Asymptotic analysis of solutions to a gauged <span class="mathjax-formula">$ \mathrm{O}(3)$</span> sigma model

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Ann. I. H. Poincaré – AN 32 (2015) 651–685

www.elsevier.com/locate/anihpc

Asymptotic analysis of solutions to a gauged O(3) sigma model

Daniele Bartolucci

, Youngae Lee

, Chang-Shou Lin

, Michiaki Onodera

aUniversity of Rome “Tor Vergata”, Department of Mathematics, Via della Ricerca Scientifica n. 1, 00133 Rome, Italy bCenter for Advanced Study in Theoretical Sciences, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei 106, Taiwan

cTaida Institute for Mathematical Sciences, Center for Advanced Study in Theoretical Sciences, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei 106, Taiwan

dInstitute of Mathematics for Industry, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan

Received 15 March 2013; received in revised form 9 January 2014; accepted 18 March 2014 Available online 4 April 2014

Abstract

We analyze an elliptic equation arising in the study of the gauged O(3)sigma model with the Chern–Simons term. In this paper, we study the asymptotic behavior of solutions and apply it to prove the uniqueness of stable solutions. However, one of the features of this nonlinear equation is the existence of stable nontopological solutions inR², which implies the possibility that a stable solution which blows up at a vortex point exists. To exclude this kind of blow up behavior is one of the main difficulties which we have to overcome.

©2014 Elsevier Masson SAS. All rights reserved.

Keywords:Gauged O(3)sigma models; Blow up analysis; Pohozaev type identity; Stable solutions

1. Introduction

The classical O(3)sigma model in 2+1 dimension originated to describe physical phenomena such as planar ferromagnet[3]. However, the solitons in this model are not suitable for particle models due to their scale invariance which makes particles have arbitrary size. This problem was overcome by Schroers in[23], where a U(1)gauge field was added and the dynamics was governed by the Maxwell term. After his work, there have been many studies on U(1)gauged O(3)sigma model, where the gauge field dynamics was governed by the Chern–Simons term[1,14,18, 19]or both of Maxwell and Chern–Simons terms[18].

In this paper, we consider another Chern–Simons gauged O(3)sigma model whose Lagrangian is defined by L=κ

4ε^μνρFμνAρ+1

2Dμφ·D^μφ− 1

2κ²(γ+n·φ)²|n×φ|².

* Corresponding author.

E-mail addresses:bartoluc@axp.mat.uniroma2.it(D. Bartolucci),youngaelee0531@gmail.com(Y. Lee),cslin@tims.ntu.edu.tw(C.-S. Lin), onodera@imi.kyushu-u.ac.jp(M. Onodera).

http://dx.doi.org/10.1016/j.anihpc.2014.03.001

0294-1449/©2014 Elsevier Masson SAS. All rights reserved.

The unknowns are the spin vectorφ=(φ₁, φ₂, φ₃):R^1,2→S²⊂R³and the gauge fieldA_ρ :R^1,2→Rwithρ= 0,1,2. The gauge covariant derivative is defined by

D_jφ=∂_jφ+A_j(n×φ), and the curvatureF_μν is given by

F_μν=∂_μA_ν−∂_νA_μ.

Moreover,n=(0,0,1)is the north pole ofS²andε^αβγ is the totally antisymmetric tensor withε⁰¹²=1. The constant κ >0 represents the strength of the Chern–Simons action, and the constant γ ∈ [−1,1] is a free parameter which determines the vacuum manifold of the potential. Since the Euler–Lagrangian equation is very complicated to study even for stationary solution, we restrict to consider energy minimizers only. The static energy from the Lagrangian is

E(φ, A)=

R²

e(φ, A) dx,

where the energy densitye(φ, A)is given by e(φ, A)=1

2

κ²F₁₂²

|n×φ|²+ |D1φ|²+ |D2φ|²+ 1

κ²(γ +n·φ)²|n×φ|²

. We see that

E(φ, A)=1 2

R²

κF12

|n×φ|±1

κ(γ +n·φ)|n×φ| 2

+ |D₁φ±φ×D₂φ|²

dx

±

R²

φ·(D₁φ×D₂φ)−F₁₂(γ+n×φ) dx.

