Singular limits for the bi-laplacian operator with exponential nonlinearity in <span class="mathjax-formula">${R}^{4}$</span>

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Singular limits for the bi-Laplacian operator with exponential nonlinearity in R ⁴

Mónica Clapp

, Claudio Muñoz

, Monica Musso

aInstituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, C.U., 04510 México DF, Mexico bDepartamento de Ingeniería Matemática, Universidad de Chile, Casilla 170, Correo 3, Santiago, Chile

cDepartamento de Matemática, Pontificia Universidad Católica de Chile, Avenida Vicuna Mackenna 4860, Macul, Santiago, Chile dDipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino, Italy

Received 29 December 2006; accepted 10 September 2007 Available online 3 December 2007

Abstract

LetΩbe a bounded smooth domain inR⁴such that for some integerd1 itsd-th singular cohomology group with coefficients in some field is not zero, then problem

²u−ρ⁴k(x)e^u=0 inΩ,

u=u=0 on∂Ω,

has a solution blowing-up, asρ→0, atmpoints ofΩ, for any given numberm.

©2007 Elsevier Masson SAS. All rights reserved.

1. Introduction and statement of main results

LetΩ be a bounded smooth domain inR⁴. We are interested in studying existence and qualitative properties of positive solutions to the following boundary value problem

²u−ρ⁴k(x)e^u=0 inΩ,

u=u=0 on∂Ω, (1.1)

wherek∈C²(Ω)¯ is a non-negative, not identically zero function, andρ >0 is a small, positive parameter which tends to 0.

In a four-dimensional manifold, this type of equations and similar ones arise from the problem of prescribing the so-calledQ-curvature, which was introduced in [7]. More precisely, given(M, g) a four-dimensional Riemannian manifold, the problem consists in finding a conformal metric g˜ for which the corresponding Q-curvatureQ_g˜ is a priori prescribed. TheQ-curvature for the metricgis defined as

Q_g= −1 2

_gR_g−R_g²+3|Ricg|² ,

* Corresponding author.

E-mail addresses:mclapp@matem.unam.mx (M. Clapp), cmunoz@dim.uchile.cl (C. Muñoz), mmusso@mat.puc.cl (M. Musso).

0294-1449/$ – see front matter ©2007 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2007.09.002

whereR_g is the scalar curvature and Ricg is the Ricci tensor of(M, g). Writingg˜=e^2wg, the problem reduces to finding a scalar functionwwhich satisfies

P_gw+2Qg=2Qg_˜e^4w, (1.2)

whereP_gis the Paneitz operator [32,10] defined as P_gw=²_gw+div

2

3R_gg−2 Ricg

dw.

Problem (1.2) is thus an elliptic fourth-order partial differential equation with exponential non-linearity. Several results are already known for this problem [9,10] and related ones [1,18,30]. When the metric g is not Riemannian, the problem has been recently treated by Djadli and Malchiodi in [19] via variational methods.

In the special case where the manifold is the Euclidean space andgis the Euclidean metric, we recover the equation in (1.1), since (1.2) takes the simplified form

²w−2Qe^4w=0.

Problem (1.1) has a variational structure. Indeed, solutions of (1.1) correspond to critical points inH²(Ω)∩H₀¹(Ω) of the following energy functional

J_ρ(u)=1 2

Ω

|u|²−ρ⁴

Ω

k(x)e^u.

For anyρsufficiently small, the functional above has a local minimum which represents a solution to (1.1) close to 0.

Furthermore, the Moser–Trudinger inequality assures the existence of a second solution, which can be obtained as a mountain pass critical point forJ_ρ. Thus, asρ→0, this second solution turns out not to be bounded. The aim of the present paper is to study multiplicity of solutions to (1.1), forρpositive and small, under some topological assumption onΩ, and to describe the asymptotic behavior of such solutions as the parameterρtends to zero. Indeed, we prove that, if some cohomology group ofΩis not zero, then given any integermwe can construct solutions to (1.1) which concentrate and blow-up, asρ→0, around some givenmpoints of the domain. These are the singular limits.

