Interior estimates for some semilinear elliptic problem with critical nonlinearity

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Interior estimates for some semilinear elliptic problem with critical nonlinearity

Estimations à l’intérieur pour un problème elliptique semi-linéaire avec non-linéarité critique

Pierpaolo Esposito

Dipartimento di Matematica, Università degli Studi “Roma Tre”, Largo S. Leonardo Murialdo, 1, 00146 Roma, Italy Received 29 September 2005; accepted 14 April 2006

Available online 25 September 2006

Abstract

We study compactness properties for solutions of a semilinear elliptic equation with critical nonlinearity. For high dimensions, we are able to show that any solutions sequence with uniformly bounded energy is uniformly bounded in the interior of the domain.

In particular, singularly perturbed Neumann equations admit pointwise concentration phenomena only at the boundary.

©2006 Elsevier Masson SAS. All rights reserved.

Résumé

On étudie les propriétés de compacité pour solutions d’une équation elliptique semi-linéaire avec non-linéarité critique. En hautes dimensions, on démontre qu’une suite de solutions avec énergie uniformément bornée est uniformément bornée dans l’intérieur du domaine. En particulier, les équations de Neumann perturbées singulièrement peuvent avoir des phénomènes de concentration seulement sur la frontière.

©2006 Elsevier Masson SAS. All rights reserved.

MSC:35J20; 35J25; 35J60

Keywords:Compactness; Critical exponent; Singular perturbations; Blow-up analysis

1. Introduction and statement of the results

The starting point in our investigation has been the study of asymptotic properties for the problem:

⎧⎨

⎩

−u+λu=u^p inΩ,

u >0 inΩ,

∂u

∂n=0 on∂Ω,

(1)

E-mail address:esposito@mat.uniroma3.it (P. Esposito).

1 The author is supported by M.U.R.S.T., project “Variational methods and nonlinear differential equations”.

0294-1449/$ – see front matter ©2006 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2006.04.004

whereΩ is a smooth bounded domain inR^N,N 3,p=^N_N⁺₋²₂ is the critical exponent from the Sobolev viewpoint andλ >0 is a large parameter. Here,n(x)is the unit outward normal ofΩ atx∈∂Ω.

Under the transformationv(x)=λ^−p−11u(x),d²= ¹_λ, problem (1) reads equivalently as a singularly perturbed Neumann problem:

⎧⎨

⎩

−d²v+v=v^p inΩ,

v >0 inΩ,

∂v

∂n=0 on∂Ω,

(2)

wherep=^N_N⁺₋²₂. For general exponentp >1, problem (2) is related to the study of stationary solutions for a chemo- taxis system (see [17]) proposed by Keller, Segel and Gierer, Meinhardt (see [18]).

Problem (1) forλlarge has been widely studied in the subcritical casep <^N_N⁺₋²₂. The asymptotic behaviour and the construction of blowing up solutions have been considered by several authors. In particular, there exist peak solutions which blow up at many finitely boundary and/or interior points ofΩ.

The critical casep=^N_N⁺₋²₂ has different features. Starting from the pioneering works of Adimurthi, Mancini and Yadava [3] (see also [1,2]), asymptotic analysis (see [13,15] for low energy solutions) and construction of solutions concentrating at boundary points ofΩ have been considered by several authors (see for example [21]). We refer to [20] for an extensive list of references about subcritical and critical case.

As far as interior concentration, the situation is quite different since in literature no results are available and it is expected that in general such solutions should not exist. A first partial result in this direction is due to Cao, Noussair and Yan [6] forN 5 and for isolated blow-up points. They show that any concentrating solutions sequence with bounded energy cannot have only interior peaks and so at least one blow-up point must lie on∂Ω. At the same time, Rey in [20] gets the same result forN=3 by removing any assumption on the nature of interior blow-up points.

