Topological Multivortex Solutions for the Chern-Simons System with Two Higgs Particles

Communications in Partial Differential Equations

ISSN: 0360-5302 (Print) 1532-4133 (Online) Journal homepage: http://www.tandfonline.com/loi/lpde20

Topological Multivortex Solutions for the Chern- Simons System with Two Higgs Particles

Zhi-You Chen & Jann-Long Chern

To cite this article: Zhi-You Chen & Jann-Long Chern (2016): Topological Multivortex Solutions for the Chern-Simons System with Two Higgs Particles, Communications in Partial Differential Equations, DOI: 10.1080/03605302.2015.1132429

To link to this article: http://dx.doi.org/10.1080/03605302.2015.1132429

Accepted author version posted online: 11 Jan 2016.

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Topological Multivortex Solutions for the Chern-Simons System with Two Higgs Particles

Zhi-You Chen¹ and Jann-Long Chern²

1Department of Mathematics, National Changhua University of Education, Changhua, Taiwan

2Department of Mathematics, National Central University, National Center for Theoretical Sciences of Taiwan, Chung-Li, Taiwan

Abstract

In this paper, we prove the uniqueness of topological multivortex solutions to the self-dual abelian Chern-Simons Model if either the Chern-Simons coupling parameter is sufficiently small or sufficiently large. In addition, we also establish the sharp region of the flux for non-topological solutions with a single vortex point.

KEYWORDS: .

2010 Mathematics Subject Classfication Primary 35J47, 35J60; Secondary 35A02.

Received December 30, 2014; Accepted December 01 2015.

Zhi-You Chen, Department of Mathematics, National Changhua University of Education, Changhua 500, Taiwan; E-mail: [email protected]

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1. INTRODUCTION In this paper, we consider the nonlinear elliptic system

⎧⎪

⎪⎨

⎪⎪

⎩

u+ ¹2e^v1−e^u=4

N1

i=1

_ip

iinR² v+¹²e^u1−e^v=4

N₂

j=1

_jq

jin R²

(1.1)

where = ²

i=1 ²

x²_i >0 N₁andN₂ are two positive integers, _i ≥0 and _j ≥0 which are called vortex numbers, and _p is the Dirac measure at p. System (1.1) arises from a relativistic Abelian Chern-Simons model with two Higgs particles. Let and be two complex scalar fields in R² representing two Higgs particles with chargesq₁ andq₂. Assume that and have zeros atp_s and q_s with corresponding order_s and _s, respectively, i.e., = z−p_s^sfor znear p_s s =1 · · · N₁

(1.2) = z−q_s^sforznearq_s s =1 · · · N₂

Let A¹r andA²r r =1 2 be two gauge fields and the corresponding covariant derivatives D_r be

D_r=_r−iq_IA¹_r D_r=_r−iq_IA²_r Set

F_rs^I=_rA^I_s −_sA^I_r (1.3)

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to be the corresponding curvatures associated with D_r on 2+1-dimensional Minkowski space with the metric g_{r s}=diag1 −1 −1, where r s =0 1 2. In (10) and (16), the Chern-Simons density takes the form

= −

4^rstA¹_r F_st²−

4^rstA²_r F_st¹+D_rD^r+D_rD^r−V with the Higgs potential density defined by

V = q₁²q²₂

^{2 22}−c²₂²+ ²²−c²₁²

where >0 is a coupling parameter, c₁ andc₂ are positive constants.

For the static case, by changing of variables, q_IA^I_j →A^I_j →c₁ and →c₂, the self- dual equation for Chern-Simons Lagrangrian can be reduced to a system of equations of first order

D₁±iD₂=0 D₁±iD₂=0 F₁₂¹± 2q₁q₂²

^{2 2}1− ²=0 (1.4)

F₁₂²± 2q₁q₂²

^{2 2}1− ²=0 where

D_r=_r−iA¹_r D_r =_r−iA²_r (1.5)

For the detailed derivation of (1.4) from the Lagrangian , we refer the readers to (10), (16), (17) and the references therein.