Then the self-dual equations for solutions minimizing the static energy are given by D₁φ+φ×D₂φ=0,

F12+ 1

κ²(γ+n·φ)|n×φ|²=0. (1.1)

If we setu=ln[(1+φ₃)/(1−φ₃)]and prescribe

φ⁻¹(s)= {p_1,1, . . . , p_d1,1}, φ⁻¹(n)= {p_1,2, . . . , p_d2,2},

wheres=(0,0,−1)is the south pole ofS², then we can reduce the system(1.1)to the following equation:

u+ 1 ε²

e^u(1−e^u) (τ+e^u)³ =4π

d1

j=1

m_j,1δ_pj,1−4π

d2

j=1

m_j,2δ_pj,2 onR²,

whereτ =¹₁⁺₋^γ_γ ∈(0,∞),m_j,i∈N∪ {0}, andδ_p stands for the Dirac measure concentrated atp. For the details of derivation of the above equation from(1.1), we refer the readers to[8].

In this paper, we want to consider the above equation in a flat 2-dimensional torusΩ: u+ 1

ε²

e^u(1−e^u) (τ+e^u)³ =4π

d1

j=1

mj,1δp_j,1−4π

d2

j=1

mj,2δp_j,2 onΩ. (1.2)

This consideration is physically meaningful, due to the theory suggested by ’t Hooft in[25]. We also refer to[4,6,7,11]

for more developments of Eq.(1.2).

Before we go further, we shall make some remarks about our nonlinear termfτ(u)≡^e_(τ^u(1₊⁻_eu^e)^u3). Asε→0, heuristi- cally, solutionsu_εof(1.2)might tend to±∞. Ifu_ε→ −∞, then(1.2)tends to:

u+e^u=a sum of Dirac measure.

On the other hand, ifu→ +∞, then Eq.(1.2)tends to u−e^−u=a sum of Dirac measure,

in other words,(−u)satisfies the Liouville equation again. Thus one of the limiting equation is the Liouville equation, which shares the same property of the well-known Chern–Simons–Higgs (CSH) equation:

u+ 1 ε²e^u

1−e^u =4π d j=1

mjδp_j onΩ. (1.3)

The CSH model has been proposed more than twenty years ago in[16]and independently in[17]to describe vortices in high temperature superconductivity. Actually,(1.3)was derived from the Euler–Lagrange equations of the CSH model via a vortex ansatz, see[16,17,27,28]. We also refer to[9,10,20–22]for more developments.

In a recent paper[26], Tarantello proved the following theorem:

Theorem A.For given{p_j}andm_j∈N, there existsε₀≡ε₀(p_j, m_j) >0such that ifε∈(0, ε0), then there exists a unique topological solutionu_ε for(1.3), i.e. a unique solution which satisfiesu_ε→0a.e. inΩasε→0.

It is natural to ask whetherTheorem Aalso holds for Eq.(1.2). In[26], Tarantello proved that ifu_εis a topological solution of(1.3), thenu_ε is strictly stable solution. As a consequence of this fact, the uniqueness of the topological solutions was established. In this paper, we study the uniqueness ofstable solutionsinstead of topological solutions, because the definition of a topological solution depends on a sequence of solutions, not only the solution itself. Here uis called a stable solution of(1.2)if the linearized equation of(1.2)atuhas nonnegative eigenvalues.

Our main purpose is to prove the equivalence of stable solutions and topological solutions under certain assump- tions. To state our result, we need the following conditions:

(H1): N₁ =N₂whereN_i≡d_i j=1m_j,i;

(H2): eitherτ=1 or, ifN_i> N_k, thenm_j,i∈ [0,1]for all 1jd_i. Then we have the following theorem.

Theorem 1.1.Letuεbe a sequence of solutions of(1.2)withε >0.

(i) ifuε→0a.e. inΩ\

j,i{pj,i}asε→0, thenuεis a strictly stable solution for sufficiently smallε >0.

(ii) if (H1)–(H2) hold andu_εis a sequence of stable solutions, thenu_ε→0a.e. inΩ\ {p_j,i}asε→0.