Let us mention that concentration phenomena of this type, in domains with topology, appear also in other prob- lems. As a first example, the two-dimensional version of problem (1.1) is the boundary value problem associated to Liouville’s equation [25]

u+ρ²k(x)e^u=0 inΩ,

u=0 on∂Ω, (1.3)

wherek(x)is a non-negative function and nowΩis a smooth bounded domain inR². In [14] it is proved that problem (1.3) admits solutions concentrating, asρ→0, around some given set ofmpoints ofΩ, for any given integerm, provided thatΩ is not simply connected. See also [5,6,21,20,11,8,24,29,31,35,38,36,37] for related results. A similar result holds true for another semilinear elliptic problem, still in dimension 2, namely

u+u^p=0, u >0 inΩ,

u=0 on∂Ω, (1.4)

wherepnow is a parameter converging to+∞. Again in this situation, ifΩis not simply connected, then forplarge there exists a solution to (1.4) concentrating around some set ofmpoints ofΩ, for any positive integerm[22].

In higher dimensions, the analogy is with the classical Bahri–Coron problem. In [2], Bahri and Coron show that, if N3 andΩ⊂R^N is a bounded domain, then the presence of topology in the domain guarantees existence of solutions to

u+u^N+2N−2 =0, u >0 inΩ, u=0 on∂Ω.

(1.5) Partial results in this direction are also known in the slightly super critical version of Bahri–Coron’s problem, namely

u+u^N+2N−2+ε=0 inΩ, u >0, u=0 on∂Ω,

(1.6)

withε >0 small. In [12] it is proved that, under the assumption thatΩ is a bounded smooth domain inR^N with a sufficiently small hole, a solution to (1.6) exhibiting concentration in two points is present. See also [3,23,34,13,33].

The main point of this paper is to show that the presence of topology in the domain implies strongly existence of blowing-up solutions for problem (1.1).

We denote byH^d(Ω) thed-th cohomology group ofΩ with coefficients in some field K. We shall prove the following

Theorem 1.Assume that there existsd1such thatH^d(Ω)=0and thatinfΩk >0. Then, given any integerm1, there exists a family of solutionsu_ρ to problem(1.1), forρsmall enough, with the property that

ρlim→0ρ⁴

Ω

k(x)e^uρ(x)dx=64π²m.

Furthermore, there arempointsξ₁^ρ, . . . , ξ_m^ρ inΩ, separated at uniform positive distance from each other and from the boundary asρ→0, for whichuρremains uniformly bounded onΩ\ ^mj=1Bδ(ξ_j^ρ)and

sup

Bδ(ξ_j^ρ)

u_ρ→ +∞,

for anyδ >0.

The general behavior of arbitrary families of solutions to (1.1) has been studied by C.S. Lin and J.-C. Wei in [26], where they show that, when blow-up occurs for (1.1) asρ→0, then it is located at a finite number of peaks, each peak being isolated and carrying the energy 64π²(at a peak,u→ +∞and outside a peak,uis bounded). See [27]

and [28] for related results.

We shall see that the sets of points where the solution found in Theorem 1 blows-up can be characterized in terms of Green’s function for the biharmonic operator inΩ with the appropriate boundary conditions. LetG(x, ξ )be the Green function defined by

²_xG(x, ξ )=64π²δ_ξ(x), x∈Ω,

G(x, ξ )=xG(x, ξ )=0, x∈∂Ω (1.7)

and letH (x, ξ )be itsregular part, namely, the smooth function defined as H (x, ξ ):=G(x, ξ )+8 log|x−ξ|.

The location of the points of concentration is related to the set of critical points of the function ϕ_m(ξ )= −

m j=1

2 logk(ξ_j)+H (ξ_j, ξ_j)

−

i=j

G(ξ_i, ξ_j), (1.8)

defined for pointsξ=(ξ1, . . . , ξ_m)such thatξ_i∈Ωandξ_i=ξ_j ifi=j.

In [4] the authors prove that for eachnon-degeneratecritical point ofϕmthere exists a solution to (1.1), for any smallρ, which concentrates exactly around such critical point asρ→0. We shall show the existence of a solution under a weaker assumption, namely, thatϕ_mhas aminimax value in an appropriate subset.

More precisely, we consider the following situation. LetΩ^mdenote the Cartesian product ofmcopies ofΩ. Note that in any compact subset ofΩ^m, we may define, without ambiguity,

ϕ_m(ξ₁, . . . , ξ_m)= −∞ ifξ_i=ξ_jfor somei=j.

We shall assume that there exists an open subsetUofΩ with smooth boundary, compactly contained inΩ, and such that infUk >0, with the following properties:

(P1) U^mcontains two closed subsetsB₀⊂Bsuch that sup

ξ∈B0

ϕ_m(ξ ) < inf

γ∈Γsup

ξ∈B

ϕ_m γ (ξ )

=:c₀,

whereΓ := {γ∈C(B,U¯^m): γ (ξ )=ξ for everyξ∈B₀}.