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Following the results in (14) by Jaffe and Taubes, (1.4) can be reduced to an elliptic system of second order. Set the operators and by

= 1

2₁−i₂ = 1

2₁+i₂ From (1.3), we have

F₁₂^I=₁A^I₂ −₂A^I₁ =+A^I₂ −i−A^I₁

(1.6)

=−iA^I₁ +iA^I₂ +−A^I₁ +iA^I₂

Without loss of generality, we consider the positive sign in (1.4). By (1.5), we have

D₁+iD₂=2−iA¹₁ +A¹₂ =2−iA¹₁ +iA¹₂ D₁+iD₂=2−iA²₁ +A²₂ =2−iA²₁ +iA²₂ Hence, from (1.4), we obtain

A¹₁ +iA¹₂ =2i

=2iln

(1.7) A²₁ +iA²₂ =2i

=2iln

Combining (1.6) and (1.7), we deduce the following relations:

F₁₂¹ =2ln+ln= 1

2ln²

(1.8)

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F₁₂²=2ln+ln= 1

2ln² Now, let

u=ln²and v=ln² (1.9)

Then, by (1.4), u and v satisfy (1.1) for x =p₁ · · · p_N1 q₁ · · · q_N2 and ² = ²/4c₁²c²₂q₁q₂². It is easy to see that the Dirac measures appearing in (1.1) come from (1.2) and (1.9).

Conversely, suppose that u vis a solution of (1.1). Let z=x¹+ix² and

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

₁z= −^N1

s=1

argz−p_{s 2}z= −^N2

s=1

argz−q_s z=e^12uz+i1z z=e^12vz+i2z

A¹₁ z= −Re2ilnz A¹₂ z= −Im2ilnz A²₁ z= −Re2ilnz A²₂ z= −Im2ilnz

(1.10)

Then A^Ir I =1 2 r =1 2, satisfy (1.4). Therefore we have shown that (1.4) is equivalent to (1.1). In the past decades, the equation derived from Chern-Simons model with one Higgs particle has been intensively studied in, e.g., (4)–(7), (8)–(10), (11)–(16), (18)–

(24) and references therein. However, for the system of coupled equations, the study of (1.1) has just begun recently (see, e.g., (2) and (17)).

For (1.1), there are two natural boundary conditions for solutions at infinity, namely,

xlim→ux= lim

x→vx=0 (1.11)

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or

xlim→ux= lim

x→vx= − (1.12)

We note that if u vis a solution with the boundary condition either (1.11) or (1.12), then, by the maximum principle, we have ux <0 and vx <0 for all x ∈R²\. In physics literature, a solution u v satisfying boundary condition (1.11) is called a topological solution and it is easy to see that both u and v decay exponentially at infinity.

In a recent paper (17), Lin-Ponce-Yang has considered the problem about the existence of the topological solutions of (1.1) for any given set of singularities and the following theorem was proved.

Theorem A.. For any given pointsp₁ p_N1 q₁ q_N2 ∈R²and_{1 N1 1 N2} ∈ R⁺∪0, (1.1) possesses a topological solution.

In addition to the existence result as stated in Theorem A, it is natural to ask the question about the uniqueness of topological solutions to (1.1). For the case of a single vortex point, the uniqueness result has been proved in (2) for any >0. Moreover, it is not difficult to see that the uniqueness of topological solutions for (1.1) in the case N₁=N₂, i=j and p_{i i}=q_{i i} is equivalent to the one for topological solutions of

u+ 1

²e^u1−e^u=4

N1

i=1

_ip

iin R² (1.13)

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Therefore, according to (1), the uniqueness result for this specific case has also been proved ifis sufficiently large or small. In this article, we provide a uniqueness result of topological solutions to (1.1) in the general case, stated in the following theorem.

Theorem 1.1. Let p₁ p_N

1 q₁ q_N

2 ∈ R² and _{1 N}

1 _{1 N}

2 ∈R⁺∪0 be

given. Then there are two positive constants ₀ and ₁, dependent on p_i q_{j i j}, such that 11possesses a unique topological solution for any ∈0 ₀∪₁ .

In (2), we call a solutionu vof (1.1) to be non-topological if u vsatisfies the boundary condition (1.12), and both e^u1−e^v and e^v1−e^u are inL¹R². For any entire solution u vof (1.1), we set

₁ u v= 1 2²

R²

e^vx1−e^ux dx and ₂ u v= 1 2²

R²

e^ux1−e^vx dx (1.14)

By virtue of (2), the existence of non-topological solutions of (1.1) has been established for the case of none of vortex point or single one. Moreover, for N₁ =N₂=N, (1.1) possesses a non-topological solution u u satisfying ₁ u u=₂ u u= for any given >

0 and >4N +1 which was obtained in (4). Besides, if N₁ =N₂ =N and p_{i i}= q_{j j}for all i j =1 · · ·N, then, based on (5), for any >0 and >4N +1 with

4N_k−^k₁k=2 · · · _M

j=1_j

there exists a non-topological solutionu uof (1.1) satisfying ₁ u u=₂ u u=. Therefore, there is an intersecting problem about finding the sharp range of flux pair ₁ u v ₂ u v for all non-topological solutions u v of

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11. In our second main result, we give an answer to the sharp region of flux pair for non-topological solutions of 11 for the case of none of vortex point or single one.