Remark 1.2.A nontopological entire solution of the CSH equation(1.3)is always unstable (seeAppendix A). Hence for a sequence of stable solutionsu_ε of the CSH equation(1.3), we can prove that u_ε is a topological solution for smallε >0. The proof is simpler than (ii) ofTheorem 1.1.

As a consequence ofTheorem 1.1, we also have the following result about the uniqueness of stable solutions of(1.2).

Theorem 1.3. Let uε be a sequence of solutions of (1.2) withε >0. If (H1)–(H2) hold, then there exists ε0:=

ε0(pj,i, mj,i) >0such that there exists a unique stable solution of(1.2)for eachε∈(0, ε0).

We remark that the uniqueness of topological solutions of (1.2) always holds even without the assumptions (H1)–(H2). Indeed, this result and (i) ofTheorem 1.1can be proved by a suitable adaptation of the argument in[26].

Roughly speaking, this is due to the fact that the behavior of a topological solution is the same no matter whether it is a solution of(1.3)or of(1.2). See either Proposition 4.8 in[26]orLemma 5.1below.

However, there are dramatic differences between these two equations when stable solutions are considered. First of all, the asymptotic analysis is relatively easier for the CSH equation(1.3). By the maximum principle, any solution uof the CSH equation(1.3)is always negative, thuse^u(1−e^u)is always positive. On the contrary, a solutionu(x)of

Eq.(1.2)could tend to either+∞or−∞asxconverges to a vortex point in caseN₁ =0 andN₂ =0. This fact readily implies that the nonlinear termf_τ(u)must change sign inΩ and this is of course the cause of a lot of difficulties in the study of the asymptotic behavior ofu_ε asε→0.

Secondly, any nontopological entire solution of the CSH equation(1.3)is always unstable. This might not be true for Eq.(1.2). Indeed, it has been proved that any nontopological radially symmetric entire solution of(1.2)is unstable provided that eitherτ =1 orm_j,i∈ [0,1]for alli, j. Hence ifτ =1 andm_j,i>1 for somei, j, then there might exist nontopological stable entire solutions for(1.2). Of course, this fact might complicate our analysis, because stable solutions might be bubbling even at a vortex pointp_j,i, whereτ =1 andm_j,i>1. Our condition (H2) partly reflects this fact. However, (H2) still allows the possibility thatm_j,k>1 as far as the global conditionN_i> N_k is satisfied, since in this case one can prove that stable solutions cannot blow up atp_j,k. But it is still an interesting open problem to see whether those conditions are necessary or not and we will discuss it in another paper.

Remark 1.4.If any one of theNi’s is zero, thenTheorems 1.1 and 1.3hold even without the assumptions (H1)–(H2).

To understand the asymptotic behavior of solutions of(1.2)asε→0, we also ask whether or not there might exist a sequence of solutionsu_εfor(1.2)such that

εlim→0

sup

K

u_ε

= ∞ and lim

ε→0

infK u_ε

= −∞, (1.4)

whereK=Ω\

i,jB_r(p_j,i)for any fixedr >0. The following theorem tells us that the kind of blow-up behavior as introduced in(1.4)cannot occur.

Theorem 1.5.Let Z≡

j,i{pj,i}and Zi ≡

j{pj,i}fori=1,2. We assume that{uε}is a sequence of solutions of(1.2). Then, up to subsequences, one of the following holds true:

(a) u_ε→0uniformly on any compact subset ofΩ\Z;

(b) for any compact subsetK⊂Ω\Z₂, there existsν_K>0such that

εlim→0

sup

K

u_ε

−ν_K;

(c) for any compact subsetK⊂Ω\Z1, there existsνK>0such that

εlim→0

infK u_ε

ν_K.

Besides the application to our analysis, we believe that the above alternative could be useful in further studies of(1.2).

We also remark that it is important to use a suitable Pohozaev type identity for handling solutions with different asymptotic behavior. The following antiderivatives off_τ(u)are used to this purpose depending on the situations at hand:

F_1,τ(u)≡ −(1−e^u)² 2(τ+1)(τ+e^u)², and

F_2,τ(u)≡e^u((1−τ )e^u+2τ ) 2τ²(τ+e^u)² .