(P2) For everyξ=(ξ₁, . . . , ξ_m)∈∂U^mwithϕ_m(ξ )=c₀,there exists ani∈ {1, . . . , m}such that

∇ξiϕ_m(ξ )=0 ifξ_i∈U,

∇ξ_iϕm(ξ )·τ=0 for someτ∈Tξ_i(∂U ) ifξi∈∂U, whereT_ξi(∂U )denotes the tangent space to∂Uat the pointξ_i.

We will show that, under these assumptions,ϕ_mhas a critical pointξ ∈U^mwith critical valuec₀. Moreover, the same is true for any small enoughC¹-perturbation ofϕ_m. Property (P1) is a common way of describing a change of topology of the sublevel sets ofϕ_mat the levelc₀, andc₀is called a minimax value ofϕ_m. It is a critical value ifU^m is invariant under the negative gradient flow ofϕ_m. If this is not the case, we use property (P2) to modify the gradient vector field ofϕmnear∂U^mat the levelc0and thus obtain a new vector field with the same stationary points, and such thatU¯^mis invariant andϕmis a Lyapunov function for the associated negative flow near the levelc0(see Lemmas 6.3 and 6.4 below). This allows us to prove Theorem 1 and the following.

Theorem 2.Letm1and assume that there exists an open subsetU ofΩ with smooth boundary, compactly con- tained inΩ, withinfUk >0, which satisfies(P1)and(P2). Then, forρ small enough, there exists a solutionu_ρ to problem(1.1)with

ρlim→0ρ⁴

Ω

k(x)e^uρ=64π²m.

Moreover, there is anm-tuple(x^ρ₁, . . . , x_m^ρ)∈U^m, such that asρ→0

∇ϕ_m

x₁^ρ, . . . , x_m^ρ

→0, ϕ_m

x₁^ρ, . . . , x_m^ρ

→c₀,

for whichu_ρ remains uniformly bounded onΩ\ ^m_j=1B_δ(x_i^ρ), and sup

Bδ x^ρ_i

u_ρ→ +∞,

for anyδ >0.

We will show that, for everym1, the setU:= {ξ ∈Ω: dist(ξ, ∂Ω) > δ}has property (P2) at a givenc₀, forδ small enough (see Lemma 6.2). Thus, if infΩk >0, and if there exist closed subsetsB₀⊂BofΩ^mwith

sup

ξ∈B0

ϕ_m(ξ ) < inf

γ∈Γsup

ξ∈B

ϕ_m γ (ξ )

,

then both conditions (P1) and (P2) hold. Condition (P1) holds, for example, ifϕ_mhas a (possibly degenerate) local minimum or local maximum. So a direct consequence of Theorem 2 is that in any bounded domainΩwith infΩk >0, problem (1.1) has at least one solution concentrating exactly at one point, which corresponds to the minimum of the regular Green functionH. Moreover if, for example,Ω is a contractible domain obtained by joining togetherm disjoint bounded domains through thin enough tubes, then the functionϕ_mhas a (possibly degenerate) local minimum, which gives rise to a solution exhibitingmpoints of concentration.

Finally, recall that problem (1.1) corresponds to a standard case ofuniform singular convergence, in the sense that the associated non-linear coefficient in problem (1.1) –ρ⁴k(x)– goes to 0 uniformly inΩ¯ asρ→0, property that is also present in problem (1.3). Non-trivial topology strongly determines existence of solutions. However, we expect that this strong influence should decay under an inhomogeneous andnon-uniformsingular behavior, where critical points of anexternalfunction determine existence and multiplicity of solutions. See [16] for a recent two-dimensional case of this phenomenon.

The paper is organized as follows. Section 2 is devoted to describing a first approximation for the solution and to estimating the error. Furthermore, problem (1.1) is written as a fixed point problem, involving a linear operator. In Section 3 we study the invertibility of the linear problem. In Section 4 we solve a projected non-linear problem. In Section 5 we show that solving the entire non-linear problem reduces to finding critical points of a certain functional.

Section 6 is devoted to the proofs of Theorems 1 and 2.

