Analysis of boundary bubbling solutions for an anisotropic Emden-Fowler equation

www.elsevier.com/locate/anihpc

Analysis of boundary bubbling solutions for an anisotropic Emden–Fowler equation

Juncheng Wei

, Dong Ye

, Feng Zhou

aDepartment of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong bDépartement de Mathématiques, UMR 8088, Université de Cergy-Pontoise, 95302 Cergy-Pontoise, France

cDepartment of Mathematics, East China Normal University, Shanghai 200062, China Received 31 October 2006; received in revised form 31 January 2007; accepted 1 February 2007

Available online 17 July 2007

Abstract

We consider the following anisotropic Emden–Fowler equation

∇(a(x)∇u)+ε²a(x)e^u=0 inΩ, u=0 on∂Ω,

whereΩ⊂R²is a smooth bounded domain andais a positive smooth function. We study here the phenomenon of boundary bubbling solutions whichdo not existfor the isotropic casea≡constant. We determine the localization and asymptotic behavior of the boundary bubbles, and construct some boundary bubbling solutions. In particular, we prove that ifx¯∈∂Ω is a strict local minimum point ofa, there exists a family of solutions such thatε²a(x)e^udxtends to 8π a(x)δ¯ _x¯inD(R²)asε→0. This result will enable us to get a new family of solutions for the isotropic problemu+ε²e^u=0 in rotational torus of dimensionN3.

©2007 Elsevier Masson SAS. All rights reserved.

Keywords:Boundary bubble; Blow-up analysis; Localized energy method

1. Introduction

The classical Emden–Fowler equation, or Gelfand equation

u+ε²e^u=0 inΩ⊂R^N, u=0 on∂Ω (1)

has motivated a lot of studies, because it has both geometrical and physical background. WhenN=2, (1) or more generally Eq. (2) below relates to the geometric problem of Riemannian surfaces with prescribed Gaussian curvature (see [7] and references therein). ForN3, it arises in the theory of thermionic emission, isothermal gas sphere, gas combustion. It is also considered in relation with Onsager’s formulation in statistical mechanics, the Keller–Segel system of chemotaxis, Chern–Simon–Higgs gauge theory and many other physical applications (see [4–6,11,13,19, 17,25] and the references therein).

It is well known that there exists a critical value ε^∗>0 such that whenε > ε^∗, no solution of (1) exists while forε∈(0, ε^∗), we have a family of minimal solutions which tend uniformly to zero asε→0. WhenN=2, for any

* Corresponding author.

E-mail addresses:wei@math.cuhk.edu.hk (J. Wei), dong.ye@u-cergy.fr, ye@math.u-cergy.fr (D. Ye), fzhou@math.ecnu.edu.cn (F. Zhou).

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

doi:10.1016/j.anihpc.2007.02.001

ε∈(0, ε^∗), we have also a second solution which is nonstable and blows up asε→0. The asymptotic behavior of nonstable solutions to (1), or to a more general equation

u+ε²k(x)e^u=0 inΩ⊂R², u=0 on∂Ω (2)

wherek(x)is a positive smooth function has been studied in [3,14,15,18,21,27]. LetG_D denote the standard Green’s function of−with Dirichlet boundary condition andH_Ddenote the regular part ofG_D, i.e.

HD(x, y)=GD(x, y)+ 1

2πlog|x−y|. (3)

Ifuεis a family of solutions to (2) satisfying Tε=ε²

Ω

k(x)e^uεdx→,

asε→0 and limε→0 uε L^∞(Ω)= ∞, then up to a subsequence, there holds either= ∞,uε→ ∞for allx∈Ω; or =8π m,m∈N^∗anduεmakesmpointssimpleblow-up onS= {x1, . . . , xm} ⊂Ω such that

ε²k(x)e^uεdx→8π m j=1

δx_j, uε→8π m j=1

GD(·, xj) inC_loc^k (Ω\S), ∀k∈N, where(x₁, . . . , x_m)is a critical point of defined by

(x)= m j=1

H_D(x_j, x_j)+

i=j

G_D(x_i, x_j)+2 m j=1

logk(x_j).

Conversely, many authors have constructed blow-up solutions, see for example [2,9,10,20]. So the solutions of Eq. (1) or (2) are now well understood in dimension two.