Using some techniques developed by Druet, Hebey and Vaugon in [12] for related problems on Riemannian man- ifolds, Castorina and Mancini in [7] were able to show, among other things, that the conclusion of previous papers holds without any restriction on the dimension. Namely, forN3 at least one blow-up point lies on∂Ω.

However, all these papers do not answer to the full question: do there exist blowing up solutions for (1) with bounded energy which do not remain bounded in the interior ofΩ asλ→ +∞? ForN >6 the answer is negative since we will show that ALL the blow-up points have to lie on∂Ω:

Theorem 1.1.LetN >6. Letλ_n→ +∞andu_nbe a solutions sequence of

⎧⎪

⎨

⎪⎩

−u_n+λ_nu_n=N (N−2)u

N+2 N−2

n inΩ,

u_n>0 inΩ,

∂un

∂n =0 on∂Ω,

(3)

with uniformly bounded energy:

sup

n∈N

Ω

u

2N N−2

n <+∞.

Then, for anyKcompact set inΩthere existsC_K such that:

maxx∈Ku_n(x)C_K for anyn∈N.

Theorem 1.1 is based on a local description of possible compactness loss and does not need any boundary condition.

In fact, we realized that Theorem 1.1 is a particular case of a more general interior compactness result, which is still more interesting that our initial question about singularly perturbed Neumann equations and becomes the main content of this paper. There holds:

Theorem 1.2.LetN >6. LetKbe a compact set inΩandΛ >0. There exists a constantC, depending onKandΛ, such that any solutionuof the problem:

⎧⎨

⎩

−u+u=N (N−2)u^N+2N−2 inΩ,

u >0 inΩ,

Ω|∇u|²+u²Λ, satisfies the bound:

maxx∈Ku(x)C.

Compactness properties of the type we are considering appear in a Riemannian context in [8,9] where a careful analysis based on theC⁰-theory developed in [10,11] for Riemannian manifolds gives the Schoen compactness result in low dimensions and provides also in high dimensions results as in Theorem 1.2. However, in this context (without homogeneous Dirichlet boundary condition) theC⁰-theory developed by Druet, Hebey and Robert is not available.

The paper is organized in the following way. In Section 2 we introduce the notion of (geometrical) blow-up set, we give a description of this set and we show by a rescaling argument that Theorem 1.1 is a particular case of Theorem 1.2. In Section 3, we provide the proof of Theorem 1.2: based on a technical result contained in [10,11] due to Druet, Hebey and Robert (which we report in Appendix A for the sake of completeness), for any interior blowing up solutions sequence we are able to prove an upper estimate (in terms of standard bubbles) which contradicts a related local Pohozaev identity.

2. The blow-up set

Letunbe a solutions sequence of

⎧⎪

⎨

⎪⎩

−un+μnun=N (N−2)u

N+2 N−2

n inΩ,

un>0 inΩ,

sup_n∈N

Ω(|∇u_n|²+u²_n) <+∞,

(4) whereΩis a domain inR^N,N3, and 0μ_n→μ∈ [0,+∞].

We define the (geometrical) blow-up set ofu_ninΩ as S=

x∈Ω: ∃xn→xs.t. lim sup

n→+∞un(xn)= +∞ ,

and, by definition ofS, clearlyu_nis uniformly bounded inC_loc⁰ (Ω\S).

Further, define the set Σc=

x∈Ω: lim sup

n→+∞

B_r(x)

u

N−2N2

n c∀r >0

,

wherec >0. LetSN be the best constant related to the immersion ofH₀¹(Ω)intoL^N2N−2(Ω):

S_N= inf

u∈H₀¹(Ω)\{0}

Ω|∇u|² (

Ω|u|^N−22N )^N−2N

. (5)

By means of an iterative Moser-type scheme, we can describe the setSin the following way:

Proposition 2.1.There existsc=c_N>0such that it holdsS=Σ_c. In particular,Sis a finite set and, ifμ= +∞, we have thatu_n→0inC_loc⁰ (Ω\S)(up to a subsequence).