Theorem 1.2. Let ux vx be a non-topological solution of the equation u+ ¹²e^v1−e^u=4_pin R²

v +¹2e^u1−e^v=4¯ _pin R² (1.15)

for any point p∈R² and constants ¯ ≥0. Then

₁ u v ₂ u v

satisfies ₁ u v∈2+2 ₂ u v∈2¯ +2

₁ u v−2+1 ₂ u v−2¯+1

>4+1¯ +1 (1.16)

Moreover, 115 possesses a non-topological solution ux vx such that ₁ u v ₂ u v

=_{1 2} for any pair_{1 2} which satisfies116.

This article is organized as follows. Section 2 is devoted to proving the consequences of Theorem 1.2. In Section 3, we will sketch the proof of Theorem 1.1 about the uniqueness of topological solutions for >0 sufficiently small. In the last section, Section 4, the uniqueness of topological solutions for >0 large enough in Theorem 1.1 will be discussed.

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2. SHARP RANGE OF FLUX FOR NON-TOPOLOGICAL SOLUTIONS

Let ux vx be a non-topological solution of 115 and ux vx=ux+ p vx+p. Then ux vx satisfies the following system

⎧⎪

⎪⎨

⎪⎪

⎩

u+e^v1−e^u=4_Oin R² v+e^u1−e^v=4¯ _Oin R²

R²e^vx1−e^uxdx <

R²e^ux1−e^vxdx <

(2.1)

where ≥0 ¯ ≥0 and O=0 0. First,we will refer to the behavior of the non-positive solutions i.e.ux≤0, vx≤0 in R²\O near infinity as following lemma.

Lemma 2.1. Ifux vx is a non-positive solution of 21, then it can be proved that

ux=

2−_{1 1}u v

lnx +C₁+o1 and vx=

2¯−_{2 1}u v

lnx +C₂+o1 near (2.2)

for some constants C₁ C₂ ∈R. Moreover, non-positive solution ux vx is a topological solution or non-topological solution.

Proof. We setwx=ux−2lnx and hence wx satisfies

w+e^v1−e^u=0 in R² Define the potential

vx= 1 2

R²

ln

x−y y

ydy

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where x =e^vx1−e^ux Thenvx satisfies

v=e^v1−e^u in R²

and we see that

2vx≤

R²

ydy

lnx +C (2.3)

where R₀1,C is a constant independent ofx andx ∈R²\B_R

0O. Sincewx≤0 inR²\ B_R

0O,wx+vx≤Clnx +1 onx x ≥1 for some constantC >0, which implies

wx+vx is a constant inR² by Lemma 4.6.1 of (24). It is not difficult to see that vx

lnx → 1 2

R²

xdx uniformly as x →

which implies ux

lnx →2− 1 2

R²

xdx uniformly as x →

and_{1 1}u v > 2+2 or_{1 1}u v=2. Similarly forvx, the result (2.3) is obtained and _{2 1}u vsatisfies

_{2 1}u v=2¯ if _{1 1}u v=2

_{2 1}u v > 2¯ +2 if _{1 1}u v >2+2

Therefore, we we complete this lemma.

Proposition 2.1. If ux vx is a non-topological solution of 21, then _{1 1}u v and _{2 1}u vsatisfy 116.

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Proof. The proof is standard by Pohozaev identity and 22. Thus we omit the detail.

Next, we consider the following ODE system ur+ ¹_rur+e^vr1−e^ur=0

vr+ ¹_rvr+e^ur1−e^vr=0 r >0 (2.4) with the initial value

ur=2lnr+₁+o1

vr=2¯lnr +₂+o1 as r →0⁺ (2.5)

where ₁∈ Rand₂ ∈R. For all entire solutions ur _{1 2} vr _{1 2} of (2.4)–(2.5), we set their flux pair

_{11 2 21 2}=

0

re^vr1−e^ur dr

0

re^ur1−e^vr dr

(2.6)

According to the flux pair, all entire solutions of (2.4) with boundary condition (1.12) can be classified into the following three types.

Type (I): lim

r→ur= − lim

r→vr = − with ₁ < and ₂ <, i.e., u v is a non- topological solution.

Type (II): lim

r→ur= − lim

r→vr= − with 2 < ₁ ≤2+2 ₂= Type (III): lim

r→ur= − lim

r→vr= − with₁ = 2 < ¯ ₂≤2¯ +2

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Then the structure of solutions sets is described as follows from paper(2).