Moreover, we denote byGthe Green’s function onΩwhich satisfies

−_xG(x, y)=δ_y− 1

|Ω|, x, y∈Ω and

Ω

G(x, y) dx=0, (1.5)

and byγ (x, y)=G(x, y)+_2π¹ ln|x−y|its regular part. We also define

u⁺₀(x)≡ −4π

d₁

j=1

m_j,1G(x, p_j,1), u⁻₀(x)≡ −4π

d₂

j=1

m_j,2G(x, p_j,2), u₀≡u⁺₀ −u⁻₀,

and therefore we see that it holds u0= −4π(N1−N₂)

|Ω| +4π

d1

j=1

mj,1δp_j,1−4π

d2

j=1

mj,2δp_j,2 onΩ. (1.6)

In this paper, we consider only a domain which is a subset of R² because of not only physical background but also mathematical tools (Pohozaev Identity, Green’s function, etc.). We note that our results cannot be generalized to higher dimensional case.

The rest of this paper is devoted to the proof of the above theorems. In Section2, we discuss some preliminary results. In Section3, we investigate the asymptotic behavior of solutions of(1.2)asε→0. In Sections4–6, we study the asymptotic behavior of stable solutions. The main purpose is to prove some identities involving data coming from different regions, one being a neighborhood of the vortex point and the other one its complement. The more subtle part is the asymptotic analysis of the bubbling behavior of stable solutions at vortex points. Finally, we proveTheorems 1.1 and 1.3.

2. Preliminaries

We consider the following limiting problem for(1.2)whenZis empty, u+e^u(1−e^u)

(τ+e^u)³ =0 inR², (2.1)

and we also define (recallf_τ(u)=^e_(τ^u(1₊⁻_eu^e)^u3)) β≡ 1

2π

R²

f_τ(u) dx. (2.2)

By applying the method of moving planes as introduced in[12]and improved in[5]and[24], we obtain the following lemma.

Lemma 2.1.Letube a solution of(2.1). Assume that there exists a constantc∈Rsuch that either uc or uc or lim sup

|x|→∞

|u|

|x|²c.

Iff_τ(u)∈L¹(R²), thenuis radially symmetric about some pointx₀∈R².

Proof. The proof ofLemma 2.1is standard and we just provide a sketch for reader’s convenience. First of all, we observe thatf_τ(u)∈L¹(R²)∩L^∞(R²). Next we define

v(x)≡ 1 2π

R²

ln|x−y| −ln

|y| +1 f_τ

u(y) dy, (2.3)

so thatv=fτ(u)and by known elliptic estimates

|xlim|→∞

v(x)

ln|x|=β. (2.4)

At this point we may defineh=u+vand then observe thath=0.

Step 1. Now we claim that h is constant in R². If uc or ucin R² for some constant c∈R, then (2.4) implies that eitherhc₁(ln(|x| +1)+1)orhc₁(ln(|x| +1)+1)for some constantc₁∈R. Then, by Liouville’s

theorem,h(x)=u(x)+v(x)≡constant. Now we consider the case lim sup_|x|→∞|^|_x^u_|^|₂ c. Then, we also see that lim sup_|x|→∞|^|_x^h_|^|₂ is bounded. By the mean value theorem, there exist constantsc₁, c₂∈Rsuch that

sup

B_R

2

(y)

D^αh c1

R² sup

B_R(y)

|h|c₂,

for anyy∈R²,R=^|y₂^| and|α| =2 (see Theorem 2.10 in [13]). Then D^αhis a constant for|α| =2 sinceD^αhis bounded and harmonic inR². After a coordinates transformation, we can assume that eitherh(x)=a(x₁²−x₂²)+b orh(x)=cx₁+dx₂+efor some constantsa, b, c, d, e∈Rwherex=(x₁, x₂). Hence(2.4)implies that either

u(x)=a

x₁²−x₂² −

β+o(1) ln|x| +b=

a+o(1) x₁²−x₂² +b as|x| → ∞, (2.5) or

u(x)=cx1+dx2−

β+o(1) ln|x| +e=

c+o(1)x1+

d+o(1) x2+e as|x| → ∞. (2.6) For a fixedδ∈(0,1)we can find a constantCδ>0 such that

