Existence of self-dual non-topological solutions in the Chern–Simons Higgs model

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www.elsevier.com/locate/anihpc

Existence of self-dual non-topological solutions in the Chern–Simons Higgs model

Kwangseok Choe

^a

, Namkwon Kim

^b,^∗

, Chang-Shou Lin

^b,c

aDepartment of Mathematics, Inha University, Incheon, 402-751, Republic of Korea bDepartment of Mathematics, Chosun University, Kwangju, 501-759, Republic of Korea

cDepartment of Mathematics, Taida Institute for Mathematical Sciences (TIMS), National Taiwan University, Taiwan Received 18 February 2011; received in revised form 24 June 2011; accepted 28 June 2011

Available online 12 July 2011

Abstract

In this paper we investigate the existence of non-topological solutions of the Chern–Simons Higgs model inR². A long standing problem for this equation is: GivenNvortex points andβ >8π(N+1), does there exist a non-topological solution inR²such that the total magnetic flux is equal toβ/2? In this paper, we prove the existence of such a solution ifβ /∈ {8π N_k₋^k₁|k=2, . . . , N}. We apply the bubbling analysis and the Leray–Schauder degree theory to solve this problem.

Résumé

L’objectif de cet article est de prouver l’existence de solutions non-topologiques du modèle de Chern–Simons Higgs dansR². Un problème de longue date existe pour cette équation : SoitN points vortex etβ >8π(N+1), existe-t-il une solution non- topologique dansR²telle que le flux magnétique total est égal àβ/2 ? Dans cet article, nous prouvons l’existence d’une solution pourβ /∈ {8π N_k₋^k₁|k=2, . . . , N}. Nous appliquons l’analyse par bulles et la theorie de Leray–Schauder pour résoudre ce problème.

Keywords:Semi-linear PDE; Non-topological vortices; Chern–Simons Higgs model

1. Introduction

In the paper, we want to show the existence of non-topological multi-vortex solutions to the (2+1)-dimensional relativistic Chern–Simons gauge field theory. The Chern–Simons gauge field theory is minimal self-dual model con- taining the Chern–Simons term and was proposed independently by Hong et al. [11] and Jackiw and Weinberg [12]

to study the anyonic superconductivity. The Chern–Simons–Higgs Lagrangian density is given by L=κ

4^μνρF_μνA_ρ+D_μφD^μφ− 1 κ²|φ|²

1− |φ|²2

, (1.1)

* Corresponding author. Tel.: +82 62 230 6608; fax: +82 62 234 4326.

E-mail addresses:[email protected] (K. Choe), [email protected] (N. Kim), [email protected] (C.-S. Lin).

doi:10.1016/j.anihpc.2011.06.003

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whereA_μ(μ=0,1,2)is the gauge field onR³,F_μν=_∂x^∂μA_ν−_∂x^∂νA_μis the curvature tensor,φis the Higgs field onR³,D_μ=_∂x^∂μ −iA_μ(i=√

−1)is the gauge covariant derivative associated withA_μ,_μνρis the skew symmetric tensor with012=1 and the constantκ is the coupling constant. When the energy for a pair (φ, A)is saturated, in [11] and [12], the authors independently derived the following Bogomol’nyi type equations.

(D₁+iD₂)φ=0, and (1.2)

F12+ 2 κ²|φ|²

|φ|²−1

=0. (1.3)

Following Jaffe and Taubes [13], we can reduce the self-dual system (1.2) and (1.3) to a single elliptic equation of second order as follows. Letp₁, . . . , p_N be any set of points inR². Introduce a real valued functionuandθby

φ=e¹²^(u⁺^{iθ )} and θ=2 N j=1

arg(z−p_j), z=x₁+ix₂∈C¹. Thenusatisfies

u+ 4 κ²e^u

1−e^u

=4π N j=1

δ(z−pj) inR², (1.4)

whereδ(z−p_j)is the Dirac measure with the total mass atp_j.

