Critical points of the Trudinger–Moser trace functional with high energy levels

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Ann. I. H. Poincaré – AN 32 (2015) 59–95

www.elsevier.com/locate/anihpc

Critical points of the Trudinger–Moser trace functional with high energy levels

Shengbing Deng

, Monica Musso

aDepartamento de Ingeniería Matemática and CMM, Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile bDepartamento de Matemática, Pontificia Universidad Catolica de Chile, Avda. Vicuña Mackenna 4860, Macul, Chile

Received 19 March 2013; received in revised form 19 September 2013; accepted 7 October 2013 Available online 18 October 2013

Abstract

LetΩbe a bounded domain inR²with smooth boundary. In this paper we are concerned with the existence of critical points for the supercritical Trudinger–Moser trace functional

∂Ω

e^kπ(1+μ)u2 (0.1)

in the set{u∈H¹(Ω):

Ω(|∇u|²+u²) dx=1}, wherek1 is an integer andμ >0 is a small parameter. For any integerk1 and for anyμ >0 sufficiently small, we prove the existence of a pair ofk-peaks constrained critical points of the above problem.

©2013 Elsevier Masson SAS. All rights reserved.

Keywords:Trudinger–Moser trace functional; Reduction methods

1. Introduction

LetΩbe a bounded domain inR²with smooth boundary, and letH¹(Ω)be the Sobolev space, equipped with the norm

u =

Ω

|∇u|²+u² dx

¹

2

.

Letαbe a positive number, the Trudinger–Moser trace inequality states that C_α(Ω)= sup

u∈H¹(Ω),u1

∂Ω

e^α|u|2

C <+∞, ifαπ,

= +∞, ifα > π (1.1)

✩ The research of the second author has been partly supported by Fondecyt Grant 1120151 and CAPDE-Anillo ACT-125, Chile.

* Corresponding author.

E-mail addresses:shbdeng65@gmail.com(S.-B. Deng),mmusso@mat.puc.cl(M. Musso).

0294-1449/$ – see front matter ©2013 Elsevier Masson SAS. All rights reserved.

http://dx.doi.org/10.1016/j.anihpc.2013.10.002

[1,2,6,7,18,22,23]. Let us mention that the early works[6,7]do not include the case when the constant in(1.1)is exactlyπ. For(1.1)there is a loss of compactness at the limiting exponentα=π. Despite of that, it has been proven in[28]that the supremumC_π(Ω)is attained by a functionu∈H¹(Ω)with

Ω[|∇u|²+u²] =1, for any bounded domainΩ inR², with smooth boundary. Also, for anyα∈(0, π ), the supremumC_α(Ω)is finite and it is attained.

But the exponentα=πis critical in the sense that for anyα > π,Cα(Ω)= ∞. See also[8,16,17]for generalizations.

The aim of this paper is to study the existence of critical points of the Trudinger–Moser trace functional E_α(u)=

∂Ω

e^αu2, (1.2)

constrained to functions

u∈M= u∈H¹(Ω): u²=1

(1.3) in the supercritical regime

α > π.

In view of the results described above, we will be interested in critical points other than global supremum. As far as we know, no results are known in the literature concerning existence of critical points for the Trudinger–Moser trace constrained problem in thesupercritical regime. Nevertheless, much more is known for the corresponding Trudinger–

Moser functional.

Let us recall that the Trudinger–Moser inequality in dimension 2 states that sup

u∈H₀¹(Ω),∇u21

Ω

e^μ|u|2dx

C <+∞, ifμ4π,

= +∞, ifμ >4π. (1.4)

Here againΩ is a bounded domain ofR², with smooth boundary. We refer the reader to[25,23,27,29]for the first works on problem(1.4), and to[3,4]for some more recent contributions. For problem(1.4)there is a loss of compact- ness at the limiting exponentμ=4π[21]. Despite of this loss of compactness, the supremum

sup

u∈H₀¹(Ω),∇u21

Ω

e^4π|u|2dx

is attained for any bounded domainΩ⊂R². This was proven first in the seminal work[5]for the ballΩ=B₁(0) (see also an alternative proof in[10]). In[26]the result was proven for domainsΩwhich are small perturbation of the ball. The general result in dimension 2 was proven by Flucher in[14], and Lin[20]extended it for the corresponding Trudinger–Moser inequality for general domain ofR^N, withN >2.