Proof. First of all, we show the following implication:

B2r(x)

u

2N N−2

n

SN

qN (N−2) ^N

2

, q2 ⇒

Br(x)

u

N N−2q n

^N−2

N 8

S_Nr²

B2r(x)

u^q_n

for anyr <¹₂dist(x,R^N\Ω).Letϕ∈C₀^∞(B2r(x))be so that 0ϕ1,ϕ≡1 inBr(x)and∇ϕ∞²_r. Multiply- ing (4) byϕ²u^q_n⁻¹and integrating by parts, by (5) and Hölder’s inequality we get that:

Ω

∇u_n∇

ϕ²u^q_n⁻¹ +μ_n

Ω

ϕ²u^q_n=N (N−2)

Ω

u

4 N−2

n

ϕu

q

n2

2

N (N−2) S_N

B2r(x)

u

N−22N

n

²

N

Ω

∇ ϕu

q

n2². On the other hand, we can write:

Ω

∇u_n∇

ϕ²u^q_n⁻¹

=(q−1)

Ω

ϕ²u^q_n⁻²|∇u_n|²+2

Ω

ϕu^q_n⁻¹∇ϕ∇u_n

= 2 q

Ω

∇ ϕu

q

n2²+q−2 2

Ω

ϕ²u^q_n⁻²|∇u_n|²−2 q

Ω

|∇ϕ|²u^q_n

2

q

Ω

∇ ϕu

q

n2²−2 q

Ω

|∇ϕ|²u^q_n.

Combining these two estimates, we get that 2

q

Ω

∇ ϕu

q

n2²2 q

Ω

|∇ϕ|²u^q_n+N (N−2) SN

B2r(x)

u

2N N−2

n

²

N

Ω

∇ ϕu

q

n2²

8

qr²

B2r(x)

u^q_n+1 q

Ω

∇ ϕu

q

n2²

in view of

B_2r(x)u

2N N−2

n (_{qN (N}^SN₋₂₎)^N2. Therefore, by (5) we obtain that

B_r(x)

u

N N−2q n

^N−2_N

Ω

ϕu

q

n2

_N^2N₋₂^N−2_N

1

S_N

Ω

∇ ϕu

q

n2² 8 S_Nr²

B_2r(x)

u^q_n.

Since _N^N₋₂q > q, we can iterate the procedure starting fromq=2 up to get a-priori bounds inL^p-norms aroundx for anyp >2 provided theL^N2N−2-norm aroundx is sufficiently small. Namely, we find 0< δ <1,p > ^N_N⁺₋²₂^N₂ and c=c_N>0, depending only onN, such that, if

B2r(x)u

2N N−2

n c, then

B_δr(x)

u^p_{n p}²

C(N, r)

B_2r(x)

u²_n,

for some constantC(N, r)depending only onNandr. Letu⁽¹⁾_n be the solution of

−u⁽¹⁾n =N (N−2)u

N+2 N−2

n inBδr(x), u⁽¹⁾_n =0 on∂B_δr(x),

andu⁽²⁾_n be an harmonic function such thatu⁽²⁾_n =u_non∂B_δr(x). Since N (N−2)u

N+2 N−2

n

L^s(B_δr(x))=O

B2r(x)

u²_n ¹

2

for somes >^N₂, by elliptic regularity theory (cf. [14]) we get that u⁽¹⁾_n

C⁰(B_δr(x))=O

B2r(x)

u²_n ¹

2

provided

B_2r(x)u

N−22N

n c. By the representation formula for harmonic function, we get that u⁽²⁾_n

C⁰(Bδr/2(x))=O

∂B_δr(x)

un

.

Since by the maximum principle 0< u_nu⁽¹⁾_n +u⁽²⁾_n , we get that u_nC⁰_(Bδr/2(x))C

B2r(x)

u²_n ¹

2 +

∂Bδr(x)

u_n

(6) for someC >0.