For the details of the derivation of Eqs. (1.2)–(1.4) and recent developments of related subjects, we refer the readers to Hong et al. [11], Jackiw and Weinberg [12], Dunne [10], Lee et al. [15,16], Caffarelli and Yang [2], Choe [7,8], Choe and Kim [9], Lin and Yan [18], Spruck and Yang [21,22], Tarantello [23,24], Yang [25], Wang [26] and references therein. Eq. (1.4) has recently attracted a lot of attention because it is closely related to the mean field equation of Liouville type, see Nolasco and Tarantello [20], Chen and Lin [5] and Lin and Wang [17].

A solution u of Eq. (1.4) is called topological if u(x)→0 as |x| → +∞, and is called non-topological if u(x)→ −∞as |x| → ∞. For a given set{p1, . . . , pN}of vortex points, the existence of topological solution had been proved by Wang [26] long time ago. However, the existence problem of non-topological solutions is much more subtle. When p1= · · · =pN, Chen et al. [6] proved that for a positive number β, there exists radially symmetric solutionuof (1.4) such that

4 κ²

R²

e^u 1−e^u

dx=β holds if and only ifβ >8π(1+N ).

Naturally, we will ask the question: Given anyβ >8π(1+N ), does Eq. (1.4) possess a non-topological solution usuch that

4 κ²

R²

e^u 1−e^u

dx=β?

Note that by (1.3), the quantityβrepresents twice of the total magnetic flux:

R²

F₁₂dx= 2 κ²

R²

e^u 1−e^u

dx=β

2. (1.5)

It was Chae and Imanuvilov [3] who obtained the first existence result of non-topological solutions. They solved the problem by viewing Eq. (1.4) as a perturbation of the classical Liouville equation. Consequently, they could find solutions such that the order parameter|φ|is very small, andβis very close to 8π(N+1). On the other hand, Chan et al. [4] could obtain non-topological solutions withβ greater than 16π N. The method of Chan et al. is to construct solutions which bubble at each vortex pointp_j, therefore, in their theory the configuration of{p₁, . . . , p_N}must have a symmetry. Those are only two existence results for multi-vortex non-topological solutions. At this moment, the answer toward understanding the structure of non-topological solutions is far from complete. In the paper, we want to answer the long standing open problem affirmatively.

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Theorem 1.1. Let p₁, . . . , p_N ∈R² be given. For any number β >8π(N +1) satisfying β /∈ {8π N_k₋^k₁ |k = 2,3, . . . , N}, there exists a solutionuof Eq.(1.4)satisfying

4 κ²

R²

e^u 1−e^u

dx=β.

We sketch our idea to prove Theorem 1.1. In Section 4, we consider the deformation of Eq. (1.4):

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

u+ 4 κ²e^u

1−e^u

=4π N j=1

δεp_j inR², 4

κ²

R²

e^u 1−e^u

dx=β

(1.6)

whereε∈ [0,1]. Forε=1, (1.6) is the same equation as (1.4), andε=0, (1.6) is reduced to

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

u+ 4 κ²e^u

1−e^u

=4π N δ0, 4

κ²

R²

e^u 1−e^u

dx=β (1.7)

where 0 is the origin inR². The major work in this paper is to establish the uniform bound of any solutionu_ε(x)of (1.6). This apriori estimate is surprising because in most nonlinear problems, the collapsing of vortices would cause the bubbling phenomenon.

After establishing the apriori bounds, we will apply the classical Leray–Schauder degree theory to solve Eq. (1.4).

Note that for any radial solutionu(z)for (1.7),z=x₁+ix₂,u_θ(z)=u(e^iθz)is also a solution, and the orbit{u_θ(z)}, θ∈ [0,2π], isS¹, whose Euler characteristic vanishes. Thus the contribution of each orbit to the topological degree is 0. Hence due to a result of Wang [27], we conclude that the computation of the degree for Eq. (1.7) is reduced to those radially symmetric solution of (1.7). By the result for radial solution of Eq. (1.7) in [4], the topological degree for Eq. (1.7), thus for (1.4) also, is equal to−1. Then Theorem 1.1 follows immediately.