∞>

R²

f_τ(u)dx

δu2δ

f_τ(u)dx

δu2δ

C_δdx=C_δx∈R²δu(x)2δ. (2.7) Therefore, by using(2.5)and(2.6), we see that|{x∈R²|δu(x)2δ}| = ∞unlesshis constant which proves the claim. Then, as a consequence of(2.4), we see that

|xlim|→∞

u(x)

ln|x| = −β. (2.8)

Step 2. We claim that ifβ=0 thenu≡0. Suppose that there existsx₀∈R²such thatu(x₀) <0. Then there exists r >0 such that

u|Br(x0)<0. (2.9)

Let us set v_δ(x)=δln(^|x−_r^x0|) on R²\ B_r(x₀). Then we see that v_δ u on ∂B_r(x₀). Since u=o(ln|x|) as

|x| → ∞ (which is of course a consequence of (2.8) and β =0), then there exists R_δ >0 such that v_δ > u on R²\B_Rδ(0). We claim that v_δu on B_Rδ(0)\B_r(x₀). If not, there exists x₁∈B_Rδ(0)\B_r(x₀) such that u(x₁)−v_δ(x₁)=maxB_Rδ(0)\Br(x0)(u−v_δ) >0. Then by the maximum principle, we see that

0(u−v_δ)(x₁)= −f_τ

u(x₁) >0 sinceu(x₁) > v_δ(x₁)0.

Thus,vδ=δln(^|x−_r^x0|)uonR²\Br(x0). Sinceδ >0 is arbitrary, we conclude that

u(x)0 onR²\Br(x0). (2.10)

Now we see that(2.9)and(2.10)contradict(2.2)withβ=0. Therefore we haveu0 onR², and then, by using(2.2) withβ=0, we conclude thatu≡0 onR².

Step 3. From now on, we consider the caseβ =0. By using the strong maximum principle and(2.8), we conclude that

u >0, fτ(u) <0 ifβ <0,

u <0, f_τ(u) >0 ifβ >0. (2.11)

In view of(2.11), we can use the maximum principle to show that u−βln|x| +C ifβ <0,

u−βln|x| +C ifβ >0, (2.12)

for large|x|and a suitable constantC∈R. By using(2.12), thenf_τ(u)∈L¹(R²)implies that|β|>2 and then we deduce the sharper estimate

u(x)= −βln|x| +O(1) as|x| → +∞. (2.13)

At this point, the method of moving planes to be used with(2.13)shows thatuis radially symmetric. Since the proof is standard we skip it here and refer to[5,12]for further details. Therefore, the proof ofLemma 2.1is completed. 2

Letu(r;s)be the solution of the following initial value problem

⎧⎨

⎩ u+1

ru+e^u(1−e^u)

(τ+e^u)³ =0 forr >0, u(0;s)=s, u(0;s)=0,

(2.14) whereudenotes ^du_dr(r;s)and let us set

β(s)≡ 1 2π

R²

fτ

u(r;s) = ∞ 0

fτ

u(r;s) r dr. (2.15)

It turns out that the solutions of(2.14)admit only three kinds of limiting conditions asr→ ∞:

⎧⎨

⎩

topological boundary condition:u→0,

nontopological boundary condition of type I:u→ −∞, nontopological boundary condition of type II:u→ ∞.

(2.16) We will use the following lemma recently obtained in[8].

Lemma 2.2.Letu(r;s)be a solution of(2.14). Then, we have

(i) β(0)=0. In this case,u(r;0)≡0is the unique topological solution of(2.14);

(ii) β:(−∞,0)→(4,∞)is strictly increasing and bijective and

slim→0₋β(s)= ∞ and lim

s→−∞β(s)=4.

In this case,u(r;s)is a nontopological solution of type I;

(iii) β:(0,∞)→(−∞,−4)is strictly increasing and bijective and

slim→0₊β(s)= −∞ and lim

s→∞β(s)= −4.

In this case,u(r;s)is a nontopological solution of type II.

3. Proof ofTheorem 1.5: the asymptotic behavior of solutions

One of the main steps in the proof ofTheorem 1.5is to obtain a uniform bound for

Ω

1 ε²

e^uε(1−e^uε) (τ+e^uε)³

dx.