By the continuous embedding ofH¹(Ω)intoL^N−22N (Ω)we get that sup_n∈N

Ωu

2N N−2

n <+∞and therefore,Σ_cis a finite set, wherec=cNis as above. Moreover, up to a subsequence we can assume thatun uweakly inH¹(Ω)and un→uinL²(Ω), in view of the compact embedding ofH¹(Ω)intoL²(Ω). Integrating (4) againstϕun,ϕ∈C₀^∞(Ω), we get thatμnu²_nis uniformly bounded inL¹_loc(Ω)and hence,u=0 ifμ= +∞.

By the compactness of the embedding ofH¹(B_2r(x))intoL²(B_2r(x))and ofH¹(B_δr(x))intoL¹(∂B_δr(x))in the sense of traces, in view of (6) we get thatS=Σ_cN is a finite set and, ifμ= +∞,u_n→0 inC_loc⁰ (Ω\S)(up to a subsequence). 2

Remark 2.2.Blowing up the sequence u_n around a point x∈S, by means of the same techniques which we will exploit strongly in Appendix A, it is easy to show that:

lim sup

n→+∞

Br(x)

u

2N N−2

n

SN

N (N−2) ^N

2

for anyr >0. Hence, the valuec=c_Nin Proposition 2.1 can be taken asc=(_{N (N}^SN₋₂₎)^N2. We are now in position to deduce Theorem 1.1 by Theorem 1.2.

Proof of Theorem 1.1. Multiplying (3) byu_nand integrating by parts, we get that:

Ω

|∇un|²+λnu²_n

Λ:=N (N−2)sup

n∈N

Ω

u

N−22N

n <+∞. (7)

We can define the blow-up setS of the sequenceun. By the validity of Theorem 1.2, we deduce thatShas to be an empty set and therefore,unis uniformly bounded inC_loc⁰ (Ω).

Otherwise, ifS= ∅, up to a subsequence, we can assume that there existsx0∈Ssuch that maxx∈B_r(x₀)un(x)→ +∞as n→ +∞, for anyr >0. By Proposition 2.1, we know thatS is a finite set. Let 0< r <dist(x0, S\ {x0}) andxn be such that un(xn)=maxx∈B_r(x₀)un(x)→ +∞ as n→ +∞. Clearly, since un is uniformly bounded in C_loc⁰ (Ω\S),xn→x0asn→ +∞.

Introduceε_n=u_n(x_n)^−N2−2 →0 and defineU_n(y)=ε^N−

2

n2 u_n(ε_ny+x_n)fory∈B_n:=B ^r

2εn(0). We have that

−U_n+μ_nU_n=N (N−2)U

N+2N−2

n inB_n, 0< U_n(y)U_n(0)=1,

where μn =λnε_n². Assume that μn=λnε²_n→ μ∈ [0,+∞]. Since Un is uniformly bounded in H_loc¹ (R^N) and inC⁰_loc(R^N), ifμ= +∞, Proposition 2.1 implies that U_n→0 in C_loc⁰ (R^N) (up to a subsequence) contradicting U_n(0)=1.

So,μ <+∞. By standard elliptic estimates (cf. [14]), we have thatU_n→UinC_loc² (R^N)whereU∈H¹(R^N)is a solution of

−U+μU=N (N−2)U^NN−2+2 inR^N,

0< U (y)U (0)=1 (8)

(in view of (7)). By a Pohozaev identity onR^N(see [19]), we must have that μ_n=λ_nε²_n→μ=0.

Now, we do the following rescaling. Letv_n(x)=λ⁻

N−2

n 4 u_n(x/√

λ_n+x_n)be defined forx∈B₁(0). The functionv_n satisfies:

⎧⎪

⎨

⎪⎩

−vn+vn=N (N−2)v

N+2 N−2

n inB1(0),

vn>0 inB1(0),

B₁(0)(|∇vn|²+v_n²)Λ, since

B₁(0)

|∇v_n|²+v_n²

=

B_1/^√_λn(xn)

|∇u_n|²+λ_nu²_n

Ω

|∇u_n|²+λ_nu²_n

Λ.