2. Preliminaries

Without loss of generality, we may assume thatκ²=4 in (1.4) and (1.6) in the sequel. Now supposeuis a solution of (1.6). Then it is easy to see that

u(x)= −2αln|x| +O(1) as|x| → +∞, (2.1) where

α= β

4π −N. (2.2)

Applying the maximum principle to Eq. (1.6), we haveu(x) <0,∀x∈R². Write

u(x)=v(x)+fε(x), (2.3)

wheref_ε(x)=N

j=1ln|x−εp_j|². Thenv(x)satisfies v+e^v⁺^f^ε

1−e^v⁺^f^ε

=0. (2.4)

Lemma 2.1.Letube a solution of (1.6). Thenusatisfies

R²

e^u 2−e^u

dx=4π

α²−N²

−4π N j=1

εpj· ∇v(εpj), (2.5)

wherev≡u−f_ε.

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Proof. In terms ofv, (2.4) becomes

−v= −e^u e^u−1

.

Multiplying the above byx· ∇(v+f_ε)and integrating overΩ=B_R, we have

∂Ω

1

2(x·ν)|∇v|²−(x· ∇v)(ν· ∇v)

dσ−

Ω

(x· ∇f_ε)v dx

=

∂Ω

(x·ν)

e^u−1 2e^2u

dσ−

Ω

2e^u−e^2u dx.

Hereν=x/|x|. Sincev(x)= −2(α+N )ln|x| +O(1)near∞, LHS= −4π(α+N )²−

Ω

N j=1

2x· ∇ln|x−εp_j|v dx+o(1)

= −4π(α+N )²+8π N (α+N )− N j=1

2

Ω

εpj· ∇ln|x−εpj|v dx+o(1)

asR→ ∞. By the repeated use of the integration by parts and the fact that ln|x−εp_j|is the Green function,

Ω

εp_j· ∇ln|x−εp_j|v dx

=lim

r→0

∂Ω

−

∂B_r(εp_j)

εp_j· (x−εpj)

|x−εp_j|²ν· ∇v−

∂Ω

−

∂B_r(εp_j)

ν· ∇

εp_j· (x−εpj)

|x−εp_j|²

v

=lim

r→0

∂B_r(εp_j)

−εpj· (x−εp_j)

|x−εp_j|²ν· ∇v+ν· ∇

εpj· (x−εp_j)

|x−εp_j|²

v

+o(1)

= −2π εpj· ∇v(εpj)+o(1).

Then we have the desired estimates. 2

The following corollary is an immediate consequence of Lemma 2.1.

Corollary 2.2.Letube a solution of(1.6). Thenusatisfies

R²

e^udx=4π(α+N )(α−N−1)−4π N j=1

εpj· ∇(u−fε)(εpj),

R²

e^2udx=4π(α+N )(α−N−2)−4π N j=1

εp_j· ∇(u−f_ε)(εp_j).

We recall some useful results on radially symmetric solutions of (1.7) withκ²=4. IfN=0 then every solution of (1.7) is called a 0-vortex solution, and after a translation this solution must be radially symmetric. Also, u is a decreasing function of |x|[21]. More generally, when N0, there exists a unique radially symmetric solution of (1.7) if and only if β >8π(N+1)(see [4]). Actually, there exists a 1-parameter family of radially symmetric solutionu(r;s)of (1.7) which satisfies

−u(r;s)−1

ru(r;s)=e^u(r^;^s)

1−e^u(r^;^s)

, r >0, u(r;s)=2Nlnr+s+o(1) nearr=0.

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There exists a numbers_∗∈Rsuch thatu(r;s)is a non-topological solution if and only ifs < s_∗, and β(s)≡

R²

e^u(r^;^s)

1−e^u(r^;^s)

dx (2.6)

is a well-defined continuously differentiable function on (−∞, s_∗). It is proved in [4] that β(·)is monotonically increasing on(−∞, s_∗). Moreover,β(s)tends to∞asss_∗, andβ(s)→8π(N+1)ass→ −∞. It is obvious that s_∗=0 ifN=0.

Lemma 2.3.Letube a non-topological solution of (1.6)withβ fixed. There exists a constantν=ν(max|p_j|) >0 such thatu <−νinR².

Proof. We know thatu <0. So,u−f_ε_n<−f_ε_nC_r on∂B_r= {x| |x| =r}for anyr >maxj|p_j| +1.