2. Preliminaries and ansatz for the solution

This section is devoted to construct a reasonably good approximationUfor a solution of (1.1). The shape of this approximation will depend on some pointsξi, which we leave as parameters yet to be adjusted, where the spikes are meant to take place. As we will see, a convenient set to selectξ=(ξ1, . . . , ξm)is

O:=

ξ∈Ω^m: dist(ξj, ∂Ω)2δ0, ∀j=1, . . . , m, and min

i=j|ξ_i−ξ_j|2δ0

(2.1) whereδ0>0 is a small fixed number. We thus fixξ∈O.

For numbersμj>0,j=1, . . . , m, yet to be chosen,x∈R⁴andε >0 we define u_j(x)=4 log μ_j(1+ε²)

μ²_jε²+ |x−ξ_j|²−logk(ξ_j), (2.2)

so thatu_jsolves

²u−ρ⁴k(ξ_j)e^u=0 inR⁴, (2.3)

with

ρ⁴= 384ε⁴

(1+ε²)⁴, (2.4)

that is,ρ∼εasε→0.

Sinceu_jandu_jare not zero on the boundary∂Ω, we will add to it a bi-harmonic correction so that the boundary conditions are satisfied. LetHj(x)be the smooth solution of

⎧⎪

⎨

⎪⎩

²Hj=0 inΩ, H_j= −u_j on∂Ω, H_j= −u_j on∂Ω.

We define our first approximationU (ξ )as U (ξ )≡

m j=1

U_j, U_j≡u_j+H_j. (2.5)

As we will rigorously prove below,(u_j+H_j)(x)∼G(x, ξ_j)whereG(x, ξ )is the Green function defined in (1.7).

Whileu_j is a good approximation to a solution of (1.1) near ξ_j, it is not so much the case for U, unless the remainderU−u_j=(H_j+

k=ju_k)vanishes at main order nearξ_j. This is achieved through the following precise choice of the parametersμ_k

logμ⁴_j=logk(ξ_j)+H (ξ_j, ξ_j)+

i=j

G(ξ_i, ξ_j). (2.6)

We thus fixμ_j a priorias a function ofξ. We write μ_j=μ_j(ξ )

for allj=1, . . . , m. Sinceξ ∈O, 1

C μjC, for allj=1, . . . , m, (2.7)

for some constantC >0.

The following lemma expandsUj inΩ. Lemma 2.1.Assumeξ ∈O. Then we have

H_j(x)=H (x, ξ_j)−4 logμj(1+ε²)+logk(ξ_j)+O(μ²_jε²), (2.8)

uniformly inΩ, and

uj(x)=4 logμj(1+ε²)−logk(ξj)−8 log|x−ξj| +O(μ²_jε²), (2.9) uniformly in the region|x−ξ_j|δ₀, so that in this region,

U_j(x)=G(x, ξ_j)+O(μ²_jε²). (2.10)

Proof. Let us prove (2.8). Definez(x)=H_j(x)+4 logμ_j(1+ε²)−logk(ξ_j)−H (x, ξ_j). Thenzis a bi-harmonic function which satisfies

⎧⎪

⎪⎨

⎪⎪

⎩

²z=0 inΩ,

z= −uj+4 logμj(1+ε²)−logk(ξj)−8 log| · −ξj| on∂Ω, z= −u_j−_|·−¹⁶_ξ

j|² on∂Ω.

Let us definew≡ −z. Thuswis harmonic inΩ and sup

Ω

|w|sup

∂Ω

|w|Cμ²_jε².

We also have sup_∂Ω|z|Cμ²_jε². Standard elliptic regularity implies sup

Ω

|z|C

sup

Ω

|w| +sup

∂Ω

|z|

Cμ²_jε²,

as desired. The second estimate is direct from the definition ofu_j. 2 Now, let us write

Ω_ε=ε⁻¹Ω, ξ_j=ε⁻¹ξ_j. (2.11)

Thenusolves (1.1) if and only ifv(y)≡u(εy)+4 logρεsatisfies ²v−k(εy)e^v=0 inΩ_ε,

v=4 logρε, v=0 on∂Ω_ε. (2.12)

Let us defineV (y)=U (εy)+4 logρε, withUour approximate solution (2.5). We want to measure the size of the error of approximation

R≡²V−k(εy)e^V. (2.13)

It is convenient to do so in terms of the following norm v∗= sup

y∈Ωε

_m

j=1

1

(1+ |y−ξ_j|²)^7/2+ε⁴ ₋1

v(y)

. (2.14)

Here and in what follows,Cdenotes a generic constant independent ofεand ofξ∈O.