Here we consider the following generalized Emden–Fowler equation

_au+ε²e^u=0 inΩ⊂R², u=0 on∂Ω (4)

whereΩis a smooth bounded domain,_ais the operator _au= 1

a(x)∇

a(x)∇u

=u+ ∇loga∇u anda(x)is a smooth function overΩsatisfying

0< a1a(x)a2<+∞. (5)

Our motivation is due to the fact that few is known for Eq. (1) in dimensionN3. As far as we know, the only explicit results in higher dimensions concern the radial solutions in spheres (see [11,13]) or in annuli (see [22]). It is worth to mention that Pacard proved in [23] (see also [16]) that for annuli, i.e.Ω=A_r0 = {x∈R^N, r₀< x <1}, there exists X⊂(0,1)of measure equal to 1 such that for allr₀∈X, there are infinitely many symmetry breaking points with bifurcation from the branch of radial solutions. Unfortunately, we do not have no more precise information about the behavior of these nonradial solutions. However, through these results, we observe already a quite different situation with the case in dimension two. Our idea here is to consider axially symmetric solutions of (1) in a torus, and try to give some precise descriptions of them. In fact, letTbe a standardN-dimensional torus (N3), i.e.

T=

x=(x_i)∈R^N;

x₁²+ · · · +x²_N−1−1 ²+x_N² r₀²

(6) with 0< r0<1. If we look for solutions of (1) in the form ofu(x)=u(r, s)where

r=

x₁²+ · · · +x_N²₋₁ and s=x_N,

a direct calculus yields that the problem (1) is transformed to

∇(r^N−2∇u)+ε²r^N−2e^u=0 inΩ_T, u=0 on∂Ω_T, (7)

whereΩ_T= {(r, s)∈R²;(r−1)²+s²< r₀²}. This is just Eq. (4) with a(r, s)=r^N−2, so problem (4) represents a special case of (1) in higher dimension.

Eq. (4) seems to be similar to (2) or (1). We can show the existence of critical valueε^∗>0 (depending onaandΩ), the existence of minimal solution and nonstable solution for allε∈(0, ε^∗). But the structure of nonstable solutions is quite different. In [28], the authors studied the asymptotic behavior of bubbling solutions to (4), they proved that if Tε=O(1), then eitheru_ε→0 uniformly on any compact subset ofΩ, or there exists a finite setS= {x_i} ⊂Ω and m_i∈N^∗such thatu_ε→u^∗weakly inW^1,p(Ω)for anyp∈(1,2), whereu^∗verifies

_au^∗+8π

i

m_iδ_xi=0 inΩ, u^∗=0 on∂Ω. (8)

Moreover,m_i∈N^∗and eachx_imust be acritical pointofa. Recently, we have constructed in [26] bubbling solutions near anytopologically nontrivial critical pointofainΩ. In particular, near anyinterior strict local maximumofa, we have solutions with arbitrary given number of bubbles, which illustrates again the contrast with the isotropic situation.

Nevertheless, if we look at Eq. (7), the functiona(r, s)=r^N−2has no critical point inΩ_T. Consequently the blow- up cannot occur in the interior of the domain, it must appear near the boundary. A natural question is to understand these boundary bubbling solutions, which is just the aim of this paper.

First, we show the localization and asymptotic behavior of boundary blow-up whenahas no critical point inΩ. Theorem 1.1.Suppose that the anisotropic coefficientahas no critical point inΩ. Letu_ε be a family of solutions to problem(4)satisfying

Tε=ε²

Ω

e^uεdx→ <∞ and max

Ω

u_ε→ ∞

asεtends to0. Then=8π mwithm∈N^∗and up to a subsequence, there exists a finite setS= {x₁, . . . , x_q} ⊂∂Ω andm₁, . . . , m_q∈N^∗such that

ε²e^uεχ_Ωdx→

1jq

8π mjδ_xj inD(R²).

Moreover, the tangential derivative∂_τa(x_j)=0for any1jpand u_ε→0 inC_loc^k (Ω\S), ∀k∈N.

Remark 1.2.We should emphasize that the boundary blow-up phenomenon does not exist for the isotropic case, or more generally whenais constant in a neighborhood of the boundary. In that case, solutions of Eq. (4) are decreasing with respect tod(x, ∂Ω)in a fixed neighborhood of∂Ω, by moving plane argument as showed in [21] (see also [18]).

Remark 1.3.We can combine Theorem 1.1 with the result in [28] to get a more general conclusion forasatisfying just (5), see Proposition 3.4.

The second part of the paper concerns the existence of boundary bubbling solutions. Let us introduce some nota- tions. LetG(x, y)be the Green’s function associated to−_a, that is, for anyy∈Ω,

_aG(x, y)+8π δy=0 inΩ and G(x, y)=0 ifx∈∂Ω. (9)

DefineHto be the regular part ofG(x, y)as

H (x, y)=G(x, y)+4 log|x−y|. (10)

Then our main results for this part can be stated as follows.

Theorem 1.4.Letx¯∈∂Ωbe a local minimum point ofaon∂Ω, i.e.