Suppose now that there exist a sequence of solutionsun:=uε_nand a sequence of points{xn}such thatun(xn)→0.

Sinceωn:=un(x)−fε_n(x)Cr on∂Br,r >maxj|pj| +1, andωnsatisfiesωn+10 inBr, by the maximum principle, we haveωn(x)Cr inBr. Thus if|xn−εnpj| →0 for somej ∈ {1,2, . . . , N}, thenun(xn)=ωn(xn)+ f_ε_n(x_n)→ −∞, which yields a contradiction. Hence we must have

lim inf

n→∞ dist

x_n,{ε_np_j|j =1, . . . , N}

> C >0.

We claim that|x_n| → ∞. Indeed, if there exists a bounded subsequence of{x_n}, then, by the locally uniform Hölder estimate ofu_noutsidep_j’s,u_nmust converge locally uniformly to a functionu^∗0 up to subsequences. But thenu^∗ must be a solution of (1.6) having zero as a maximum, which contradicts the strong maximum principle. This proves our claim.

Now, choose a numbers₀<0 such thatβ₀(s₀) > β=4π(α+N ), whereβ₀(·)isβ(·)in (2.6) for the 0-vortex solution. Since|x_n| → ∞andu_n(x_n)→0, there exists a sequence{y_n}such that|y_n||x_n|andu_n(y_n)=s₀. This holds true becauseu_n(x)→ −∞as|x| → ∞.

Letu_n(x)=u_n(x+y_n)for|x||y_n|/2.u_nsatisfies

−u_n=e^uⁿ

1−e^uⁿ

− N j=1

4π δεnpj−yn.

Then, along a subsequence,u_nconverges inC²_loc(R²)tou0 withu(0)=s₀<0, satisfyingu+e^u(1−e^u)=0 inR², and

R²

e^u 1−e^u

dxlim inf

n→∞

R²

e^uⁿ 1−e^uⁿ

dx=β.

Sinceuis a 0-vortex non-topological solution of (1.7), it is radially symmetric [21]. SinceM_∗:=max_R²uu(0)=s0, β < β₀(s₀)β₀(M_∗)=

R²

e^u 1−e^u

dxβ,

which leads to a contradiction and the lemma is proved. 2

Now, letϕ(r;s)=_ds^du(r;s)fors < s_∗. We borrow the following lemma forϕ(r;s)from [4].

Lemma 2.4.ϕ(·;s)has only one zero inR+=(0,∞)andlimr→∞ϕ(r;s)

lnr = −c0for some constantc0=c0(s) >0.

Lemma 2.4 will be used in the calculation of degree in the final section.

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3. Boundedness of solutions

In this section, we show that the solutions of the problem (1.6) are uniformly bounded by the blow-up analysis if β >8π(N+1)andβ=8π N_k₋^k₁for any integerk=2, . . . , N. In what follows we always assume thatN1. Recall that

f_ε(x)= N j=1

ln|x−εp_j|² with 0ε1.

Theorem 3.1.Letp₁, . . . , p_N∈R²,β >8π(N+1)andβ /∈ {8π N_k₋^k₁|k=2, . . . , N}, andube a solution of (1.6).

Then, for each compact subsetK, there exists a constantC=C(β, N,maxj|p_j|, K)independent ofεsuch that

u−f_εL^∞(K)C. (3.1)

We will show Theorem 3.1 by contradiction. Sinceu(x) <0 inR², (1.6) can be rewritten as u(x)+d(x)u(x)=0

inR²\{p₁, . . . , p_N}, where|d(x)| = |êû⁽¹_u(x)⁻êû⁾|C. Thus for any compact setK⊂R²\{p₁, . . . , p_N}, by the Harnack inequality, there exists a constantC(K) >0 such that

sup

K

u(x)Cinf

K

u(x).

Therefore, if there exists a sequenceu_n:=u_ε_n of solutions which do not satisfy (3.1), thenu_n→ −∞uniformly on each compact subset. We will show that{u_n}exhibits a specific concentration near∞, which leads to a contradiction.

In what follows,u_nis assumed to satisfyu_n(x)= −2αln|x| +O(1)near∞, whereαis given in (2.2). We start with the following lemma.