Toward this goal we have the following lemma.

Lemma 3.1.Letu_εbe a sequence of solutions of(1.2). Then, there exists a constantM₀∈(0,∞)such that

Ω

1 ε²

e^uε(1−e^uε) (τ+e^uε)³

dxM₀.

Proof. We observe that, for anya∈(0,∞), it holds uε

1−e^uε a+e^uε

=div

∇uε

1−e^uε a+e^uε

+(a+1)|∇u_ε|²e^uε

(a+e^uε)² . (3.1)

Then, multiplying both sides of Eq.(1.2)by_a¹⁻₊^e_e^uεuε and integrating overΩ, we conclude that

Ω

(a+1)|∇u_ε|²e^uε (a+e^uε)² + 1

ε²

e^uε(1−e^uε)²

(τ+e^uε)³(a+e^uε)dx=4π N₁

a +N2

. (3.2)

Let us fixa=1. Then there exist some constantsM1, M20 such that

Ω

|∇uε|²e^uε

(1+e^uε)²dxM₁, (3.3)

and

Ω

1 ε²

e^uε(1−e^uε)²

(τ+e^uε)³(1+e^uε)dxM₂. (3.4)

We also see that there existsδε,1∈(1,2)such that

{uε=−δε,1}

|∇u_ε|dS=

−1

−2 {uε=r}

|∇u_ε|dS

dr, (3.5)

and there exists a constantc₀>0 such that

{−2uε0}

|∇uε|²dxc0

{−2uε0}

|∇u_ε|²e^uε

(1+e^uε)²dxc0M1. (3.6)

Hence we also have

{u_ε=−δ_ε,1}

|∇u_ε|dS=

−1

−2 {u_ε=r}

|∇u_ε|dS

dr=

{−2uε−1}

|∇u_ε|²dxc₀M₁. (3.7)

Letνbe an exterior unit normal vector to∂{x∈Ω| −δ_ε,1u_ε0}. By using^∂u_∂ν^ε|uε=00 and(3.7), we see that 0

{−δ_ε,1uε0}

1 ε²

e^uε(1−e^uε) (τ+e^uε)³ dx= −

{−δ_ε,1uε0}

u_εdx

= −

{u_ε=−δ_ε,1}

∂u_ε

∂ν dS−

{uε=0}

∂u_ε

∂ν dS

{u_ε=−δ_ε,1}

|∇u_ε|dSc₀M₁. (3.8)

The same argument with minor changes shows that we can find constantsδ_ε,2∈(1,2)andc₁>0 such that

{0u_εδ_ε,2}

1 ε²

e^uε(1−e^uε) (τ+e^uε)³

dxc₁M₁. (3.9)

Moreover, there exist constantsc2, c3>0 such that

{u_ε−δ_ε,1}

1 ε²

e^uε(1−e^uε) (τ+e^uε)³

dxc₂

{u_ε−δ_ε,1}

1 ε²

e^uε(1−e^uε)²

(τ+e^uε)³(1+e^uε)dxc₂M₂, (3.10) and

{uεδε,2}

1 ε²

e^uε(1−e^uε) (τ+e^uε)³

dxc₃

{uεδε,2}

1 ε²

e^uε(1−e^uε)²

(τ+e^uε)³(1+e^uε)dxc₃M₂. (3.11) The desired conclusion follows by using(3.8),(3.9),(3.10)and(3.11). 2

Let us recall the following form of the Harnack inequality which will be widely used in the sequel (see[2]and[13]).

Lemma 3.2.LetD⊆R²be a smooth bounded domain andvsatisfy:

−v=f inD,

with f ∈L^p(D), p >1. For any subdomainD⊂⊂D, there exist two positive constantsσ ∈(0,1)and γ >0, depending onDonly such that:

(a) if sup_∂DvC,then sup_DvσinfDv+(1+σ )γfL^p+(1−σ )C, (b) if inf∂Dv−C, thenσsup_DvinfDv+(1+σ )γfL^p+(1−σ )C.

Moreover, we have the following lemmas.

Lemma 3.3.Letuεbe a sequence of solutions of(1.2). LetKbe a compact subset such thatK⊂Ω\Z. Then there exist constantsa, b >0such that|uε(xε)−uε(zε)|ar²+bfor anyr >0andzε∈Bεr(xε)⊆K.