Theorem 2.1. There exist strictly increasing functions_{u v} − ⁰₁→− ⁰₂satisfying _v1≥_u1, and

1lim→−_i1= − lim

1→⁰_1i1=⁰₂ fori=u v such that

T = =_{1 2} r _¯r is a topological solution=⁰₁ ⁰_{2 NT} = =_{1 2} r _¯r is a non-topological solution

=_{1 2 1} < ⁰_{1 u1} < ₂< _v1

and

⎧⎪

⎪⎨

⎪⎪

⎩

_up =_{1 2 1} < ⁰_{1 2} =_v1

= =_{1 2} r _¯r is a solution of type (III) _low =_{1 2 1} < ⁰_{1 2} =_u1

= =_{1 2} r _¯r is a solution of type (II) Moreover, if _{1 2}∈_NT, then _{11 2 21 2} satisfies 116.

Remark 2.1.

(i) From continuity and Theorem 2.1, there exist strictly increasing continuous functions ^c₁ − ⁰₁→− ⁰₂ and ^d₂ − ₁⁰→− ⁰₂ such that

lim

₁→⁰₁

^c₁₁= lim

₁→⁰₁

^d₂₁=⁰₂ lim

₁→−^c₁₁= lim

₁→−^d₂₁= −

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and ₁ ^c_{11 2} ^d₂₁∈ _NT for all ₁< ₁⁰ implies

₁₁ ^c₁₁=c and ₂₁ ^d₂₁=d on − ⁰₁

for anyc d∈2+2 ×2¯ +2 . Moreover, we see that ifc₁< c₂ resp.,d₁ >

d₂, then

^c₁¹₁ < ^c₁²₁resp ^d₂¹₁ < ^d₂²₁

for any₁ ∈− ⁰₁ by Lemma 2.3 of (2).

(ii) _{11 2} and_{21 2} are continuous on _NT from (2).

Lemma 2.2. Suppose≥0 and ¯ ≥0. Then the following statements are valid.

(i) Let ₁ ∈2+2 . If there exists a sequence ₁ⁱ ₂ⁱ_i∈N ⊆_NT such that ⁱ₁ ⁱ₂→ ⁰₁ ⁰₂ asi → and ₁ⁱ₁ ⁱ₂=₁ for all i∈N, then ₂₁ⁱ ₂ⁱ→ asi → . (ii) Let ₂ ∈2¯ +2 . If there exists a sequence ₁ⁱ ₂ⁱ_i∈N ⊆_NT such that ⁱ₁ ⁱ₂→

⁰₁ ⁰₂ asi → and ₂ⁱ₁ ⁱ₂=₂ for all i∈N, then ₁ⁱ₁ ⁱ₂→ asi → . (iii) Let M >0, there exists a point ^M₁ ^M₂ ∈ _NT such that ^M₁ < ⁰₁ and ^M₂ < ⁰₂ implies

_{11 2}−2+1

_{21 2}−2¯ +1

> M for any _{1 2}∈_NT ∩^M₁ ⁰₁×^M₂ ⁰₂

Proof. By Pohozaev identity, we have ₁₁ⁱ ₂ⁱ−2+1 ₂ⁱ₁ ⁱ₂−2¯ +1

−4+1¯ +1≥6 R

0

re^{ur1i i2+vr1i i2}dr

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for any R >0. It follows that the results (i), (ii) and (iii) hold, and hence we complete this

lemma.

Lemma 2.3. The following statements are valid.

(i) Let ₁ ∈2+2 . If there exists a strictly increasing continuous function − ⁰₁→− ⁰₂ such that ₁→ − as ₁→ −, _{1 1}⊆_NT and _{11 1}=₁ for all ₁ ∈− ⁰₁, then ₁− ¹₁^+¯₊₁ →L

1 as ₁ → − for some L

1 ∈ R.

(ii) Let ₂ ∈2¯+2 . If there exists a strictly increasing continuous function − ⁰₁→− ⁰₂ such that ₁→ − as ₁→ −, _{1 1}⊆_NT and _{21 1}=₂ for all ₁ ∈− ⁰₁, then ₁− ¹₁^+¯₊₁ →L

2 as ₁ → − for some L

2 ∈ R.