Lemma 2.2.The errorRin(2.13)satisfies R_∗Cε asε→0.

Proof. We assume first|y−ξ_k|< δ₀/ε, for some indexk. We have ²V (y)=ρ⁴

m j=1

k(ξj)e^uj(εy)= 384μ⁴_k

(μ²_k+ |y−ξ_k|²)⁴+O(ε⁸).

Let us estimatek(εy)e^{V (y)}. By (2.8) and the definition ofμ_js, H_k(x)=H (ξ_k, ξ_k)−4 logμ_k+logk(ξ_j)+O(μ²_kε²)+O

|x−ξ_k|

= −

j=k

G(ξ_j, ξ_k)+O(μ²_kε²)+O

|x−ξ_k| ,

and ifj=k, by (2.10)

U_j(x)=u_j(x)+H_j(x)=G(ξ_j, ξ_k)+O

|x−ξ_k|

+O(μ²_jε²).

Then

H_k(x)+

j=k

U_j(x)=O(ε²)+O

|x−ξ_k|

. (2.15)

Therefore,

k(εy)e^{V (y)}=k(εy)ε⁴ρ⁴exp

u_k(εy)+H_k(εy)+

j=k

U_j(εy)

= 384μ⁴_kk(εy) (μ²_k+ |y−ξ_k|²)⁴k(ξ_k)

1+O

ε|y−ξ_k|

+O(ε²)

= 384μ⁴_k (μ²_k+ |y−ξ_k|²)⁴

1+O

ε|y−ξ_k| . We can conclude that in this region

R(y)C ε|y−ξ_k|

(1+ |y−ξ_k|²)⁴+O(ε⁴).

If|y−ξ_j|δ0/εfor allj, using (2.8), (2.9) and (2.10) we obtain ²V =O(ε⁴ρ⁴) and k(εy)e^{V (y)}=O(ε⁴ρ⁴).

Hence, in this region, R(y)=O(ε⁸) so that finally

R_∗=O(ε). 2

Next we consider the energy functional associated with (1.1) J_ρ[u] =1

2

Ω

(u)²−ρ⁴

Ω

k(x)e^u, u∈H²(Ω)∩H₀¹(Ω). (2.16)

We will give an asymptotic estimate of J_ρ[U], where U (ξ ) is the approximation (2.5). Instead of ρ, we use the parameterε(defined in (2.4)) to obtain the following expansion:

Lemma 2.3.With the election ofμ_j’s given by(2.6),

J_ρ[U] = −128π²m+256π²m|logε| +32π²ϕ_m(ξ )+εΘ_ε(ξ ), (2.17) whereΘ_ε(ξ )is uniformly bounded together with its derivatives ifξ∈O, andϕ_mis the function defined in(1.8).

Proof. We have

J_ρ[U] =1 2

m j=1

Ω

(U_j)²+1 2

j=i

Ω

U_jU_i−ρ⁴

Ω

k(x)e^U

≡I1+I2+I3;

Note that²Uj=²uj=ρ⁴k(ξj)e^uj inΩ andUj=Uj=0 in∂Ω. Then I₁=1

2ρ⁴ m j=1

k(ξ_j)

Ω

e^ujU_j and I₂=1 2ρ⁴

j=i

k(ξ_j)

Ω

e^ujU_i.

Let us define the change of variablesx=ξj+μjεy, wherex∈Ωandy∈Ωj≡(μjε)⁻¹(Ω−ξj). Using Lemma 2.1 and the definition ofρin terms ofεin (2.4) we obtain

I1=192 m j=1

Ω_j

1 (1+ |y|²)⁴

4 log 1

1+ |y|²−8 logμjε+H (ξj, ξj)+O

μjε|y|

=32π² m j=1

H (ξ_j, ξ_j)−8 logμjε

−64π²m+O

εμ_j

Ωj

|y| (1+ |y|²)⁴

=32π² m j=1

H (ξ_j, ξ_j)−8 logμjε

−64π²m+εΘ(ξ ),

whereΘ_ε(ξ )is bounded together with its derivatives ifξ∈O. Besides we have used the explicit values

R⁴

1

(1+ |y|²)⁴ =π² 6 , and

R⁴

log(1+ |y|²) (1+ |y|²)⁴ =π²

12. We consider nowI2. As above,

1 2ρ⁴

Ω

e^ujU_i=

Ωj

192 (1+ |y|²)⁴

u_i(ξ_j+μ_jεy)+H_i(ξ_j+μ_jεy)