Concerning the supercritical regime for the Trudinger–Moser functional, namely Iμ(u)=

Ω

e^μ|u|2dx, u∈H₀¹(Ω),∇u²2=1, withμ >4π, (1.5) some results are known. In the works[26]and[15]it has been proven that a local maxima and saddle point solutions in the supercritical regimeμ∈(4π, μ0)for the functional(1.5)do exist, for someμ₀>4π.

Our first result is an extension of the existence of a local maxima for the Trudinger–Moser trace functional in the supercritical regime α∈(π, α₀). Namely, a local maximizer for problem(1.2)–(1.3)exists when the value of αis slightly to the right ofπ.

Theorem 1.1.LetΩbe a bounded domain inR². Then there existsα0> π, such that for anyα∈(0, α0), there exists a functionu_α∈Mwhich locally maximizes ofE_αonM.

This result is proved in Section2.

Much more is known for problem(1.5)andμ >4π. Recently in[12](see also[11]), the authors obtained several results concerning critical points for problem(1.5)also in averysupercritical regime. They found general conditions on the domainΩ under which there is a critical point forI_μ(u)with

Ω|∇u|²dx=1 whenμ∈(4π k, μk), for any

integerk1 and for someμ_kslightly bigger than 4π k. In particular, for any bounded domainΩ, they found a critical point forI_μ(u)with

Ω|∇u|²dx=1 whenμ∈(4π, μ1), for someμ₁>4π. TheL^∞-norm of this solution converges to∞asμ→4πand its mass is concentrated, in some proper sense, asμ→4π, around a point in the interior ofΩ. On the other hand, ifΩ has a hole, namely it is not simply connected, they proved the existence of a critical point forI_μ(u)with

Ω|∇u|²dx=1 also in the supercritical rangeμ∈(8π, μ2), for someμ₂>8π. Again in this case, theL^∞-norm of these solutions converges to∞asμ→8π, but now its mass concentrates, asμ→8π, around two distinct points insideΩ. Furthermore, ifΩ is an annulus, taking advantage of the symmetry, a critical point forIμ(u) with

Ω|∇u|²dx=1 andμ∈(4π k, μk)does exist. In this latter case, theL^∞-norm of the solution converges to∞ asμ→4π kand its mass concentrates, asμ→4π k, aroundkpoints distributed along the vertices of a proper regular polygon withksides lying insideΩ.

The second result of this paper establishes the counterpart of the above situation for the Trudinger–Moser trace functional in the supercritical regime: we will show the existence of critical points for E_α constrained to M, for α∈(kπ, α_k), for anyk1 integer and for someα_k slightly to the right ofkπ. We next describe our result.

LetG(x, y)be the Green’s function of the problem

⎧⎨

⎩

−xG(x, y)+G(x, y)=0, x∈Ω;

∂G(x, y)

∂νx =2π δy(x), x∈∂Ω, (1.6)

andHits regular part defined as H (x, y)=G(x, y)−2 log 1

|x−y|. (1.7)

Our second result reads as follows.

Theorem 1.2.LetΩ be any bounded domain inR²with smooth boundary. Fix a positive integerk1. Then there existsαk > kπ such that forα∈(kπ, αk), the functionalEα(u)restricted toM has at least two critical pointsu¹_α andu²_α. Furthermore, for anyi=1,2there exist numbersmⁱ_j,α>0and pointsξ_j,αⁱ ∈∂Ω, forj=1, . . . , ksuch that

αlim→kπmⁱ_j,α=mⁱ_j∈(0,∞), (1.8)

ξ_j,αⁱ →ξ_jⁱ∈∂Ω, withξ_jⁱ =ξ_lⁱ forj =l,asα→kπ (1.9)

and

uⁱ_α(x)=

α−kπ α

k j=1

mⁱ_j,αG x, ξ_j,αⁱ

+o(1)

, i=1,2, (1.10)

where o(1)→0 uniformly on compact sets of Ω¯ \ {ξ₁ⁱ, . . . , ξ_kⁱ}, as α→kπ. In particular,(ξⁱ, mⁱ)=(ξ₁ⁱ, . . . , ξ_kⁱ, mⁱ₁, . . . , mⁱ_k)in(∂Ω)^k×(0,∞)^k, fori=1,2, are two distinct critical points for the function

f_k(ξ, m)=2 k

2 k j=1

m²_jlog 2m²_j

− k j=1

m²_jH (ξ_j, ξ_j)−

i =j

m_im_jG(ξ_i, ξ_j)

.