Proof. By using the Green’s representation formula for a solutionuεof(1.2), we see that forx∈K⊂⊂Ω\Z, u_ε(x)= 1

|Ω|

Ω

u_ε(y) dy+

Ω

G(x, y)

−u_ε(y) dy

= 1

|Ω|

Ω

u_ε(y) dy+

Ω

G(x, y)

1 ε²

e^uε(1−e^uε) (τ+e^uε)³ −4π

d1

j=1

m_j,1δ_pj,1+4π

d2

j=1

m_j,2δ_pj,2

dy

= 1

|Ω|

Ω

u_ε(y) dy+

Ω

G(x, y)1 ε²

e^uε(1−e^uε)

(τ+e^uε)³ dy+O(1). (3.12)

Then,

uε(x)−uε(z)=

Ω

G(x, y)−G(z, y) 1 ε²

e^uε(1−e^uε)

(τ+e^uε)³ dy+O(1) forx, z∈K. (3.13) In view ofLemma 3.1, we see that

uε(x)−uε(z)= 1 2π ε²

Ω

ln

|z−y|

|x−y|

e^uε(1−e^uε)

(τ+e^uε)³ dy+O(1) forx, z∈K. (3.14) For fixedr >0, we assume thatz_ε∈B_εr(x_ε)⊆K. By the mean value theorem, there existsθ=θ (ε, y)∈(0,1)such that

ln|zε−y| −ln|xε−y|= ||z_ε−y| − |x_ε−y||

θ|z_ε−y| +(1−θ )|x_ε−y| |x_ε−z_ε|

θ|z_ε−y| +(1−θ )|x_ε−y|. (3.15) For anyy∈Ω\B_2εr(x_ε), we have|z_ε−y|εrand|x_ε−y|2εr. Thus, we see that

ln|z_ε−y| −ln|x_ε−y| εr

θ εr+(1−θ )2εr = 1

2−θ1 onΩ\B_2εr(x_ε). (3.16)

At this point,Lemma 3.1implies that

1 2π ε²

Ω\B2εr(xε)

ln

|zε−y|

|xε−y|

e^uε(1−e^uε) (τ+e^uε)³

dy=O(1). (3.17)

We also see that

B2εr(xε)

ln

|zε−y|

|xε−y|

dy

B2εr(xε)

|xε−zε|

θ|zε−y| +(1−θ )|xε−y|dy

B_2εr(x_ε)

|x_ε−z_ε|

min{|z_ε−y|,|x_ε−y|}dy

B2εr(xε)

|xε−zε|

|zε−y| +|xε−zε|

|xε−y| dy

B_4εr(z_ε)

|x_ε−z_ε|

|z_ε−y| dy+

B_2εr(x_ε)

|x_ε−z_ε|

|x_ε−y| dy

2

B4εr(0)

|x_ε−z_ε|

|y| dy16r²ε²π. (3.18)

Therefore we conclude that 1

2π ε²

B_2εr(xε)

ln

|zε−y|

|x_ε−y|

e^uε(1−e^uε) (τ+e^uε)³

dy8r²sup

t∈R

e^t(1−e^t) (τ+e^t)³

, (3.19)

and we readily obtain constantsa, b >0 such that for anyr >0, it holds

u_ε(x_ε)−u_ε(z_ε)ar²+b forz_ε∈B_εr(x_ε)⊆K. 2 (3.20)

Lemma 3.4.LetKbe a connected compact set such thatK⊂Ω\Z. Suppose that there exists a sequence of solutions {u_ε}of(1.2)such that

εlim→0

infK |u_ε|

=0.

Then, we haveu_εL^∞(K)→0asε→0.

Proof. Choose a sequence of points{x_ε} ⊆Ksuch that|u_ε(x_ε)| =infK|u_ε|. Passing to a subsequence (still denoted by u_ε), we may assume that limε→0x_ε=x₀∈K. We argue by contradiction. Suppose that there exists a positive constant c_K>0 and a sequence {z_ε} ⊆K such that sup_K|u_ε| = |u_ε(z_ε)|c_K for small ε >0. We will use the constantM00 obtained inLemma 3.1. Ifuε(zε)−cKthen, by usingLemma 2.2, we can chooses1<0 such that

β(s1) >M₀

π and −cK< s1<0.