=

Ω_j

192 (1+ |y|²)⁴

u_i(ξ_j+μ_jεy)−4 logμ_i(1+ε²)+logk(ξ_i)+8 log|ξ_j−ξ_i|

+

Ω_j

192 (1+ |y|²)⁴

Hi(ξj+μjεy)−Hi(ξj)

+

Ωj

192 (1+ |y|²)⁴

H_i(ξ_j)−H (ξ_j, ξ_i)+4 logμ_i(1+ε²)−logk(ξ_i)

+G(ξ_j, ξ_i)

Ω_j

192 (1+ |y|²)⁴

=32π²G(ξi, ξj)+O

εμj

Ω_j

|y| (1+ |y|²)⁴

+O(μ²_jε²)

=32π²G(ξ_i, ξ_j)+εΘ_ε(ξ ).

Thus

I₂=32π²

j=i

G(ξ_i, ξ_j)+εΘ_ε(ξ ). (2.18)

Finally we considerI₃. Let us denoteA_j≡B(ξ_j, δ₀)andx=ξ_j+μ_jεy. Then using again Lemma 2.1 I₃= −ρ⁴

m j=1

Aj

k(x)e^U+O(ε⁴)

= −ρ⁴ m j=1

B(0,^δ0

μj ε)

k(ξj+μjεy) k(ξ_j)(1+ |y|²)⁴

(1+ε²)⁴ ε⁴

1+O

εμ_j|y|

+O(ε⁴)

= −384m

R⁴

1

(1+ |y|²)⁴+O

εμ_j

R⁴

|y| (1+ |y|²)⁴

= −64π²m+εΘε(ξ ),

uniformly inξ∈O. Thus, we can conclude the following expansion ofJ_ρ[U]:

J_ρ[U] = −128mπ²+256mπ²|logε| +32π²ϕ_m(ξ )+εΘ_ε(ξ ), (2.19) where Θ_ε(ξ ) is a bounded function together with is derivatives in the region ξ ∈O, ϕ_m defined as in (1.8) and ρ⁴=₍₁^384ε_+ε2⁴)⁴. 2

In the subsequent analysis we will stay in the expanded variable y∈Ωε so that we will look for solutions of problem (2.12) in the formv=V+ψ, whereψwill represent a lower order correction. In terms ofψ, problem (2.12) now reads

Lε(ψ )≡²ψ−W ψ= −R+N (ψ ) inΩ_ε,

ψ=ψ=0 on∂Ωε, (2.20)

where

N (ψ )=W[e^ψ−ψ−1] and W=k(εy)e^V. (2.21)

Note that W (y)=

m j=1

384μ⁴_j (μ²_j+ |y−ξ_j|²)⁴

1+O

ε|y−ξ_j|

fory∈Ωε. (2.22)

This fact, together with the definition ofN (ψ )given in (2.21), give the validity of the following Lemma 2.4.Forξ∈O,W_∗=O(1)andN (ψ )_∗=O(ψ²_∞)asψ_∞→0.

3. The linearized problem

In this section we develop a solvability theory for the fourth-order linear operator Lε defined in (2.20) under suitable orthogonality conditions. We consider

Lε(ψ )≡²ψ−W (y)ψ, (3.1)

whereW (y)was introduced in (2.20). By expression (2.22) and settingz=y−ξ_j, one can easily see that formally the operatorLεapproaches, asε→0, the operator inR⁴

Lj(ψ )≡²ψ− 384μ⁴_j

(μ²_j+ |z|²)⁴ψ, (3.2)

namely, equation²v−e^v=0 linearized around the radial solutionv_j(z)=log ^384μ

4 j

(μ²_j+|z|²)⁴. Thus the key point to develop a satisfactory solvability theory for the operatorLε is the non-degeneracy ofv_j up to the natural invariances of the equation under translations and dilations. In fact, if we set

Y0j(z)=4|z|²−μ²_j

|z|²+μ²_j, (3.3)

Y_ij(z)= 8zi

μ²_j+ |z|², i=1, . . . ,4, (3.4)

the only bounded solutions ofLj(ψ )=0 inR⁴are linear combinations ofYij,i=0, . . . ,4; see Lemma 3.1 in [4] for a proof.

We define fori=0, . . . ,4 andj=1, . . . , m, Zij(y)≡Yij(y−ξ_j), i=0, . . . ,4.