Lemma 3.2.Letx_n be a maximum point ofu_n. Then,|x_n| → ∞asn→ ∞and u_n(x_n) >−C uniformly for some constantC >0.

Proof. Suppose that|xn|is bounded up to subsequences. Then,un(xn)→ −∞sinceun→ −∞locally. Now, we let s_n=e¹²^uⁿ^(xⁿ⁾, u˜_n(x)=u_n(x/s_n)−lns_n², v_n= ˜u_n−f_s_n_ε_n.

Thenu˜_nu˜_n(s_nx_n)=0 inR², and

−vn=e^vⁿ⁺^f^snεn

1−s_n²e^vⁿ⁺^f^snεn .

Clearly,vn(x)=(u˜n−fs_nε_n)(x)−fs_nε_n(x)Cfor|x|maxj|pj|+1. Also, since the right-hand side of the above equation is bounded, vnC onBR(0)forR=maxj|pj| +1. Thus, we havevnC inR² for some constantC.

Then it follows that

−2Nlns_n−f_ε_n(x_n)= −f_s_n_ε_n(s_nx_n)=v_n(s_nx_n)C.

Sinces_n→0, it follows thatf_ε_n(x_n)→ +∞asn→ ∞, which yields a contradiction. Therefore|x_n| → ∞.

Next, since u_n converge locally uniformly to −∞, Green’s representation formula for {u_n} implies that

∇(u_n−f_ε_n)→0 inC_loc¹ (R²). Then by (2.2), Corollary 2.2 and the above convergence, we have 4π(α+N )(α−N−2)+o(1)=

R²

e^2uⁿdxe^uⁿ^(xⁿ⁾

R²

e^uⁿdx

=e^uⁿ^(xⁿ⁾

4π(α+N )(α−N−1)+o(1) . Sinceα > N+2,u_n(x_n)is bounded from below uniformly. 2

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From now on, we let

wn(x)=un(x/rn)−lnr_n², rn=1/|xn| →0,

wherex_nis a maximum point ofu_nas before. Thenw_nsatisfies

−w_n=e^wⁿ

1−r_n²e^wⁿ

− N j=1

4π δrnεnpj inR². (3.2)

Note thatr_nε_np_j→0 asn→ +∞. By Lemma 2.3, the Brezis–Merle type alternatives for{w_n}in [9] are still valid in any open setDsatisfyingDR²\{0}. Sincew_n(r_nx_n)→ ∞by Lemma 3.2,{w_n}has a blow-up pointq on the unit circle, namely, there exists a sequence{y_q,n}such thaty_q,n→q andw_n(y_q,n)→ ∞. Thus,{w_n}satisfies the blow up case in the alternatives: along a subsequence, there exists a non-empty finite setSof nonzero points such that wn→ −∞uniformly on eachKR²\(S∪ {0})and

e^wⁿ

1−r_n²e^wⁿ

→

q∈S

2π Mqδ_q,

on anyDR²\{0}withS⊂Din the distribution sense.

For anyq∈S, letd be small, and M_q,n= 1

2π

B_d(q)

e^wⁿ

1−r_n²e^wⁿ dx.

ClearlyMq,n→Mq. By Pohozaev’s identity, we have

Bd(q)

e^wⁿdx=π M_q(M_q−2)+o(1),

Bd(q)

r_n²e^2wⁿdx=π M_q(M_q−4)+o(1). (3.3) Indeed, multiplying by(x−q)· ∇wn both sides of (3.2) and integrating over Bd(q), we can repeat the Pohozaev argument as in Lemma 2.1. ThereforeM_q4 for allq∈S. Then we have|S|(α+N )/2. Furthermore, we can prove that allM_q’s are the same in the following lemma.

Lemma 3.3.Let{w_n}andM_qas before. ThenM_p=M_qfor allp, q∈S. Moreovermax_|x−q|<dw_n(x)+lnr_n²>−C uniformly for any small constantd >0.