By Theorem 1.2 we get that there existsC >0 such that

x∈maxB_1/2(0)v_n(x)C.

So, we reach a contradiction since we have already shown that vn(0)=λ⁻

N−24

n un(xn)= 1

λ_nε_n^{2 N−2}

4 → +∞

asn→ +∞. The proof is now complete. 2 3. Nonexistence of interior blow-up points

The proof of Theorem 1.2 is based on a contradiction argument. In view of Proposition 2.1, let us assume the existence of a solutions sequenceunof the following problem:

⎧⎪

⎪⎨

⎪⎪

⎩

−u_n+u_n=N (N−2)u

N+2

nN−2 inB₁(0),

u_n>0 inB1(0),

sup_n∈N

B₁(0)u

N−2N2

n <+∞,

which blows up in B₁(0) only at 0: maxx∈B1(0)u_n(x)→ +∞ as n→ +∞ and u_n is uniformly bounded in C_loc⁰ (B₁(0)\ {0}).

By means of Propositions A.1, A.2 and by elliptic regularity theory (cf. [14]), up to a subsequence, we will assume throughout this section the existence of sequencesx_n¹, . . . , x_n^k→0,ε¹_n, . . . , ε_n^k→0 andx₁, . . . , x_k∈R^Nsuch that for anyi=1, . . . , k:

U_nⁱ(y)= ε_n^iN−2

2 un

εⁱ_ny+x_nⁱ

→ 1

(1+ |y−xi|²)^N−22

inC²_loc R^N\Si

asn→ +∞, (9)

u_n→u₀ inC_loc⁰

B₁(0)\ {0}

asn→ +∞, (10)

dk(x)^N−22un(x)C for anyn∈N,|x|<1, (11)

R→+∞lim lim sup

n→+∞max

x∈Bⁿ_R

d_k(x)^N−22u_n(x)−u₀(x)=0, (12)

for some constantC >0 and for some smooth solutionu₀0 of the equation:

−u₀+u₀=N (N−2)u

N+2 N−2

0 inB₁(0),

whered_k(x)=min{|x−x_nⁱ|: i=1, . . . , k},B_Rⁿ= {|x|<1: |x−x_nⁱ|Rε_nⁱ ∀i=1, . . . , k}and S_i=

y_j= lim

n→+∞

x_n^j−x_nⁱ

εⁱ_n : j < is.t. |x_n^j−x_nⁱ| ε_nⁱ =O(1)

.

Let nowxbe so that|x−x_nⁱ| =Rεⁱ_nfor somei=1, . . . , k. We have thaty=^x−_εi^xni

n satisfies:|y| =Rand|y−y_j| R− |y_j|1, forR large. Hence, by (9) we get that for anyR >0 large andC >1 there existsN₀such that for any nN₀and|x−x_nⁱ| =Rεⁱ_n

u_n(x)CU_εi

n,xⁱ_n+ε_nⁱxi(x), where

U_ε,y(x)=ε^−N2−2U x−y

ε

= ε^N−22 (ε²+ |x−y|²)^N−22

.

Since|x−(xⁱ_n+εⁱ_nx_i)|(1−^maxi=1,...,k_R ^|xi|)|x−x_nⁱ|for|x−x_nⁱ| =Rεⁱ_n, we obtain that for anyR >0 large there existsN0such that

u_n(x)2U_εi

n,x_nⁱ(x) (13)

for anynN0and|x−x_nⁱ| =Rε_nⁱ.

