppose there exist sequences ρ→0, ξ→ξ₀∈(Ω)^k\∆, φ∈K_ξ^⊥: φ =1 and L^ρ

ξ(φ) =o 1

|logρ|

. (3.2)

Write, φ−i^∗

ρ²V (x)e^{ki=1P Ui}φ

=ψ+w, (3.3)

whereψ∈K_ξ^⊥,w∈Kξandψ =o(1/|logρ|)→0. Equivalently (in a weak sense), there holds −φ=ρ²V (x)e^ki=1P Uiφ−(ψ+w) inΩ,

φ=0 on∂Ω.

Step 1. For anyp∈(1,2), w =O

ρ^(2−p)/p

→0. (3.4)

Letw=_k

h=1

₂

l=1c_hlP ψ_h^l.We multiply (3.3) byP ψ_s^j,j=1,2 ands=1, . . . , k, and get k

h=1

2 l=1

c_hl

P ψ_h^l, P ψ_s^j

H¹₀ = −ρ²

Ω

V (x)e^{ki=1P Ui}φP ψ_s^j. (3.5)

By Lemma A.4 the L.H.S. of (3.5) is estimated as follows:

L.H.S.= D

τ²ρ²c_sj+O _k

h=1

2 l=1

|c_hl|

. (3.6)

Moreover, the R.H.S. of (3.5) takes the form:

−ρ²

Ω

V (x)e^{ki=1P Ui}φP ψ_s^j

=

Ω

ρ²

k i=1

e^Ui−ρ²V (x)e

k i=1P Ui

φP ψ_s^j−ρ²

Ω

e^Usφ[P ψ_s^j−ψ_s^j] −ρ²

i=s

Ω

e^UiφP ψ_s^j, (3.7) where we have used (2.5) andφ∈K_ξ^⊥. Fix p∈(1,2)and use Lemmata A.4 and B.1, together with Hölder’s inequality (here 1/q+1/p <1), to get

Ω

ρ²V (x)e

k

i=1P Ui−ρ² k i=1

e^Ui

φP ψ_s^j

ρ²V (x)e^{ki=1P Ui}−ρ² k i=1

e^Ui

L^p

φL^qP ψ_s^j =O(ρ^2(1−p)/p). (3.8)

Using Lemma A.2, Proposition A.3 and Hölder’s inequality (with 1/q+1/p=1), we get, ρ²

Ω

e^Usφ[P ψs^j−ψs^j] =O

ρ²e^UsL^pφL^q

=O(ρ^2(1−p)/p). (3.9)

Using Lemma C.6 and Hölder’s inequality (with 1/q+1/p=1), we obtain, ρ²

i=s

Ω

e^UiφP ψ_s^j =O

i=s

ρ²e^UiP ψ_s^jL^pφL^q

=O(ρ^2/p). (3.10)

Then, by (3.7)–(3.10) it follows that the R.H.S. of (3.5) satisfies:

R.H.S.=O(ρ^2(1−p)/p). (3.11)

Inserting the estimates (3.6) and (3.11) into (3.5), we deduce that k

h=1

2 l=1

|c_hl| =O(ρ^2/p). (3.12)

Finally, by Lemma A.4 and by (3.12) we deduce that w =O(ρ^(2−p)/p)→0,

and claim (3.4) is proved.

Step 2. For anyi=1, . . . , k (φ, P ψ_i⁰)_H1

0→0. (3.13)

Leti=1, . . . , k. Note that, forτ_i>0 andξ_i∈Ω, the functions w_i(x)= 4

3τi

log

τ_i²ρ²+ |x−ξ_i|²τ_i²ρ²− |x−ξ_i|² τ_i²ρ²+ |x−ξ_i|² +8

3

τiρ² τ_i²ρ²+ |x−ξ_i|² and

ti(x)= −2 τ_iρ² τ_i²ρ²+ |x−ξ_i|²

satisfy:−w_i −ρ²e^{Uτi ,ξi}w_i =ρ²e^{Uτi ,ξi}ψ_τ⁰

i,ξ_i and−t_i −ρ²e^{Uτi ,ξi}t_i =ρ²e^{Uτi ,ξi} inR²respectively. A straight- forward calculation shows that,

Ω

|∇w_i|²=M_i²

1+o(1)

(logρ)²,

Ω

|∇t_i|²=O(1) asρ→0 (3.14)

withM_i=_3τ³²_i(

R² |y|²

(1+|y|²)⁴)^1/2, and (3.14) holds uniformly in dist(ξ_i, ∂Ω)ε,ε >0, andτ_i bounded away from zero.