Additionally, let us considerR₀ a large but fixed number andχ a radial and smooth cut-off function withχ≡1 in B(0, R0)andχ≡0 inR⁴\B(0, R0+1). Let

χ_j(y)=χ

|y−ξ_j|

, j =1, . . . , m.

Givenh∈L^∞(Ω_ε), we consider the problem of finding a functionψsuch that for certain scalarsc_ij one has

⎧⎪

⎨

⎪⎩

Lε(ψ )=h+₄

i=1

_m

j=1c_ijχ_jZ_ij, inΩ_ε, ψ=ψ=0, on∂Ω_ε,

Ωεχ_jZ_ijψ=0, for alli=1, . . . ,4, j=1, . . . , m.

(3.5) We will establish a priori estimates for this problem. To this end we shall introduce an adapted norm inΩ_ε, which has been introduced previously in [15]. Givenψ:Ω_ε→Randα∈N^mwe define

ψ_∗∗≡ m j=1

ψC^4,α(r_j<2)+ m j=1

|α|3

r_j^|α|D^αψL^∞(rj2), (3.6)

withr_j= |y−ξ_j|.

Proposition 3.1.There exist positive constantsε0>0andC >0such that for anyh∈L^∞(Ωε), withh_∗<∞, and anyξ∈O, there is a unique solutionψ=T (h)to problem(3.5)for allεε0, which defines a linear operator ofh.

Besides, we have the estimate T (h)

∗∗C|logε|h_∗. (3.7)

The proof will be split into a series of lemmas which we state and prove next. The first step is to obtain a priori estimates for the problem

⎧⎪

⎨

⎪⎩

Lε(ψ )=h inΩ_ε, ψ=ψ=0 on∂Ω_ε,

Ω_εχ_jZ_ijψ=0 for alli=0, . . . ,4, j=1, . . . , m,

(3.8) which involves more orthogonality conditions than those in (3.5). We have the following estimate.

Lemma 3.1.There exist positive constants ε0>0 and C >0such that for any ψ solution of problem(3.8)with h∈L^∞(Ωε),h∗<∞, andξ∈O, then

ψ_∗∗Ch_∗ (3.9)

for allε∈(0, ε0).

Proof. We carry out the proof by a contradiction argument. If the above fact were false, then, there would exist a sequence εn→0, points ξⁿ=(ξ₁ⁿ, . . . , ξ_mⁿ)∈O, functions hn withhn∗→0 and associated solutions ψn with ψ_n∗∗=1 such that

⎧⎪

⎨

⎪⎩

Lε_n(ψn)=hn inΩε_n, ψ_n=ψ_n=0 on∂Ω_εn,

Ω_εnχ_jZ_ijψ_n=0, for alli=0, . . . ,4, j=1, . . . , m.

(3.10)

Let us setψ˜_n(x)=ψ_n(x/ε_n),x∈Ω. It is directly checked that for anyδ>0 sufficiently smallψ˜_nsolves the problem ²ψ˜_n=O(ε_n⁴)+ε⁻_n⁴h_n=o(1), uniformly inΩ\ ^mk=1B(ξ_jⁿ, δ),

˜

ψ_n=ψ˜_n=0 on∂Ω,

together with ˜ψ_n∞1 andψ˜_n∞C_δ, in the considered region. Passing to a subsequence, we then get that ξⁿ→ξ^∗∈Oandψ˜_n→0 in theC^3,α sense over compact subsets ofΩ\{ξ₁^∗, . . . , ξ_m^∗}. In particular

|α|3

1 ε^|_n^α|

D^αψ_n(y)→0, uniformly iny−(ξ_jⁿ) δ 2εn

,

for anyδ>0 andj∈ {1, . . . , m}. We obtain thus that m

j=1

|α|3

r_j^|α|D^αψnL^∞r_jδ/ε_n,→0, (3.11)

for anyδ>0. In conclusion, theexterior portionofψn∗∗goes to zero, see (3.6).