The aim will be to estimate from aboveun(x)in terms of the standard bubblesU_εi

n,xⁱ_n(x),i=1, . . . , k, inBδ(0)\ k

i=1B_Rεi

n(x_nⁱ), 0< δ <1. By performing some simple asymptotic analysis we get the following result (see also the techniques developed by Schoen in [22] and exploited in [15,16]):

Lemma 3.1.Letα∈(0,^N−₂²). There existR >0,0< δ <1andN0∈Nsuch that u_n(x)

k i=1

εⁱ_n^N−2

2 −αx−x_n^i2−N+α+M_nx−x_n^i−α ,

for anynN₀and|x|δwith|x−x_nⁱ|Rεⁱ_n,i=1, . . . , k, whereM_n=2δ^αsup_|x|=δu_n(x).

Proof. Let us introduce the operatorLn= −+1−N (N−2)u

4 N−2

n . Sinceunis a positive solution ofLnun=0 in Bδ(0), we have thatLnsatisfies the minimum principle inBδ(0)for any 0< δ <1 (see [14]). Sinceu0is a smooth function, by (12) we have that there existR >2^α1, 0< δ <1 andN0∈Nsuch that

dk(x)^N−22un(x)

α(N−2−α) kN (N−2)

^N−2

4

(14) for anynN0andx∈Bδ(0):|x−x_nⁱ|Rε_nⁱ,i=1, . . . , k.

Define now a comparison functionϕnin the formϕn=k

i=1ϕⁱ_n, where ϕⁱ_n(x)=

εⁱ_n^N−2

2 −αx−x_n^i2−N+α+M_nx−x_n^i−α, and computeL_nonϕ_n−u_n:

Ln(ϕn−un)= k i=1

Lnϕ_nⁱ = k i=1

α(N−2−α)x−x_nⁱ⁻²+1−N (N−2)u

N−24

n

ϕⁱ_n.

Letx∈Bδ(0)be such that|x−xⁱ_n|Rεⁱ_n,i=1, . . . , k. There existsj∈ {1, . . . , k}so that|x−x_n^j| =min{|x−x_nⁱ|:

i=. . . , k}. Since|x−x_n^j||x−xⁱ_n|, we have thatϕ^j_n(x)ϕ_nⁱ(x)for anyi=1, . . . , kand therefore, L_n(ϕ_n−u_n)(x)=

k i=1

α(N−2−α)x−x_nⁱ⁻²+1−N (N−2)un(x)^N4−2 ϕ_nⁱ(x)

α(N−2−α)x−x_n^j−2ϕ_n^j(x)−N (N−2)un(x)^N4−2

k i=1

ϕ_nⁱ(x)

α(N−2−α)−kN (N−2)x−x_n^jN−22 u_n(x) ⁴

N−2x−x_n^j−2ϕ_n^j(x)0

in view of (14), for any nN0 andx ∈B_δ(0): |x −x_nⁱ|Rεⁱ_n,i=1, . . . , k. In view of the validity of (13) on

∂B_Rεi

n(x_nⁱ), we can always assume thatRandN0are such that u_n(x)2

ε_n^iN−2

2 x−x_n^i2−N

for anynN₀and|x−x_nⁱ| =Rεⁱ_n. Therefore, we have that u_n(x)2R^−α

ε_n^iN−2

2 −αx−x_n^i2−N+α εⁱ_n^N−2

2 −αx−x_n^i2−N+αϕ_nⁱ(x)ϕ_n(x) fornN₀and|x−x_nⁱ| =Rεⁱ_nfor somei=1, . . . , k. Since

u_n(x) 1

2δ^αM_nM_n k i=1

x−x_n^i−αϕ_n(x)

for|x| =δ andnlarge, by the minimum principle forLn we get the desired estimate in the regionx∈Bδ(0)with

|x−x_nⁱ|Rε_nⁱ ∀i=1, . . . , k. 2

We have to combine the estimate contained in Lemma 3.1 with the following Pohozaev-type inequality (“essen- tially” proved in [7], for the Pohozaev identity refer to [19]):