Let nowτ_i=τ_i(ξ )andξ as specified in (3.2). The projectionP u_i∈H¹₀(Ω)of the function ui=wi+16π

3τ_i H (ξi, ξi)ti

satisfies

−P u_i−ρ²V (x)e^{kh=1P Uh}P u_i=ρ²e^Uiψ_i⁰−ρ²

h=i

e^Uh16π

3τ_i G(ξ_h, ξ_i)+F_i,

where

Fi(x)= −16π 3τ_i

H (x, ξi)−H (ξi, ξi)

ρ²e^Ui+

ui−P ui+16π

3τ_i H (x, ξi)

ρ²e^Ui +

ρ²

k h=1

e^Uh−ρ²V (x)e^{kh=1P Uh}

P u_i+ρ²

h=i

e^Uh 16π

3τ_i G(ξ_h, ξ_i)−P u_i

. (3.15)

Sinceu_i−P u_i+^16π_3τiH (x, ξ_i)is an harmonic function with boundary values satisfying u_i−P u_i+16π

3τ_i H (x, ξ_i) 2,α,∂Ω

Cρ²

for anyα∈(0,1), by elliptic regularity theory it follows that ui−P ui+16π

3τ_i H (x, ξi) 2,α,Ω

Cρ²

for anyα∈(0,1). Hence, we get that P u_i(x)=16π

3τi

G(x, ξ_i)+O(ρ²)inC_loc⁰ Ω\ {ξ_i}

, (3.16)

P u_i =M_i

1+o(1)

|logρ|, (3.17)

and by means of Lemmata A.2, B.1, from (3.15) we deduce thatfi=i^∗(Fi)satisfies:

f_i =o(1), asρ→0.

Therefore, P u_i−i^∗

ρ²V (x)e^{kh=1P Uh}P u_i

=P ψ_i⁰−16π 3τ_i

h=i

G(ξ_h, ξ_i)P U_h+f_i (3.18)

withfi →0. Multiply (3.3) byP ui and use (3.18) to obtain:

(φ, P ψ_i⁰)_H1

0=16π

3τ_i

h=i

G(ξh, ξi)(φ, P Uh)_H1 0+o

P u_i

|logρ|

+o(1)

=16π 3τi

h=i

G(ξ_h, ξ_i)(φ, P U_h)_H1

0+o(1),

where we have taken in account (3.2), (3.17) and the estimate in Step 1. In conclusion, we need to show that (φ, P U_h)_H1

0=o(1)for anyh=1, . . . , k. Multiply (3.3) byP ψ_i⁰and use Lemmata A.4, B.1 to obtain:

ρ²

Ω

e^Uiφ(ψ_i⁰−P ψ_i⁰)=ρ²

h=i

Ω

e^UhφP ψ_i⁰+o(1).

By (A.3), (A.4) and Lemmata A.2, A.4, finally we deduce that (φ, P U_i)_H1

0=ρ²

Ω

e^Uiφ=o(1), and (3.13) is completely established.

Define the spaces

L:=

Φ:

Φ 1+ |y|²

L²(R²)

<+∞

, H:=

Φ: ∇ΦL²(R²)+ Φ

1+ |y|²

L²(R²)

<+∞

endowed respectively with the normΦL= Φ/(1+ |y|²)L²(R²)andΦH=(∇Φ²_L2(R²)+ Φ²_L)^1/2. IfS² denotes the unit sphere inR³ with the standard metric andπN is the stereographic projection through the north pole, let us point out that the mapΦ→Φ◦πN is an isometry from L toL²(S²)and from H toH¹(S²). Hence, by the compactness of the embeddingH¹(S²) →L²(S²), we get the compactness of the embedding H→L.