∃δ >0 such that a(x) < a(y),¯ ∀y∈B_δ(x)¯ ∩Ω, y= ¯x.

We assume also ∂_νa(x) <¯ 0. Then forε >0 sufficiently small, problem (4) has a family of solutionsu_ε such that ε²e^uεχ_Ωdx→8π δx_¯inD(R²). More precisely, we have

uε(x)=log 1

(ε²μ²_ε+ |x−ξ_ε|²)²+H (x, ξε)+o(1) inΩ (11) whereξ_ε,μ_εsatisfy

ξ_ε→ ¯x, (.ξ_ε, ∂Ω)∼ 1

|logε| and μ_ε∼ 1

|logε|² asε→0. (12)

Here we use the symbolf ∼gto mean the existence ofC >0 such that 1

C lim inf

ε→0

f

g lim sup

ε→0

f g C.

Throughout the work, the symbol Cdenotes always a positive constant independent ofε, it could be changed from one line to another.

The following result shows the reason why we construct the bubbling solutions near a local minimum point ofa on the boundary, but not near a maximum point.

Theorem 1.5.Assume thatTε=O(1)andx¯∈∂Ωis a nondegenerate local maximum point ofa, thenx /¯∈S. If we return to the original equation (1) overT, we get then a family of solutions which blows up at a(N−2)- dimensional submanifold on∂T.

Theorem 1.6.LetTbe the torus defined by(6), then there exists a family of solutionsu_εfor(1)such that asε→0,

εlim→0

T

ε²e^uεdx=0 (13)

andu_εblows up exactly on ST=

(x_i)∈T;

x₁²+ · · · +x_N²₋₁=1−r₀, x_N=0 .

The paper is organized as follows. First we consider the behavior of boundary bubbles, we prove Theorems 1.1 and 1.5 by potential analysis, Pohozaev identity, combined with blow-up arguments used in [15] and [28]. Then we prove Theorem 1.4 via thelocalized energy method, a combination of Liapunov–Schmidt reduction method and variational techniques similar to our previous paper [26]. The difficulties for proving all these results come from the fact that the distance between the bubbles and the boundary will tend to zero, therefore some refinements, in particular some precise informations for the Green’s function and the corresponding Robin function need to be developed to make our approach successful. Theorem 1.6 can be shown as a direct consequence of Theorems 1.5 and 1.4.

2. Behavior of the Green’s functionG(x, y)near∂Ω

In order to study the bubbles of (4) which tend to the boundary, we need to have a good understanding of the Green’s functionG(x, y)associated to−_a, whenyis near∂Ω, and its regular partH, especially the corresponding Robin functionx→H_R(x)=H (x, x).

Lemma 2.1.There exist positive constantsd0andC such that for anyx∈Ω,d(x, ∂Ω)d0, there exists a unique pointx_ν∈∂Ωsatisfyingd(x, ∂Ω)= |x−x_ν|and ifx^∗=2xν−xdenotes the reflection point ofxwith respect to∂Ω, then G(y, x)+4 log|x−y| −4 log|x^∗−y|C, ∀y∈Ω. (14) Moreover,

d(x,∂Ω)lim→0

G(y, x)−4 log|x^∗−y|

|x−y|

L^∞(Ω)

=0. (15)

Proof. ConsiderL(y, x)=G(y, x)+4 log|x−y| −4 log|x^∗−y|. We have

−_a(y)L(y, x)= −4∇loga(y)·

y−x

|y−x|²− y−x^∗

|y−x^∗|²

inΩ.

It is clear that the right-hand side is uniformly bounded inL^p(Ω)for anyp∈ [1,2). Furthermore, by the regularity of the domainΩ, we know that L(·, x) _L∞(∂Ω)=O(d(x, ∂Ω))whend(x, ∂Ω)d₀. The standard elliptic theory implies then L(·, x) _L∞(Ω)=O(1).

More precisely, remark that for 1p <2, a(y)L(y, x) L^p(Ω)tends to 0 as|x−x^∗| →0. So L(·, x) L∞(Ω)→0 asd(x, ∂Ω)tends to 0. 2

Recall the following expansion ofx→H (x, y)proved in [26].

Lemma 2.2. Let H_y(x)=H (x, y) for any y ∈Ω. Then y →H_y is a continuous map from Ω into C^0,γ(Ω),

∀γ∈(0,1). LetH_D be the regular part of the standard Green’s function defined by(3), we have H (x, y)=8π HD(x, y)+ ∇loga(y)· ∇

|x−y|²log|x−y| +H₁(x, y), (16) wherey→H₁(·, y)is a continuous map fromΩintoC^1,γ(Ω)for allγ∈(0,1). Furthermore, the function(x, y)→ H₁(x, y)∈C¹(Ω×Ω), in particularx→H (x, x)∈C¹(Ω).