Proof. Choose a small constantd >0 such thatB_2d(q)∩(S∪{0})= {q}for anyq∈S. Due to Green’s representation formula forw_n, the local estimates for periodic blow-up solutions of the Chern–Simons Higgs equation in [7] are still valid. Actually, following the proof of existence of profiles for the mean field equation [1], we obtain

wn(x)−wn(qn)+M_q,n 2 ln

1+e^wⁿ^(qⁿ⁾|x−qn|²C (3.4)

in|x−q_n|d for some constantC. Herew_n(q_n)=max_|x−q|dw_n(x)→ ∞withq_n∈B_d(q). Clearlyq_n→q.

We claim that

w_n(q_n)= −lnr_n²+O(1) ifM_q>4.

Indeed, we note thatwn+lnr_n²0 inR². Then our claim is an immediate consequence of the following inequality.

Bmaxd(q)

r_n²e^wⁿ

B_d(q)

e^wⁿdx

B_d(q)

r_n²e^2wⁿdx=π Mq(Mq−4)+o(1).

We now prove thatM_qare all the same. For this purpose, we suppose thatM_p> M_q for somep, q∈S. Choose two pointsx∈∂B_d(p)andy∈∂B_d(q), respectively. It also follows from Green’s representation formula forw_nthat there exists a constantC such that|w_n(x)−w_n(y)|C. SinceM_p>4, we havew_n(p_n)= −lnr_n²+O(1), where p_nis a maximum point ofw_ninB_d(p). Sincew_n(q_n)−lnr_n², it follows from (3.4) that

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w_n(x)−w_n(y)=

1−M_p,n 2

w_n(p_n)−

1−M_q,n 2

w_n(q_n)+O(1)

1−M_p,n 2

−lnr_n²

−

1−M_q,n 2

−lnr_n² +C

1

2(M_q,n−M_p,n)

−lnr_n²

+C→ −∞, which yields a contradiction. ThereforeM_p=M_qfor allp, q∈S.

Finally, we showM_q>4 for allq∈S. Letx_nbe a maximum point ofu_ninR². Since the limit points ofx_n/|x_n| belong toS, it is enough to show

lim inf

n→∞

|x−xn|<R

e^uⁿ

1−e^uⁿ

dx >8π (3.5)

for some large enoughR >0. Actually, sinceun(xn)is bounded from below, it follows from Harnack’s inequality that {un}is bounded inL^∞(Br(xn))for anyr >0. Along a subsequence,un(x+xn)converges inC²_loc(R²)to a 0-vortex non-topological solution V of (1.7). Then (3.5) immediately follows from the fact that e^V(1−e^V)_L¹₍_R²₎>8π [4,6,21]. This finishes the proof. 2

In what follows, we letM denote the mass atq∈S instead of M_q. In the following two lemmas, we show that concentration may occur only for special values ofα. In particular, the following lemma implies that the origin is not a blow-up point for{w_n}.

Lemma 3.4.For each constant0< s <min_q_∈S|q|, we have

nlim→∞

|x|s

e^wⁿ

1−r_n²e^wⁿ

dx=0, (3.6)

andwn→ −∞uniformly onBs(0). Moreover,|S|2andM=4N/(|S| −1).

Proof. Lets <min_q_∈S|q|be a small positive number. We first show (3.6). To see this, we argue by contradiction and suppose that, along a subsequence,

M₀:= lim

n→∞

|x|s

1 2πe^wⁿ

1−r_n²e^wⁿ dx >0.

Letξ_n=w_n−f_r_n_ε_n. Note thatξ_n→ −∞uniformly on any compact subset ofR²\(S∪ {0}). By Eq. (3.2), Green’s representation formula forξ_nimplies that

ξ_n(x)−b_n→ −M₀ln|x| −

q∈S

Mln|x−q| inC_loc¹ R²\

S∪ {0}

for some real sequencebn→ −∞. The Pohozaev-type identity shows that

∂Ω

1

2(x·ν)|∇ξn|²−(x· ∇ξn)(ν· ∇ξn)

dσ

=

∂Ω

(x·ν)

e^wⁿ−r_n² 2e^2wⁿ

dσ−

Ω

2e^wⁿ−r_n²e^2wⁿ dx−

Ω

x· ∇f_r_n_ε_n

e^wⁿ−r_n²e^2wⁿ dx

whereΩis an open set with smooth boundary. To simplify the notations, we let

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I_n=

∂Ω

1

2(x·ν)|∇ξ_n|²−(x· ∇ξ_n)(ν· ∇ξ_n)

dσ,

Jn=

Ω

x· ∇fr_nε_n

e^wⁿ−r_n²e^2wⁿ dx.