Assume thatξ∈Oεfor someε >0, where Oε=

(q₁, . . . , q_k)∈Ω^k: dist(q_i, ∂Ω)2ε, |q_i−q_j|2εfori=j .

Letχ:R→ [0,1]be a smooth cut-off function such that χ (x)=1 if |x|ε/2, χ (x)=0 if |x|ε. For any i=1, . . . , kset

φ˜ⁱ(y)=φⁱ(y)χⁱ(y), y∈Ωⁱ:=Ω−ξi

τ_iρ ,

whereφⁱ(y)=φ(τ_iρy+ξ_i)andχⁱ(y)=χ (τ_iρy). We will always considerφ˜ⁱ extended to be zero outsideΩⁱ. Step 3. For anyi=1, . . . , k

φ˜ⁱ0 weakly in H. (3.19)

First of all, we remark that the functionφ˜ⁱ satisfies (in a weak sense), −φ˜ⁱ=aⁱ(y)φ˜ⁱ−z˜ⁱ+ ˜vⁱ inΩⁱ,

φ˜ⁱ=0 on∂Ωⁱ, (3.20)

where

aⁱ(y)=τ_i²ρ⁴V (τiρy+ξi)e^{kh=1P Uh(τiρy+ξi)},

˜

zⁱ(y)=zⁱ(y)χⁱ(y), zⁱ(y)=(ψ+w)(τ_iρy+ξ_i),

˜

vⁱ(y)= −χⁱ(y)(φⁱ−zⁱ)−2∇χⁱ(y)∇(φⁱ−zⁱ).

Observe that by Lemma B.1 we have, ρ²

k h=1

Ω

e^Uhφ²=ρ²

Ω

V (x)e

k

h=1P Uhφ²+O

Ω

ρ²

k h=1

e^Uh−ρ²V (x)e

k h=1P Uh

φ²

=ρ²

Ω

V (x)e^{kh=1P Uh}φ²+o(1). (3.21)

Multiplying (3.3) byφ, we obtain that ρ²

Ω

V (x)e

_k

h=1P Uhφ²=

Ω

|∇φ|²−

Ω

∇ψ∇φ=O(1). (3.22)

By (3.21), (3.22) we get thatφ˜ⁱ is bounded in H, since

Ωⁱ

|∇ ˜φⁱ|²=O

Ω

|∇φ|²

=O(1)

and

Ωⁱ

(φ˜ⁱ(y))²

(1+ |y|²)²dy=ρ² 8

Ω

e^Uiφ²χ²(x−ξ_i)=O(1).

Hence, we can assume (up to a subsequence) thatφ˜ⁱ φⁱ₀in H and strongly in L. On the other hand, by Proposi- tion A.1 it follows that

aⁱ(y)=V (τ_iρy+ξ_i)

τ_i²(1+ |y|²)²e^{8π(H (τiρy+ξi,ξi)+}

h=iG(τ_iρy+ξ_i,ξ_h))+O(ρ²)→ 8

(1+ |y|²)² (3.23)

uniformly on compact sets ofR². Furthermore,

∇ ˜zⁱ²_L2(R²)=O

Ω

|∇ψ|²+ |∇w|²

→0 (3.24)

and, for anyΨ ∈C₀^∞(R²)we have,

R²

˜ vⁱΨ =

Ω

−χ (x−ξ_i)(φ−ψ−w)(x)−2∇χ (x−ξ_i)∇(φ−ψ−w)(x) Ψ

x−ξ_i τ_iρ

dx=0 (3.25) ifε/(2τ_iρ)dist(SuppΨ,0). Finally, forj=1,2 we see that,

R²

|y|²−1

(|y|²+1)³φ₀ⁱdy=

R²

yj

(|y|²+1)³φⁱ₀dy=0. (3.26)