Moreover, for anyi=1,2, for anyδ >0small, for anyj=1, . . . , k, sup

x∈B(ξ_jⁱ,δ)

uⁱ_α(x)→ +∞ asα→kπ. (1.11)

There are two important differences between the result stated inTheorem 1.2and the corresponding result obtained in[12]for the Trudinger–Moser functional(1.5). A first difference is that for problem(1.2)–(1.3)existence of critical points in the rangeα∈(kπ, α_k)is guaranteed inanybounded domainΩwith smooth boundary, at any integer levelk.

No further hypothesis onΩis needed, unlike the Trudinger–Moser case(1.5). The second difference is that, we do find twofamilies of critical points for problem(1.2)–(1.3)whenα∈(kπ, α_k), and not only one as in the Trudinger–Moser case(1.5).

In recent years a very successful method has been developed for studying elliptic equations in critical or supercrit- ical regimes. The main idea is to try to guess the form of the solution (using the shape of the “standard bubble”), then linearize the equation at this approximate solution and use a Lyapunov–Schmidt reduction to arrive at a reduced finite dimensional variational problem, whose critical points yield actual solutions of the equation. In this paper we use this method to study problem(1.2)–(1.3)in the supercritical regime. We explain this in Section3, where we also provide the proof ofTheorem 1.2. Some technical results are postponed to Section4and Section5.

Let us just mention that through out the paper,C will always denote an arbitrary positive constant, independent ofλ, whose value changes from line to line.

2. The local maximizer: proof ofTheorem 1.1 We set

E(u)=

∂Ω

e^u2, (2.1)

and

M_α= u∈H¹(Ω): u²=α

. (2.2)

We note that by the obvious scaling property, finding critical points ofE_α onM (see(1.2)and(1.3)) is equivalent to finding critical points of E onM_α (see (2.1)and(2.2)). In this section, we study the local maximizer for the functionalEconstrained on the setM_αwithαin the right neighborhood ofπ.

We start with the following Lion’s type lemma. The proof is quite standard, but we reproduce it here for complete- ness.

Lemma 2.1.Letu_mbe a sequence of functions inH¹(Ω)withu_m =1. Suppose thatu_m u₀weakly inH¹(Ω).

Then either (i) u0=0, or

(ii) there existsα > πsuch that the familye^u2mis uniformly bounded inL^α(∂Ω).

In particular, in case(ii), we have that

∂Ω

e^{π u2m}→

∂Ω

e^{π u20} asm→ ∞.

Proof. Sinceu_m =1 andu_m u₀weakly inH¹(Ω), we have

Ω

(∇u_m∇u0+u_mu0)→

Ω

|∇u0|²+u²₀

asm→ ∞. Thus we find that

mlim→∞u_m−u₀²= lim

m→∞

Ω

∇(u_m−u₀)²+(u_m−u₀)²

= lim

m→∞

um²−2

Ω

(∇um∇u0+umu0)+ u0²

=1− u₀². Assume u₀ =0. Takep∈(1,₁₋¹

u0²), and chooseq₁andq₂such that 1< pq₁< ¹

um−u0² and _q¹

1 +_q¹₂ =1. By Hölder inequality we have

∂Ω

e^πpu2m=

∂Ω

e^{πp(um−u0+u0)2}=

∂Ω

e^{πp[(um−u0)2+2(um−u0)u0+u20]}

=

∂Ω

e^{πp[(um−u0)2+2umu0−u20]}

∂Ω

e^{πp[(um−u0)2+2umu0]}

=

∂Ω

e^{πp(um−u0)2}e^2πpumu0

∂Ω

e^{πpq1(um−u0)2 1}

q1

∂Ω

e^{2πpq2umu0 1}

q2

.

We now recall that π=sup

θ: sup

u∈H¹(Ω),u1

∂Ω

e^{θ u2}dσ <∞

, (2.3)

see for instance[2,6,7,18]. Hence, given the choice ofp andq₁, we get that there exists a constantC, independent ofm, such that

∂Ω

e^{πpq1(um−u0)2}< C.