Using the equation satisfied byH_y, we can get, for anyγ∈(0,1), H_{y C}0,γ(Ω)=O

1 d(y, ∂Ω)

uniformly inΩ. (17)

Now we show the behavior of the Robin functionx→H (x, x)near the boundary.

Lemma 2.3.LetH_R denote the Robin functionx→H (x, x), then HR(x)=4 logd(x, ∂Ω)+O(1), ∇HR(x)=O

1 d(x, ∂Ω)

uniformly inΩ. (18)

Sketch of Proof. Using the equation ofx→H (x, y), clearly H (x, y)=8π HD(x, y)+O(1)inΩ×Ω. By the behavior of H_D (see for example [1]), we haveH (x, x)=4 logd(x, ∂Ω)+O(1)inΩ. For the estimate of∇H_R, using the equation satisfied byH_D(·, y), we obtain

H_D(·, y)

C¹(Ω)=O 1

d(y, ∂Ω)

fory∈Ω.

Consider the equation ofx→H₁(x, y), _a(x)H1(x, y)=4

∇loga(x)− ∇loga(y)

· x−y

|x−y|²−8π∇loga(x)∇xHD(x, y)

− ∇_x²

|x−y|²log|x−y| ·

∇loga(x),∇loga(y) inΩandH₁(x, y)= −∇loga(y)· ∇x(|x−y|²log|x−y|)ifx∈∂Ω. We get then

H1(·, y)

C¹(Ω)=O 1

d(y, ∂Ω)

uniformly inΩ. (19)

Moreover, for 1p <2, we can prove, by direct calculus, x−y

|x−y|²

W^{1−p ,p1} (∂Ω)

=O 1

d(y, ∂Ω)

uniformly fory∈Ω,

hence ∇yH_D(·, y) _W1,p(Ω)=O(d(y, ∂Ω)⁻¹) inΩ. Checking carefully the equation satisfied by ∇yH₁(·, y), we obtain

∇yH1(·, y)

C⁰(Ω)=O 1

d(y, ∂Ω)

uniformly inΩ. (20)

AsH_D(x, y)=H_D(y, x)inΩ×ΩandH_R(x)=8π HD(x, x)+H₁(x, x), we are done. 2

Remark 2.4.Here we havea(x)G(x, y)=a(y)G(y, x)inΩ×Ω, but not the usual symmetry for the standard Green’s functionG_Dand thanks to the expansion (16), we see thatH_yis not inC¹(Ω)in general (except if∇a(y)=0). These facts make our estimate for the Robin functionH_Rmore involved.

3. Boundary blow-up analysis

Now we are in position to prove Theorems 1.1 and 1.5. IfTεis bounded, we know thatuεis bounded inW^1,p(Ω) for anyp∈ [1,2). We get first a Brezis–Merle type result.

Lemma 3.1.LetΩ be a smooth bounded domain inR². There existsα >0 (depending onΩ anda)such that if a solution of (4)uεsatisfies, forx∈Ωandδ >0

Bδ(x)∩Ω

ε²e^uεdyα,

then u_{ε L∞(B}

δ/2(x)∩Ω)C.

By the results in [3], we need just to consider the situation near the boundary, i.e. whenx∈∂Ω. Indeed, we can use conformal transformation to changeΩ locally as a part ofR×R₊(see the proof of Lemma 3.2 below), then we use a reflection argument in order to apply methods in [3] (see also [28]). We leave the detail for interested readers.

We define the blow up set foru_εas follows:

S^def=

x∈Ω; ∃εk→0 andxε_k→xsuch thatuε_k(xε_k)→ ∞ . Thanks to Lemma 3.1, we obtain

S=Σ^def=

x∈Ω; ∀δ >0, lim inf

ε→0

Bδ(x)∩Ω

ε²e^uεdy > α

.

Consequently, up to a subsequence, we have thatε²e^uεχΩdxtends to

xj∈Sηjδx_j inD(R²)and #(S) <∞, since Tε=O(1). By the result in [28], if we assume thatahas no critical point inΩ, thenS⊂∂Ω.