Indeed, 16

τ_i

R²

|y|²−1

(1+ |y|²)³φ˜ⁱdy=ρ²

Ω

e^Uiψ_i⁰φχ (x−ξ_i)=(φ, P ψ_i⁰)_H1

0+O(ρ²)→0 (by Step 2), and the orthogonality conditions give,

32

R²

yj

(1+ |y|²)³φ˜ⁱdy=τ_iρ³

Ω

e^Uiψ_i^jφχ (x−ξ_i)=O(ρ³)→0, j =1,2,

and so (3.26) follows by observing thatφ˜ⁱ/(1+ |y|²)→φ₀ⁱ/(1+ |y|²)in L²(R²). In conclusion, by (3.20)–(3.23), (3.25) we obtain thatφ₀ⁱ∈His a (distributional) solution for

−φ₀ⁱ= 8

(1+ |y|²)²φ₀ⁱ inR² (3.27)

satisfying:

R²

|y|²−1

(|y|²+1)³φ₀ⁱdy=

R²

yj

(|y|²+1)³φⁱ₀dy=0 forj =1,2.

The isometry betweenHandH¹(S²)and elliptic regularity theory (see [24]) imply thatφ₀ⁱ is actually a regular solution of (3.27). By Lemma D.1 we get that necessarilyφ₀ⁱ=0 and (3.19) is established.

Step 4. A contradiction arises!

By the compactness of the embeddingL →H, we have thatφ˜ⁱ →0 in L for anyi=1, . . . , k. By (3.21) and ψ =o(1)we have,

ρ²

Ω

V (x)e^{ki=1P Ui}φ²+

Ω

∇ψ∇φ=ρ² k

i=1

Ω

e^Uiφ²χ²(x−ξ_i)+o(1)

= k i=1

R²

8

(1+ |y|²)²(φ˜ⁱ)²dy+o(1)→0, in contradiction with the relation:

1=

Ω

|∇φ|²=ρ²

Ω

V (x)e^{ki=1P Ui}φ²+

Ω

∇ψ∇φ.

This completes the proof of Proposition 3.1. 2

Remark 3.2. Let us remark that the result in Proposition 3.1 is “sharp" in the sense that in [22] it is shown that for k=1 we have also(L^ρ_ξ)⁻¹c₁|logρ|for some positive constantc₁< c.

4. The reduced problem and the functional expansion

Our first goal will be to prove that, for anyρ >0 small enough and for any pointξ ∈(Ω)^k\∆, there exists φ_ξ^ρ∈K_ξ^⊥such that

Π_ξ^⊥ _k

i=1

P U_i+φ_ξ^ρ−i^∗

ρ²V (x)e

k

i=1P U_i+φ^ρ_ξ

=0. (4.1)

Proposition 4.1. Letξ =(ξ₁, . . . , ξ_k)in a compact set of(Ω)^k \∆and τ_i =τ_i(ξ )be given in (2.6). For any p∈(1,⁴₃) there existsρ₀>0 and R >0 (uniformly inξ) such that for any ρ∈(0, ρ₀)there exists a unique φ_ξ^ρ∈K_ξ^⊥such that

Π_ξ^⊥ _k

i=1

P U_i+φ_ξ^ρ−i^∗

ρ²V (x)e

k

i=1P Ui+φ^ρ_ξ

=0 (4.2)

and

φ^ρ_ξRρ^(2−p)/p|logρ|. (4.3)

Proof. According to Proposition 3.1, for ρ small (L^ρ_ξ)⁻¹ is a linear operator from K_ξ^⊥ into itself such that (L^ρ_ξ)⁻¹c2|logρ| uniformly inξ. Let us point out that φ is a solution of (4.2) if and only if it is a fixed point of the operatorT_ξ^ρ:K_ξ^⊥→K_ξ^⊥defined by

T_ξ^ρ(φ)=

(L^ρ_ξ)⁻¹◦Π_ξ^⊥◦i^∗ M_ξ^ρ(φ), M_ξ^ρ(φ)=ρ²V (x)e

k

i=1P Ui[e^φ−1−φ] +

ρ²V (x)e

k

i=1P Ui−ρ² k i=1

e^Ui

.

We prove that, forρsmall enough andRlarge enough (but independent ofρ),T_ξ^ρdefines a contraction map from {φRρ^(2−p)/p|logρ|}into itself.