On the other hand, Young’s inequality implies that 2|u_mu₀|ε²u²_m+_ε¹2u²₀, withε >0 small. Then from(2.3), we have

∂Ω

e^2πpq2umu0 <

∂Ω

e^{πpq2[ε2u2m+}

1 ε2u²₀]=

∂Ω

e^πpq2ε2u2me^πpq2

1 ε2u²₀

< C

by choosing ε so that pq2ε²<1. Here againC is a constant, independent ofm. Thus, we have that there exists α=pπ > πsuch that the familye^u2mis uniformly bounded inL^α(∂Ω).

We shall now show that

∂Ω

e^{π u2m}→

∂Ω

e^{π u20} asm→ ∞. (2.4)

Indeed, letlbe a positive number andp >1. We have

∂Ω

e^{π u2m}−

∂Ω∩{|um|l}

e^{π u2m} =

∂Ω∩{|u_m|>l}

e^{π u2m} 1

l

2(p−1) p

∂Ω

e^{π u2m}u

2(p−1) p

m

1

l

2(p−1)

p ∂Ω

e^{πpu2m 1}

p

∂Ω

u²_m ^p−1

p C

l

2(p−1) p

.

From the above relation, we conclude that

∂Ω

e^{π u2m}|∂Ω|e^{π l2}+ C l

2(p−1) p

.

Hence dominated convergence theorem implies(2.4).

Suppose now thate^u2mis not bounded inL^α(∂Ω)for anyα > π. Using Stokes theorem, forα > π we have

∂Ω

e^αu2mdσ=

Ω

div e^αu2m

dxC

Ω

|∇u_m||u_m|e^αu2mdx

C

Ω

|∇u_m|²dx ¹

2

Ω

|u_m|^qdx ¹

q

Ω

e^βu2mdx ^α

β

whereq >1 satisfies¹₂+¹_q+_β^α =1 withβ >2π. Then we get that

Ωe^βu2mdxis unbounded for allβ >2π.

Observe now that we can assume that

Ωu_mdx=0, since otherwise we setu¯_m=u_m−_|Ω¹_|

Ωu_mdx and obtain

Ωu_mdx=0. We can also assume that

Ω|∇u_m|²=1. Furthermore, by Poincaré inequality, (u_m)is bounded in H¹(Ω), and also(|u_m|)is bounded inH¹(Ω). Hence there existsu∈H¹(Ω)such that|u_m| u₀weakly inH¹(Ω).

We claim that

mlim→∞

Ω

∇(um−η)⁺²dx=1, ∀η >0. (2.5)

By contradiction, assume there existsη >0 such that limm→∞

Ω|∇(u_m−η)⁺|²dx =1. Defineγ=infm

Ω|∇(u_m− η)⁺|²dx <1 and choose a sufficiently smallε >0 such thatα:=_γ^2π_+ε >2π. Let us recall that

2π=sup

θ: sup

u∈H¹(Ω),

Ω|∇u|²1,

Ωu=0

Ω

e^{θ u2}dx <∞

(2.6) (see[2,6,7,28]). From(2.6), there exists a positive constantCsuch that

Ω

e^{α[(|um|−η)+−|Ω|1}

Ω(|um|−η)⁺]²dx=

Ω

e^2π[

(|um|−η)+− 1

|Ω|

Ω (|um|−η)+

√γ+ε ]²

dx < C,

where we use the fact that

Ω|∇^(u√m_γ⁻₊^η)_ε⁺|²dx <1.

Defined_m=_|Ω¹_|

Ω(|u_m| −η)⁺. Choosingε>0 small such thatα˜:=₁₊^α_ε >2π, and by Young’s inequality, u²_m(η+dm)²+2(η+dm)

|um| −η₊

−dm

+

|um| −η₊

−dm

2

1+ε

|um| −η₊

−dm

2

+ 1

ε +1

(η+dm)². Thus, since theredm=O(1)asm→ ∞,

Ω

e^{αu˜ 2m}dx=

Ω

e

α 1+εu²_m

dxC1

Ω

e^{α[(|um|−η)+−|Ω1|}

Ω(|um|−η)⁺]²dxC2, for some positive constantsC₁andC₂. This is a contradiction, thus(2.5)holds.

Setv_m=min{|u_m|, η}, thenv_mis bounded inH¹(Ω)and, up to subsequence, we have thatv_m v. Observe now that|u_m| =v_m+(|u_m| −η)⁺, and

1=

Ω

|∇u_m|²

Ω

∇|u_m|²dx=

Ω

|∇v_m|²dx+

Ω

∇

|u_m| −η₊²dx.