Let us consider now a smooth radial cut-off functionηˆwithη(s)ˆ =1 ifs < ¹₂,η(s)ˆ =0 ifs1, and define ψˆ_n,j(y)= ˆη_j(y)ψ_n(y)≡ ˆη

εn

δ₀y−(ξ_jⁿ) ψ_n(y), such that

suppψˆ_n,j⊆B

(ξ_jⁿ),δ0

εn

. We observe that

Lε_n(ψˆ_n,j)= ˆη_jh_n+F (ηˆ_j, ψ_n), where

F (f, g)=g²f+2f g+4∇(f )· ∇g+4∇f · ∇(g)+4 4 i,j=1

∂²f

∂yi∂yj

∂²g

∂yi∂yj

. (3.12)

Thus we get

²ψˆn,j=Wn(y)ψˆn,j+ ˆηjhn+F (ηˆj, ψn) inB (ξ_jⁿ),^δ_ε⁰

n

, ψˆ_n,j=ψˆ_n,j=0 on∂B

(ξ_jⁿ),^δ_ε⁰

n

. (3.13)

The following intermediate result provides an outer estimate. For notational simplicitywe omitthe subscriptnin the quantities involved.

Lemma 3.2.There exist constantsC, R₀>0such that for largen

|α|3

r_j^|α|D^αψˆ_jL^∞(rjR0)C

ˆψ_jL^∞(rj<2R0)+o(1)

. (3.14)

Proof. We estimate the right-hand side of (3.13). If 2< rj< δ0/εwe get ²ψˆ_j=O

1 r_j⁸

ψˆ_j+ 1

r_j⁷o(1)+O(ε⁴)+O ε³

r_j

+O ε²

r_j²

+O ε

r_j³

. From (3.13) and standard elliptic estimates we have

|α|3

|D^αψˆ_j|C 1

r_j⁸ ˆψ_jL^∞(r_j>1)+ 1

r_j⁷o(1)+O ε

r_j³

, in 2r_jδ0

ε.

Now, ifr_j2

|r_j^|α|D^αψˆ_j|C 1

r_j⁵ ˆψ_jL^∞(r_j>1)+o(1)

, |α|3.

Finally 1

r_j⁵ ˆψ_jL^∞(r_j>1) ˆψ_jL^∞(1<r_j<R₀)+ 1

R⁵₀ ˆψ_jL^∞(r_j>R₀), thus fixingR0large enough we have

|α|3

r_j^|α|D^αψˆ_jL^∞(r_jR₀)C

ˆψ_jL^∞(1<rj<R0)+o(1)

, 2< r_j<δ0

ε, and then (3.14). 2

We continue with the proof of Lemma 3.1.

Sinceψ_n∗∗=1 and using (3.11) and Lemma 3.2 we have that there exists an indexj∈ {1, . . . , m}such that lim inf

n→∞ ψnL∞(r_j<R₀)α >0. (3.15)

Let us setψ˜_n(z)=ψ_n((ξ_jⁿ)+z). We notice thatψ˜_nsatisfies ²ψ˜_n−W

(ξ_jⁿ)+zψ˜_n=h_n

(ξ_jⁿ)+z

, inΩ_n≡Ω_ε−(ξ_jⁿ).

Sinceψ_n,ψ_n are bounded uniformly, standard elliptic estimates allow us to assume thatψ˜_n converges uniformly over compact subsets ofR⁴to a bounded, non-zero solutionψ˜ of

²ψ− 384μ⁴_j

(μ²_j+ |z|²)⁴ψ=0.

This implies thatψ˜ is a linear combination of the functionsY_ij, i=0, . . . ,4. But orthogonality conditions overψ˜_n pass to the limit thanks to ˜ψn_∞1 and dominated convergence. Thusψ˜ ≡0, a contradiction with (3.15). This conclude the proof. 2

Now we will deal with problem (3.8) lifting the orthogonality constraints

Ωεχ_jZ_0jψ=0, j=1, . . . , m, namely

⎧⎪

⎨

⎪⎩

Lε(ψ )=h inΩε, ψ=ψ=0 on∂Ω_ε,

ΩεχjZijψ=0, for alli=1, . . . ,4, j=1, . . . , m.

(3.16)

We have the following a priori estimates for this problem.

Lemma 3.3. There exist positive constants ε₀ and C such that, if ψ is a solution of (3.16), with h∈L^∞(Ω_ε), h_∗<∞and withξ∈O, then

ψ_∗∗C|logε|h_∗ (3.17)

for allε∈(0, ε0).

Proof. LetR > R0+1 be a large and fixed number. Let us considerZˆ0j be the following function

Zˆ_0j(y)=Z_0j(y)−1+a_0jG(εy, ξ_j), (3.18)

wherea_0j=(H (ξ_j, ξ_j)−8 log(εR))⁻¹. It is clear that ifεis small enough