If we takeΩ= {x| |x|s}, then we obtain that I_n= −π M₀²+o(1)

asn→ ∞. Moreover J_n=4π N M0+

Ω

N j=1

2rnε_np_j·(x−r_nε_np_j)

|x−r_nε_np_j|²

e^wⁿ−r_n²e^2wⁿ

dx+o(1)

=4π N M0+

|x|s/r_n

N j=1

2εnpj·(x−εnpj)

|x−ε_np_j|²

e^uⁿ−e^2uⁿ

dx+o(1)

=4π N M0+o(1)

asn→ ∞. Therefore we obtain that

−π M₀²+o(1)= −4π M0−

|x|s

r_n²e^2wⁿdx−4π N M0+o(1).

In particularM04+4N.

Next, we takeΩ=

q∈SBr(q)with small enoughr >0. Note that

∇ξ_n(x)→ ∇ξ(x)= −M(x−q)

|x−q|² + ∇H_q(x), ∇H_q(x)= −M₀x

|x|² −

p∈S\{q}

M(x−p)

|x−p|²

uniformly on any compact subset ofΩ\S. Denotingx≡q+rνon|x−q| =r, we have∇ξ= −^M_r ν+ ∇Hq on

|x−q| =r. Asn→ ∞, I_n→

q∈S

|x−q|=r

1

2(x·ν)|∇ξ|²−(x· ∇ξ )(ν· ∇ξ )

dσ

=

q∈S

|x−q|=r

−M² 2r +M

r q· ∇H_q(x)

dσ

= −π|S|M²−2π|S|MM₀−2π M²

q∈S

p∈S\{q}

q·(q−p)

|q−p|² +O(r)

= −π|S|M

|S|M+2M0

+O(r).

Moreover, it follows from (3.3) that

Ω

2e^wⁿ−r_n²e^2wⁿ dx=

Ω

r_n²e^2wⁿdx+

Ω

2

e^wⁿ−r_n²e^2wⁿ dx

=

q∈S

|x−q|r

r_n²e^2wⁿdx+4π|S|M+o(1)

=π|S|M(M−4)+4π|S|M+o(1).

Finally we obtain that

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J_n=4π N|S|M+ N j=1

Ω

2rnε_np_j·(x−r_nε_np_j)

|x−r_nε_np_j|²

e^wⁿ−r_n²e^2wⁿ

dx+o(1)

=4π N|S|M+o(1).

Then we conclude that 4N=2M0+M

|S| −1

8+8N+M

|S| −1

, (3.7)

which leads to a contradiction. Therefore (3.6) is proved. By letting M0=0 in (3.7), we conclude that 4N = M(|S| −1).

Next, we show thatwn→ −∞uniformly onBs(0). It follows from Lemma 3.2, (3.4) and Green’s representation formula forwnthat, for|x|s/rn,

u_n(x)=f_ε_n(x)+C_n+ 1 2π

R²

ln |y|

|x−y|e^uⁿ

1−e^uⁿ dy

=f_ε_n(x)+C_n+ 1 2π

|y|s/rn

ln |y|

|x−y|e^uⁿ 1−e^uⁿ

dy+O(1).

Standard argument ([1]) shows that

u_n(x)−f_ε_n(x)−C_n=o(1)lnrn+O(1) for|x|s/r_n.

Then it follows from (3.4) thatC_n=(M+2N+o(1))lnr_n+O(1), and consequently,w_n(x)=f_r_n_ε_n(x)+(M−2+ o(1))lnrn+O(1)for|x|s. This completes the proof of Lemma 3.4. 2

Lemma 3.5.For any constantR >maxq∈S|q|,w_n→ −∞uniformly for|x|R, and

nlim→∞

|x|R

e^wⁿ

1−r_n²e^wⁿ

dx=0. (3.8)

Moreover(|S| +1)M=4α.