Therefore(2.5)implies that

Ω|∇vm|²dx→0 asm→ ∞, sovis constant. On the other hand,

mlim→∞

Ω

|∇v_m|²dx= lim

m→∞

Ω∩{|um|η}

∇|u_m|²dx=0.

This implies that|{x: |u_m|η}| →0 asm→ ∞. By Fatou Lemma, {x: u₀η}lim inf

m→∞ x: |u_m|η=0,

then|{x: u0η}| =0 for anyη >0. Hence we getu0=0. 2

We denoteβ:=sup_u∈MπE(u)=sup_u∈ME_π(u). A direct consequence of the previous lemma is the following Proposition 2.1.Letumbe a bounded sequence inH¹(Ω)withum =1. Suppose thatum u0weakly inH¹(Ω).

Suppose E_π(u_m)→β withβ >|∂Ω|. Then there exists α > π such that the familye^u2m is uniformly bounded in L^α(∂Ω). In particularE_π(u_m)→E_π(u₀)andu₀ =0.

Proof. Supposee^u2mis unbounded inL^α(∂Ω)for allα > π, and assume the supremum ofE_π onMis not attained.

Then byLemma 2.1, we have thatu₀=0, which is impossible becauseE_π(u_m)→β >|∂Ω|. 2 LetK_πbe the set defined by

K_π= u∈M: E_π(u)=β .

Lemma 2.2.The setK_π is compact.

Proof. Let{u_m} ⊂K_πbe such thatu_m u₀weakly inH¹(Ω), then byProposition 2.1, Eπ(um)→Eπ(u0).

Moreover,u0um =1, then E_π(u₀)E_π

u0

u₀

sup

v∈M

E_π(v)=β.

Then we getE_π(u₀)=β, andu₀ =1, henceu_m→u₀strongly inH¹(Ω), henceK_π is compact. 2 The property ofK_π of being compact implies that the family of norm-neighborhoods

Nε= u∈M∃v∈Kπ: u−v< ε constitutes a basic neighborhood forKπ inM.

Lemma 2.3.For sufficiently smallε >0, one has sup

N2ε\Nε

Eπ< β=sup

Nε

Eπ. (2.7)

Proof. We argue by contradiction. We suppose that there is a sequenceu_m∈N_2ε\N_εsuch thatE_π(u_m)→β. Then we haveu_m∈H¹(Ω)withu_m²=1. Up to subsequence, we can assume thatu_m u₀weakly inH¹(Ω). By the definition ofN_2ε, there isz_m∈K_π such thatz_m−u_m<2ε. By the compactness ofK_π, we have thatz_m→z strongly, withz∈K_π, andzsatisfies

−z+z=0 inΩ, ∂z

∂ν= π ze^z2

∂Ωz²e^z2 on∂Ω.

By the maximum principle, we havez∈L^∞(Ω).

By the lower-semi continuity, we havez−u₀2ε. Then z− u₀

u₀

z−u₀ +

u₀− u₀ u₀

= z−u₀ +1− u₀4ε.

Thus ^u_u⁰

0∈N_4ε, and soE_π(u₀)E_π(^u_u⁰

0)β. IfE_π(u₀)=β thenu₀ =1, andu_m→u₀. On the other hand, our assumption implies thatu₀∈/N_ε, thusu₀does not belong toK_π andu₀cannot be relatively maximal. Thus we necessarily getE_π(u₀) < β.

Setw_m=u_m−z_m+z, so we havew_m u₀weakly inH¹(Ω). Since e^π|wm|2=e^{π|um−zm+z|2}e^{2π|um−zm|2}e^2π|z|2

=e^{2πum−zm2(um−zmum−zm )2}e^2π|z|2e^{8π ε2(um−zmum−zm )2}e^2π|z|2.

Choosing ε small such that 16ε²1, then from (2.3)we have thate^π|wm|2 is uniformly bounded in L²(∂Ω), as m→ ∞. Thus limm→∞Eπ(wm)=Eπ(u0). On the other hand, we have wm−um→0 strongly in H¹(Ω). By uniform local continuity ofEπ, and compactness ofKπ, we obtain thatEπ(wm)−Eπ(um)→0, andEπ(u0)=β. This is a contradiction. 2

Lemma 2.4.There existsα^∗> π,ε >0such that for allα∈ [π, α^∗), then we have

(i) sup

N_2ε\N_ε

E_α<sup

N_ε

E_α. (2.8)

(ii) β_α:=sup_NεE_αis achieved inN_ε. (iii) K_α= {u∈N_ε|E_α(u)=β_α}is compact.