Lemma 3.2.For anyk∈N,u_ε→0inC_loc^k (Ω\S). Moreover, for anyx₀∈S, we have∂_τa(x₀)=0andη₀8π. Proof. First, by the definition of the Green’s functionG,

uε(x)= 1 8π a(x)

Ω

a(z)G(z, x)ε²e^uε(z)dz, ∀x∈Ω. (21)

Fix any subsetK⊂Ω such thatK∩S= ∅. Givenλ >0, sinceS⊂∂Ω, applying (15) and Remark 2.4, there exists δ >0 such that G(y, x) _L∞(K)λ, ifd(y,S)δ. Therefore, if we decomposeΩasΩ₁= {y∈Ω;d(y,S)δ}and

Ω₂ =

{y∈Ω;d(y,S) > δ}, for anyx∈K,

Ω₁

a(y)G(y, x)ε²e^uε(y)dya₂λ×

Ω₁

ε²e^uε(y)dya₂Tελ=O(λ).

On the other hand, the definition ofS implies thatuis uniformly bounded in any compact set away fromS. Hence for fixedδ >0, there existsC >0 such that

Ω2

a(y)G(y, x)ε²e^uε(y)dyCε²

Ω2

G(y, x) dy=O(ε²),

because G(·, x) _L1(Ω)is uniformly bounded forx∈Ω. By (21), we get limε→0 u_{ε L}∞(K)=0. The usual elliptic theory shows thenuε→0 inC^k(K)for allk∈N.

Letx₀∈S. Choosing a smooth open neighborhoodU ofx₀such thatU∩S= {x₀}, we can assume that there exists a conformal diffeomorphismΦfromB_δintoUsatisfyingΦ(0)=x₀,

Φ⁻¹(∂Ω∩U )= [−δ, δ] × {0} and Φ⁻¹(Ω∩U )=(R×R+)∩B_δ^def=Ω₀. It is not difficult to check thatvε=uε◦Φis a solution of

−_bv_ε=ε²|∇Φ|²e^vε inΩ₀, withb=a◦Φ (22) andv_ε=0 onΓ₀=∂Ω₀∩(R× {0}). Takingb(x)∂₁v_εas a test function, sinceν₁=0 andv_ε=0 onΓ₀, we get

Ω₀

∂b(x)

∂x₁

|∇v_ε|² 2 dx=

Ω₀

ε²∂(b(x)|∇Φ|²)

∂x₁ (e^vε−1) dx−ε²

Γ

b(x)|∇Φ|²(e^vε−1)ν1dσ

+

Γ

b(x)

2 |∇v_ε|²ν₁dσ−

Γ

b(x)∂v_ε

∂x1

∂v_ε

∂ν dσ,

whereΓ =∂Ω₀\Γ₀. Thus all the right-hand side terms are uniformly bounded. If∂_τa(x₀)=0, by takingδ small enough, we can assume without loss of generality that∂1b(x) >0 inΩ0, because|∂x₁b(0)| = |∇Φ(0)| × |∂τa(x0)| =0.

Therefore

Ω0

|∇v_ε|²dx=O(1),

this deduces thatu_εis bounded inH¹(U∩Ω). The Moser–Trudinger inequality shows then the boundedness ofe^uε inL^p(U∩Ω)for allp1, which yields uε L^∞(U∩Ω)=O(1)by the equation, hence contradicts withx0∈S=Σ. Furthermore, letx_ε∈Urealize max_U∩Ωu_ε(x). So limε→0x_ε=x₀. Letd_ε=d(x_ε, ∂Ω)and−2 logλε=u_ε(x_ε)+ 2 logε. We claim

εlim→0

d_ε

λε = ∞. (23)

First, we have limε→0λ_ε =0, since otherwise ε²e^uε is uniformly bounded in U∩Ω and no blow-up will occur.

Suppose now (23) is not true, up to a subsequence, we may assume thatd_ε/λ_εC. Define w_ε(y)=u_ε(x_ε+λ_εy)+2 logε+2 logλ_ε inΩ_ε=

y∈R²;x_ε+λ_εy∈U∩Ω . Clearlywε(y)0 inΩε,

−_a(xε+λεy)w_ε=e^wε inΩ_ε and

Ωε

e^wε(y)dy=ε²

U∩Ω

e^uε(x)dx=O(1).

Takingx=x_εin (21) and applying Lemma 2.1 (asd(x_ε, ∂Ω)→0), we obtain

0= 1

8π a(xε)

Ω

a(z)G(z, x_ε)ε²e^uε(z)dz−u_ε(x_ε)

= 1

2π a(xε)

Ω

log|x_ε^∗−z|

|x_ε−z|+O(1)

a(z)ε²e^uε(z)dz−uε(xε)

= 1

2π a(xε)

U∩Ω

a(z)ε²e^uε(z)log|x_ε^∗−z|

|x_ε−z|dz−u_ε(x_ε)+O(1). (24)

Here, we have used the fact that |x^∗_ε−z|/|x_ε−z| →1 uniformly in Ω\U, as x_ε →x₀. Let z=x_ε+λ_εy and y^∗_ε=(x_ε^∗−x_ε)/λ_ε, the equality (24) leads to