Ifu_ε(z_ε)c_K then, by usingLemma 2.2, we can chooses₁>0 such that β(s₁) <−M0

π and 0< s₁< c_K.

We can also choose y_ε∈K such thatu_ε(y_ε)=s₁by the intermediate value theorem. Letu¯_ε(x)=u_ε(εx+y_ε)for x∈Ω_ε,yε≡ {x∈R²|εx+y_ε∈K₁}whereK₁is a compact subset such thatK⊂int(K1)⊂Ω\Z. Thenu¯_ε satisfies

⎧⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎩

u¯_ε+e^u¯ε(1−e^u¯ε)

(τ+e^u¯ε)³ =0 onΩ_ε,yε,

¯

uε(0)=s1,

Ω_ε,yε

e^u¯ε(1−e^u¯ε) (τ+e^u¯ε)³

dxM0.

(3.21)

By usingLemma 3.3, we see thatu¯_ε is bounded inC_loc⁰ (Ω_ε,yε). Passing to a subsequence, we may assume thatu¯_ε converges inC_loc² (R²)to a functionu_∗which is a solution of

⎧⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎩

u_∗+e^u∗(1−e^u∗)

(τ+e^u∗)³ =0 onR², u_∗(0)=s₁,

R²

e^u∗(1−e^u∗) (τ+e^u∗)³

dxM0.

(3.22)

By usingLemma 3.3andLemma 2.1, we conclude thatu_∗is radially symmetric with respect to some pointp¯inR² andu_∗does not change sign. HenceLemma 2.2shows that

M0

R²

e^u∗(1−e^u∗) (τ+e^u∗)³ dx

=2πβ

u_∗(p) ¯ 2πβ(s1)>2M0, (3.23)

which is the desired contradiction. Therefore, limε→0u_εL^∞(K)=0. 2 As a corollary ofLemma 3.4, we obtain the following proposition.

Proposition 3.5.Letuεbe a sequence of solutions of(1.2). Then, up to subsequences, one of the following holds true:

(a) u_ε→0uniformly on any compact subset ofΩ\Z;

(b) for any compact subsetK⊂Ω\Z, there existsν_K>0such that

εlim→0

sup

K

u_ε

−ν_K;

(c) for any compact subsetK⊂Ω\Z, there existsν_K>0such that

εlim→0

infK u_ε

ν_K.

Proof. In view ofLemma 3.4, it suffices to show that (a) holds whenever both (b) and (c) fail to hold. Suppose that (b) and (c) do not hold. Then, we can take compact setsK₁, K₂⊂Ω\Zand sequences{x_1,ε} ⊂K₁,{x_2,ε} ⊂K₂such that

εlim→0u_ε(x_1,ε)0 and lim

ε→0u_ε(x_2,ε)0.

For any compact setK⊂Ω\Z, taking a connected compact setK˜ ⊂Ω\Zsuch that K˜ ⊇K∪K1∪K2,

and using the intermediate value theorem, we can obtain a sequence{x_ε} ⊆ ˜Ksatisfying

εlim→0

uε(xε)=0.

Hence,Lemma 3.4yields that limε→0uε_L∞(K)˜ =0, which completes the proof. 2

Proof of Theorem 1.5 completed. First of all, we assume that (b) in Proposition 3.5 holds. In this case, we also suppose that there exists r ∈(0,¹₃dist(Z1, Z2)) such that B2r(pi,1)∩B2r(pj,1)= ∅ when i =j and limε→0(supd1

j=1Br(pj,1)u_ε)0. By using limx→pj,1u_ε(x)= −∞and the intermediate value theorem, we see that there existsx_ε∈d₁

j=1B_r(p_j,1)such that|u_ε(x_ε)| =infd1

j=1Br(pj,1)|u_ε| →0 asε→0. Letx₀∈d₁

j=1B_r(p_j,1)be the limit point ofx_ε. Passing to a subsequence, only one of the following two possibilities can be satisfied: either x₀∈/Z₁orx₀∈Z₁.