Proof. (i) SinceK_π is compact, there is a neighborhoodN ofK_π such that, for anyς >0 there existsδ>0 such that for all|α−π|< δthen|E_α(u)−E_π(u)|ς, for allu∈N. Chooseε >0 such that(2.7)holds andN_ε⊂N, then(2.8)will be valid for allαin a small neighborhood ofπ.

(ii) For such α, let u_m∈N_ε be a maximizing sequence of E_α, that is,E_α(u_m)→β_α and let v_m∈K_π satisfy u_m−v_mε. We may assume thatv_m→vstrongly inH¹(Ω)withv∈L^∞, andu_m→uweakly inH¹(Ω). Set w_m=u_m−v_m+v, as in the proof ofLemma 2.3, we obtain that forε >0 small,αin a neighborhood ofπ we have that

E_α(w_m)→E_α(u), E_α(u_m)−E_α(w_m)→0 asm→ ∞.

ThenE_α(u)=β_α. Moreover, by the lower-semi continuity, we havev−uε. Then v− u

u

v−u + u− u

u

= v−u +1− u2ε.

We get that ^u_u∈ ¯N2εandEα(^u_u)βα. Furthermore, sinceu1, we can getEα(^u_u)Eα(u)andu =1. It implies thatu∈M, that isu∈Nεandβαis attained. Moreover,um→ustrongly inH¹(Ω).

(iii) As in the proof of (ii), ifum∈Kα, we may assume thatum uweakly inH¹(Ω), we then getu∈Kα, that isKα is compact. 2

Proof of Theorem 1.1. From(2.3), we have that sup_MαEis achieved forα < π. Moreover, since sup_u∈MπE(u) >

|∂Ω|, fromLemma 2.4we have that forαsufficiently close toπ, thenEhas relative maximizers onMα. 2 3. The proof ofTheorem 1.2

In this section, we consider critical points of functionalE(u)constrained on the setM_α (which is equivalent to consider critical points of E_α(u) constrained on the setM withα=kπ(1+μ), whereμ >0 small). We define a critical point ofE_α constrained onMto be a solution of the following problem

−u+u=0 inΩ;

∂u

∂ν =λue^u2 on∂Ω, (3.1)

where

λ= α

∂Ωu²e^u2 =kπ(1+μ)

∂Ωu²e^u2. (3.2)

In this section we shall prove the existence of solutions to problem(3.1)–(3.2)with the properties described in Theorem 1.2. In fact, we will construct a solution to(3.1)–(3.2)of the form

u=U+φ, (3.3)

whereU is the principal part while φ represents a lower order correction. In what follows we shall first describe explicitly the functionU (x). The definition of this function depends on several parameters: some pointsξ on the boundary ofΩ and some positive numbersm. Next we find the correctionφso thatU+φsolves our problem in a certainprojected sense(seeProposition 3.1). Finally we select proper pointsξ and numbersmin the definition ofU to get an exact solution to problem(3.1)–(3.2).

To define the functionU, first we introduce the following limit problem

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎩

w=0 inR²₊;

∂w

∂ν =e^w on∂R²₊;

∂R²₊

e^w<∞.

(3.4)

A family solution to(3.4)is given by

wt,μ(x)=wt,μ(x1, x2)=log 2μ

(x₁−t )²+(x₂+μ)², (3.5)

wheret∈Randμ >0 are parameters. See[19,24,30]. Set w_μ(x):=w_0,μ(x)=log 2μ

x₁²+(x₂+μ)². (3.6)

Letξ₁, . . . , ξ_kbekdistinct points on the boundary andm₁, . . . , m_kbekpositive numbers. We assume there exists a sufficiently small but fixed numberδ >0 such that

|ξi−ξj|> δ fori =j, δ < mj<1

δ. (3.7)

For notational convenience through out the paper we will use the notation (ξ, m)=(ξ₁, . . . , ξ_k, m₁, . . . , m_k).

For anyj=1, . . . , k, we defineεj to be the positive numbers given by the relation 2λm²_j

log 1

ε²_j +2 log 2m²_j

=1. (3.8)

Since the parametersm_j satisfy assumption(3.7), it follows that limλ→0ε_j=0. Define moreoverμ_jto be the positive constants given by

log(2μj)= −2 log 2m²_j

+H (ξ_j, ξ_j)+

i =j

m_im⁻_j¹G(ξ_i, ξ_j). (3.9)

Using once more assumption(3.7), we get that there exist two positive constantscandC, such thatcμ_j C, as λ→0.