0= 1

2π a(xε)

Ωε

a(x_ε+λ_εy)e^wε(y)log|y^∗_ε−y|

|y| dy−u_ε(x_ε)+O(1). (25)

Notice that|y_ε^∗|2C and e^wε _L1∩L^∞(Ωε)=O(1). By decomposing the last integral over domains{|y|C} ∩Ω_ε and{|y|C} ∩Ω_ε, we obtain clearly

Ω_ε

a(x_ε+λ_εy)e^wε(y)log|y_ε^∗−y|

|y| dya₂×

Ω_ε

e^wε(y)log

1+2C

|y|

dy=O(1).

But limε→0uε(xε)= ∞, we reach a contradiction with (25), so the claim (23) holds.

Thus,Ω_εtends to the whole spaceR²and standard blow-up analysis implies that up to a subsequence,w_εconverges towinC_loc² (R²)where

−w=e^w inR² and

R²

e^wdy <∞.

It is well known from [8] that

R²

e^wdy=8π.

Fatou’s lemma yields thenη₀8π. 2 More precisely, we will quantifyη₀.

Proposition 3.3.For anyx₀∈S, we haveη₀∈8πN^∗.

Proof. Let x₀∈S, under conformal transformation, we can assume that x₀=0 and there existsδ >0 such that B_δ∩S= {0},

Ω₀=B_δ(0)∩Ω=B_δ∩(R×R^∗₊) and B_δ(0)∩∂Ω= [−δ, δ] × {0} =Γ₀. It suffices to prove that (recall thatv_ε=u_ε◦Φ verifies (22)),

εlim→0

Ω₀

ε²|∇Φ|²e^vεdx∈8πN^∗. (26)

Defineζεby

−ζ_ε= ∇logb· ∇v_ε inΩ0, ζ_ε=v_ε on∂Ω0. Theng_ε=v_ε−ζ_εsatisfies

−g_ε=ε²V_εe^gε inΩ₀, g_ε=0 on∂Ω₀,

whereV_ε= |∇Φ|²e^ζε. Since v_{ε C(∂Ω0)}→0 andv_εconverges to 0 weakly inW^1,p(Ω₀)for 1< p <2 by Lemma 3.2, soζ_ε is a family of continuous function satisfying ζ_{ε C(Ω}

0)→0, thusV_ε is a family of continuous functions which converges uniformly to the positive functionV = |∇Φ|².

We will prove a Li–Shafrir type result as in [15]. Indeed, we can get a first bubble by considering max_Ω

0g_ε= g_ε(x_ε)as in the proof of Lemma 3.2. We claim:

If max

Ω₀

gε(x)+2 logε+2 log|x−xε|

=O(1),thenη0=lim

ε→0

Ω₀

ε²Vεe^gεdx=8π. (27) Recall thatdε=d(xε, ∂Ω), let

w_ε(y)=v_ε(x_ε+d_εy)+2 logε+2 logd_ε

be defined inB₁. Then−w_ε=V_ε(x_ε+d_εy)e^wε. Since all the assumptions of Proposition 2 in [15] are verified by w_ε, we conclude then

εlim→0

B_dε(x_ε)

ε²Vεe^gεdx=lim

ε→0

B₁

Vε(xε+dεy)e^wεdy=8π. (28)

LetD_ε=Ω₀\B_dε(x_ε), we will prove g_{ε L}∞(Dε)=O(1). Unfortunately we do not have the sup+inf inequality as in [15], so we need some new arguments. In fact, we will use the potential analysis. Write

g_ε(x)=

Ω0

G₀(y, x)ε²V_ε(y)e^gε(y)dy,

where G₀ is the standard Green’s function associate to − over the domain Ω₀. Let x_ε =(x_ε¹, x_ε²), thanks to Lemma 3.2, we can assume that x_ε¹=0 by translation. By Lemma 2.1, we can also fixδ₀∈(0, δ) small enough such that for any|x|< δ₀,

2π G0(y, x)+log|x−y| −log|x^∗−y|C, ∀y∈Ω₀.

Letx∈B_δ0∩D_ε. If|x|/4|x−y|3|x|, as|x−x^∗|2|x|,|y−x^∗||y−x| + |x^∗−x|5|x|and 2π G0(y, x) log 20+O(1); if|x−y|3|x|, then|y||x−y| − |x|2|x|hence|x−y||y| − |x||y|/2. Now|y−x^∗|

|y|+|x|3|y|/2, we get 2π G0(y, x)log 3+O(1). SoG0(·, x)is uniformly bounded in the domainΩ0\B_|x|/4(x).