Lemma 3.2.There existsC >0, depending only on the dimensionN, such that for any|x|<1and0< h <^1−|₄^x|

Bh(x)

u²_nC

B2h(x)\Bh(x)

1 h²u²_n+u

2N N−2

n

. (15)

Proof. Since Lemma 3.2 is written in a slightly different way with respect to [7], let us outline why some difference appears. By [7] we get that:

Bh(x)

u²_n−2 N

Ω

x,∇ϕϕu

2N N−2

n +R_n,

whereϕ∈C₀^∞(B_2h(x))is such that 0ϕ1,ϕ=1 onB_h(x), and R_n= −

Ω

u²_n

∇ϕ,∇x,∇ϕ +N

2ϕϕ+1 2

x,∇(ϕϕ)

+N−2 2 |∇ϕ|²

.

Assuming that|∇ϕ|_h², we have thatx,∇ϕϕvanishes outsideB2h(x)\Bh(x)and x,∇ϕϕ4

inB2h(x)\Bh(x). Hence, we get that

Ω

x,∇ϕϕu

2N N−2

n

4

B_2h(x)\B_h(x)

u

2N N−2

n .

Similarly, assuming that|∇²ϕ|_h²2 we show that

|R_n| C h²

B_2h(x)\B_h(x)

u²_n

for some constantC >0. The proof is now complete. 2

LetN >6 and fix 0< α <^N₃⁻⁶. Let us define the following sequence:

rn=max

ε_nⁱ: i=. . . , k_N−²⁺₂₋^α_2α .

Up to a subsequence and a re-labeling, let us assume that ε^k_n=max{εⁱ_n: i=. . . , k} and that, for some integer s∈ {1, . . . , k−1}, there hold:

|x_nⁱ −x_n^k|

r_n → +∞, i=1, . . . , s, |x_nⁱ −x_n^k|

r_n D−1, i=s+1, . . . , k, (16)

asn→ +∞, whereD >1 is a constant. By (16), we obtain that forx∈B_2Drn(x_n^k)\B_Drn(x^k_n)there hold:

|x−x_nⁱ||x_nⁱ −x_n^k| − |x−x_n^k||x_nⁱ −x^k_n| −2Drnr_n ifi=1, . . . , s

|x−x_nⁱ||x−x_n^k| − |x^k_n−x_nⁱ|Dr_n−(D−1)rn=r_n ifi=s+1, . . . , k. (17) We apply now (15) onBDr_n(x_n^k). Letr=¹₂min{|yj|: yj ∈Sk, yj =0}ifSk\ {0} = ∅andr=1 otherwise. Since r_nε^k_nfornlarge in view ofα <^N₃⁻⁴, we have that

B_Drn(x^k_n)

u²_n

Brεkn(x_n^k)

u²_n= ε^k_n2

Br(0)

U_n^k2

ε_n^k2

Br(0)\Br/2(0)

U_n^k2

.

SinceS_k∩ {^r₂|y|r} = ∅, by (9) we get that

Br(0)\B_r/2(0)

U_n^k2

→

Br(0)\B_r/2(0)

1

(1+ |y−x_k|²)^N−2 >0 and hence,

B_Drn(x^k_n)

u²_n1 2

ε_n^k2

Br(0)\Br/2(0)

1

(1+ |y−xk|²)^N−2

for n large. Let us remark that for n large _4D¹ δ > r_nRε_n^kRε_nⁱ for any i=1, . . . , k. Therefore, we can use Lemma 3.1 and (17) to provide that

un(x) k i=1

εⁱ_n^N−2

2 −αx−x_n^i2−N+α+Mnx−x_n^i−α

k

ε_n^kN−2

2 −α

r_n^2−N+α+kMnr_n^−α, (18) for anynN₀andx∈B_2Drn(x_n^k)\B_Drn(x_n^k). Hence, by (18) we get that