Proof. Let ϕ_n(x)=w_n

x/|x|²

−2αln|x|.

Choose a constantR > (minq∈S|q|)⁻¹. Thenϕ_nsatisfies

−ϕn= |x|^2α⁻⁴e^ϕⁿ

1−r_n²|x|^2αe^ϕⁿ

for|x|< R, (3.9)

andϕ_n→ −∞uniformly on any closed subset ofR²\(S˜∪ {0}), where we setS˜= {q/|q|²|q∈S}. Eachϕ_nis of classC²in a neighborhood of the origin.

We now show that ϕ_n → −∞ uniformly on any compact subset of R²\ ˜S. Choose a number 0< s <

(maxq∈S|q|)⁻¹. We argue by contradiction and suppose that there exists a subsequence still denoted by{ϕ_n}such that sup_|_x_|_sϕ_n(x)is bounded below. Elliptic estimates show that sup_|_x_|_sϕ_n(x)→ ∞. Otherwise, along a subsequence, the right-hand side of (3.9) would be uniformly bounded onB_s(0). Sinceϕ_n(x)→ −∞uniformly for|x| =s, we would have sup_|_x_|_sϕ_n(x)→ −∞, which yields a contradiction.

We claim that lim inf

n→∞

|x|s

|x|^2α⁻⁴e^ϕⁿ

dx8π (3.10)

under the assumption that sup_|_x_|_sϕ_n(x)→ ∞. To see this, we choose a point y_n∈B_s(0) such that ϕ_n(y_n)= sup_|_x_|_sϕ_n(x). It is obvious that|y_n| →0. Choose a numbert_nsuch that

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ϕ_n(y_n)+(2α−2)lnt_n=0.

Along a subsequence, we have two cases:

Case 1. If|y_n|/t_nis bounded above, then we consider the function ϕ_n(x)=ϕ_n(t_nx)−ϕ(y_n) for|x|s/t_n.

Note thatϕ_n(x)ϕ_n(y_n/t_n)=0 for|x|s/t_n. Then elliptic estimates show that, along a subsequence,ϕ_nconverges inC_loc² (R²)to a solutionϕof

−ϕ= |x|^2α⁻⁴e^ϕ inR²,|x|^2α⁻⁴e^ϕ∈L¹ R²

. Consequently we find that

lim inf

n→∞

B_s(0)

1−r_n²|x|^2αe^ϕⁿ dx

R²

|x|^2α⁻⁴e^ϕdx=8π(α−1).

Case 2. If|y_n|/t_n→ ∞then we consider the function w_n(x)=w_n

x/|y_n|

−ln|y_n|². (3.11)

Thenw_nsatisfies

−w_n=e^wⁿ

1−r_n²|y_n|²e^wⁿ

− N j=1

4π δrn|yn|εnpj inR².

Sincew_n(y_n/|y_n|)=ϕ_n(y_n)+(2α−2)ln|y_n| → ∞, along a subsequence,{w_n}has a blow-up pointq_∗on the unit circle. Consequently we have

lim inf

n→∞

Bs(0)

(−ϕ_n) dxlim inf

n→∞

Br(q_∗)

e^wⁿ

1−r_n²|y_n|²e^wⁿ

dx8π for any small constantr >0. This proves (3.10).

Without loss of generality we may assume that

nlim→∞

|x|s

dx=2π M1, M₁4.

Then the local estimates for{w_n}show that

→2π M1δp=0+

q∈ ˜S

2π Mδq

in the distribution sense. Green’s representation formula forϕ_nimplies that ϕ_n(x)−c_n→ −M₁ln|x| −

p∈ ˜S

Mln|x−p|

inC¹_loc(R²\(S˜∪ {0}))for some real sequencec_n→ −∞. We repeat the calculations used in the proof of Lemma 3.4.

The Pohozaev-type identity implies that

∂Ω

(x· ∇ϕ_n)(ν· ∇ϕ_n)−1

2(x·ν)|∇ϕ_n|²

dσ+a_n

=(2α−2)

Ω

|x|^2α⁻⁴e^ϕⁿdx−(2α−1)

Ω

r_n²|x|^4α⁻⁴e^2ϕⁿdx (3.12)