It remains to consider|y−x||x|/4. In this case, since|x||x−x_ε| +d_ε andx∈D_ε,

|y−x_ε||x−x_ε| − |y−x||x−x_ε| −|x|

4 3|x−x_ε| −d_ε

4 d_ε

2,

so|y||y−x_ε|+|x_ε|3|y−x_ε|. By the hypothesis in (27), we getg(y)+2 logε+2 log|y|C, if|y−x||x|/4, y∈Ω0andx∈Dε. Hence

I=

B_|x|/4(x)∩Ω0

G0(y, x)ε²Vεe^gε(y)dy

B_|x|/4(x)

log|y−x^∗|

|y−x|

+O(1)

ε²× C ε²|y|²dy.

Using polar coordinatesy=x+ |x|re^iθ, as|y−x^∗|9|x|/4 and|y|3|x|/4 inB_|x|/4(x),

IC 1/4 0

log

9 4r

+1

r dr <∞.

Finally, forx∈B_δ0∩D_ε, gε(x)=

Ω₀

G0(y, x)ε²Vεe^gεdy

=

Ω0\B_|x|/4(x)

G0(y, x)ε²Vεe^gεdy+

Ω0∩B_|x|/4(x)

G0(y, x)ε²Vεe^gεdy

O(1)×

Ω0\B_|x|/4(x)

ε²Vεe^gεdy+I

=O(1).

Applying Lemma 3.2, we conclude then g_{ε L}∞(D_ε)=O(1), which implies ε²V_εe^gε _L1(Dε)=O(ε²). Combining with (28), the claim (27) is proved.

If now maxΩ0

g_ε(x)+2 logε+2 log|x−x_ε|

→ ∞, (29)

we prove similar results as Lemmas 4 and 5 in [15] by induction, that is, up to a subsequence, we can get a family {xε,k}1kmsuch that

maxΩ0

gε(x)

2 +logε+ min

1kmlog|x−x_ε,k|

=O(1) and η₀=8π m. (30)

The main idea is to compare the distance between the different bubbles and their distances to the boundary. By suitable gauge transformation, we can either transform some of them into interior bubbles and then use the result in [15]; or

split them into boundary bubbles which concentrate in different places of∂Ω and then use the induction hypothesis.

We show here just the situation withm=2 and leave the complete proof for interested readers.

Assume that (29) holds. Then by considering the point y_ε which realizes the maximum of g_ε(x)+2 logε+ 2 log|x −x_ε| over Ω₀, we get a second bubble by repeating the argument in [15] and the proof of Lemma 3.2.

Suppose that the first estimate in (30) holds withx_ε,1=x_εandx_ε,2=y_ε. Define d˜_ε=min

|x_ε−y_ε|, d(x_ε, ∂Ω), d(y_ε, ∂Ω) .

Up to a subsequence and a permutation of index, we have the following possibilities:

(i) d˜_ε=d(x_ε, ∂Ω)and|y_ε−x_ε|C₀d˜_ε. Under translation, we can assumed˜_ε= |x_ε|. TakeM_ε=(C₀+1)d˜_εand considerw_ε(x)=g_ε(M_εx)+2 logε+2 logM_εinB₄∩(R×R₊). Up to a subsequence, we have then two interior bubbles where we may use the result in [15] to claim that

εlim→0

B_4Mε∩Ω0

ε²V_εe^gεdx=16π. (31)

(ii) d˜ε=d(xε, ∂Ω)and|yε−xε|/d˜ε→ ∞. So|yε|/|xε| tends to infinity. TakeMε= |yε| and consider wε(x)= gε(Mεx)+2 logε+2 logMεinB4∩(R×R₊). The two bubbles are now split away, either we have one interior bubble and one boundary bubble, or we have two separated boundary bubbles. So we turn back to the simple boundary bubble case and again (31) holds.

(iii) d˜ε= |yε−xε|, it suffices to take Mε = |xε| andwε as above, we transform then the bubbles to two interior bubbles as in case (i), so the result in [15] yields also (31).

Meanwhile, for all three cases, since|x−xε||x−yε|/2 inΩ0\B4M_ε (for correspondingMε), the first estimate in (30) impliesgε(x)+2 logε+2 log|x−yε| =O(1)inΩ0\B4M_ε. We can show always gε L^∞(Ω₀\B_4Mε)=O(1)by using the Green’s function as for (27). Now it is easy to conclude thatη0=16π. 2

Combining with the result in [28], we get

Proposition 3.4.Letu_εbe a family of solutions to problem(4)satisfying Tε=ε²

Ω

e^uεdx=O(1) and lim

ε→0max

Ω

u_ε= ∞.