We define the functionUin(3.3)to be given by U (x)=√

λ k j=1

m_j

u_j(x)+H_j(x)

, (3.10)

where

u_j(x)=log 1

|x−ξ_j−ε_jμ_jν(ξ_j)|², (3.11)

ν(ξ_j)denoting the unit outer normal to∂Ω at the pointξ_j, and whereH_j is a correction term given as the solution

of ⎧

⎨

⎩

−H_j+H_j= −u_j inΩ;

∂H_j

∂ν =2εjμ_je^uj −∂u_j

∂ν on∂Ω. (3.12)

Arguing as in Lemma 3.1 in[9], one can show that the maximum principle allows a precise asymptotic description of the functionsH_j, namely we have that

H_j(x)=H (x, ξ_j)+O ε^σ_j

for 0< σ <1 (3.13)

uniformly inΩ, asλ→0. Recall thatHis the regular part of the Green’s function, as defined in(1.6). Therefore, the functionUcan be described as follows

U (x)=√ λ

k j=1

m_j

G(x, ξ_j)+O ε^σ_j

(3.14) uniformly on compact sets ofΩ¯ \ {ξ1, . . . , ξ_k}, asλ→0. On the other hand, if we consider a region close toξ_j, for somej fixed, say for|x−ξj|< δ, with sufficiently small but fixedδ, we can rewrite

U (x)=√ λmj

wj(x)+logε_j⁻²+βj+θ (x)

, (3.15)

where

w_j(x)=w_μj

x−ξ_j εj

=log 2μj

|y−ξ_j−μ_jν(ξ_j)|², y= x εj

, ξ_j =ξ_j εj

, (3.16)

and

βj= −log(2μj)+H (ξj, ξj)+

i =j

m⁻_j¹miG(ξj, ξi), θ (x)=O

|x−ξj| +

k j=1

O ε^α_j

.

Define on the boundary∂Ωthe error of approximation R:=f (U )−∂U

∂ν. (3.17)

Here and in what followsf denotes the nonlinearity f (u)˜ =λue˜ ^u˜2.

The choice we made ofμ_j in(3.9)and ofε_j in(3.8)gives that in the region|x−ξ_j|< δ, the error of approximation can be described as follows

R=m_j√

λ 1+2λm²_j

w_j+O(1) e^λm2jw2j

1+O(λw_j)

−1

ε⁻_j¹e^wj, (3.18)

wherew_j is defined in(3.16). Indeed, forx∈∂Ω with|x−ξ_j|< δ, we have that λ⁻¹²f (U )=λ

mj

wj(x)+logε_j⁻²+βj+θ (x)

e^{λ[mj(wj(x)+logε−}

2

j +βj+θ (x))]²

=

λm_j

log 1 ε²_j +β_j

+λm_j

w_j+O(1)

×e

λm²_j(log ¹

ε2 j

+βj)²

e

2λm²_j(log ¹

ε2 j

+βj)wj

e

2λm²_j(log ¹

ε2 j

+βj)θ (x)

e^{λm2j(wj+θ (x))2}

=λm_j

log 1 ε²_j +β_j

1+

log 1

ε²_j +β_{j −}1

w_j+O(1)

×e

λm²_j(log ¹

ε2 j

+βj)²

e

2λm²_j(log ¹

ε2 j

+βj)wj

e

2λm²_j(log ¹

ε2 j

+βj)θ (x)

e^{λm2j(wj+θ (x))2}

= 1 2mj

1+2λm²_j

w_j+O(1) e

1 2(log ¹

ε2 j

+βj)

e^wje^{θ (x)}e^{λm2j(wj+θ (x))2}

= 1 2mj

ε_j⁻¹e^βj/2

1+2λm²_j

w_j+O(1)

e^wje^{θ (x)}e^λm2jw2j

1+O(λ)w_j thanks to the definition ofε_jin(3.8). On the other hand, in the same region, we have

λ⁻¹²∂U

∂ν = ∂

∂ν m_j

w_j(x)+logε⁻_j²+β_j+θ (x)

=m_jε_j⁻¹e^wj + k j=1

O ε_j²

asλ→0.