Then up to a subsequence, u_ε tends tou^∗ inD(Ω), where u^∗ is the solution given by(8) with a finite set S= {x_i} ⊂Ω, composed by critical points ofa. On the other hand, there exists a finite setS= {z_j} ⊂∂Ω such that in D(R²),

ε²e^uεχ_Ωdx→

x_i∈S

8π miδ_xi+

z_j∈S

8π λjδ_zj. Moreoverm_i, λ_j∈N^∗,∂_τa(z_j)=0for anyz_j∈Sand

u_ε→u^∗ inC_loc^k

Ω\(S∪S), ∀k∈N.

Remark 3.5.As we have mentioned earlier, whenais constant in a neighborhood of∂Ω,S= ∅.

Proof of Theorem 1.5. If it is not true, we havex¯∈S which is a nondegenerate local maximum point ofa. As in the proof of Lemma 3.2, we will take a conformal transformation to change a small neighborhood ofx¯ inΩ into Ω₀=B_δ∩(R×R^∗₊)witha replaced byb=a◦Φ andu_ε replaced byv_ε=u_ε◦Φ. Moreover, 0=Φ(x)¯ becomes a nondegenerate local maximum point ofb. Choosingδ >0 small enough, we may assume thatΩ₀∩S= {0}and x· ∇b(x)0 inΩ₀. Using the Pohozaev identity (see [24]) obtained by multiplyingb(x)x· ∇v_εto Eq. (22), we get (Γ =∂Ω₀\(R× {0}))

Ω₀

div

b(x)|∇Φ|²x

ε²e^vεdx=

Γ

ε²b(x)|∇Φ|²(x·ν)e^vεdσ−1 2

Γ

b(x)(x·ν)|∇v_ε|²dσ +

Γ

b(x)∂vε

∂ν(x· ∇v_ε) dx+1 2

Ω0

(x· ∇b)|∇v_ε|²dx,

sincex·ν=x· ∇v_ε=0 over[−δ, δ] × {0}. Thanks to Lemma 3.2, we see that the first three terms in the right-hand side tend to zero asε→0 while the last term is nonpositive. However, the left-hand side tends to 16a(x)π m¯ with m∈N^∗, this is just impossible. 2

Remark 3.6.The nondegeneracy condition can be erased if we know thatx· ∇b(x)0 near 0 inΩ0. It is worth to mention that Theorem 1.5 cannot be proved by the usual moving plane method, however we state here a special case which is a direct consequence of this method.

Proposition 3.7.Assume thata(x)≡a(r)forx=(r, s)∈Ω anda is nondecreasing with respect tor. Assume also Tε=O(1). ThenS∩ {r=r_max}is empty, wherer_max=max_x∈Ωr.

Proof. In this case, we see that Eq. (4) is in the formu+c1(x)∂1u+ε²e^u=0 withc10 inΩ, thus the moving plane argument (see Theorem 2.1 in [12]) implies that for anyx₀=(r_max, s)∈∂Ω, there exists a neighborhoodUof x₀, independent ofε, such thatu_ε is decreasing with respect tor inU. Hence no blow up can occur atx₀because Tε=O(1). 2

4. Existence of boundary bubbling solution

From now on, we will construct a family of blow up solutions stated in Theorem 1.4. We should define some approximate solutions, and choose an appropriate configuration space for the parameters. Then we need to understand the behavior of linearized operator around these approximate solutions, to get suitable functional settings and the inverse of these linearized operators. Finally, we solve the nonlinear problem and conclude by variational techniques.

This procedure has been used successfully in constructing interior bubbles for Eq. (2) in [9,10] and for Eq. (4) in [26].

Here we construct boundary bubbles. Our difficulty is to estimate precisely the distance between the bubble and the boundary.

4.1. Approximate solution

Givenξ∈Ω andμ >0, we define u(x)=log 8μ²

(ε²μ²+ |x−ξ|²)².

The configuration space forξ is chosen as the following Λ^def=

ξ∈B_δ(x)∩Ω; C1

|logε| (.ξ, ∂Ω) C2

|logε|

(32) whereC1, C2>0 will be determined later on andμis chosen as

log(8μ²)=H_R(ξ )=H (ξ, ξ ). (33)

Using the choice ofΛand the estimate (18), there existsC >0 such that forε >0 small, 1

C|logε|²μ C

|logε|². (34)

We get alsou(x)=log(8μ²)−4 log|x−ξ| +O(ε²μ)on∂Ω. The ansatz is thenU (x)=u(x)+H^ε(x)whereH^εis a correction term defined as the solution of

_aH^ε+ ∇loga∇u=0 inΩ, H^ε= −u on∂Ω. (35)