Proof. The proof of (ii) is similarly to (i), and hence we only prove the result (i). On the contrary. We may assume that there exists a sequence ₁ⁱ_i∈N such that lim_i→1ⁱ−

1+¯

1+ⁱ₁ = . Now we set U r ⁱ₁=ueⁱ¹r ⁱ₁ ⁱ₁−2lneⁱ¹r−ⁱ₁ and V r ₁ⁱ= veⁱ¹r ⁱ₁ ⁱ₁−2¯lneⁱ¹r−ⁱ₁, where

ⁱ₁=

−_2+2¯ⁱ¹ if lim_i→ⁱ₁− ¹₁^+¯₊ⁱ₁=

−₂₊ⁱ¹₂ if lim_i→ⁱ₁− ¹₁^+¯₊ⁱ₁= −

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Then we obtain thatU r ⁱ₁andV r ⁱ₁converge toU randV rinC²0 Rasi→ for anyR >0, where

⎧⎪

⎪⎨

⎪⎪

⎩

Ur+ ¹_rUr+r^2¯e^{V r} =0 on 0 Vr+¹_rVr=0 on 0

U 0 =V 0=0 U0=V0=0 if lim_i→ⁱ₁− ¹₁^+¯₊ⁱ₁= and

⎧⎪

⎨

⎪⎩

Ur+ ¹_rUr=0 on 0

Vr+¹_rVr+r²e^{U r} =0 on 0 U 0 =V 0=0 U0=V0=0 if lim_i→ⁱ₁− ¹₁^+¯₊ⁱ₁= −.

For the case lim_i→ⁱ₁−¹₁^+¯₊ⁱ₁= , we see that there exist two constants I R >0 such that

eⁱ¹Rueⁱ¹R < 2−2₁for all i≥I

which contradicts to₁ⁱ₁ ⁱ₁=₁ for alli ∈N. On the other hand, we see thatVr=

−₂₊¹_2¯r^1+2¯ for r ≥0, and it follows that

0=

0

i→limr^1+2¯

e^21+¯i1+i1e^{V ri1 i1}−e^{21++¯i1+i1+i1}r²e^{U+V ri1 i1}

dr

= lim

i→

0

r^1+2¯

e^21+¯i1+i1e^{V ri1 i1}−e^{21++¯i1+i1+i1}r²e^{U+V ri1 i1}

dr

= lim

i→₁₁^{i i}₁

which contradicts to ₁ⁱ₁ ⁱ₁=₁ for all i∈N. Thus we complete the proof of (i).

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Lemma 2.4. For any _{1 2}∈ 2+2 ×2¯ +2 and satisfies

₁−2+1

₂−2¯ +1

=4+1¯ +1 (2.7)

Then there exists a strictly increasing continuous function − ⁰₁→− ⁰₂ such that

1lim→−₁→ − lim

1→−_{11 1}=₁ and lim

1→−_{21 1}=₂

Proof. Let₁ ∈2+2 . Then there exists a strictly increasing continuous function₁¹ − ⁰₁→− ⁰₂ such that ₁¹₁→ − as₁ → −, lim

1→−_{11 1}¹₁=₁ and

1lim→−_{21 1}¹₁≤ . From Lemma 2.3, it follows that lim

1→−₁¹₁−¹₁^+¯₊₁=L

1 for

some L

1 ∈R. Now we set

U r ₁=ue⁻²⁺¹²r _{1 1}¹₁−2lne⁻²⁺¹²r−₁ V r ₁=ve⁻²⁺¹²r _{1 1}¹₁−2¯lne⁻²⁺¹²r−₁¹₁

and then we obtain that U r ₁≤0 V r ₁≤0 on 0 and U r ₁ V r ₁ converges to U r V r

inC²0 R×C²0 R as ₁→ − for anyR >0, where

⎧⎪

⎨

⎪⎩

Ur+ ¹_rUr+e^L1r^2¯e^{V r} =0 on 0 Vr+ ¹_rVr+r²e^{U r} =0 on 0 U 0=V 0=0 U0=V0=0 For the case

0 r²⁺¹e^{V r}dr = , it follows that there exist M R >0 such that e⁻²⁺¹²Rue⁻²⁺¹²R _{1 1}¹₁ <2−2₁ for all₁ ≤ −M

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which contradicts to _{11 1}¹₁=₁ for all ₁ < ⁰₁. On the other hand, if

0 r²⁺¹e^{U r} dr = , then we see that

0 r^2¯+1e^{V r}dr =2+2 andrVr→ −asr → . Hence we have

2+2= 0

r^2¯+1e^{V r}dr = lim

1→−

0

re^vr1

1

1 1

1−r²e^{U r 1+2+221}

dr =₁₁¹₁

which contradicts to _{11 1}¹₁=₁ for all ₁ < ⁰₁. Thus it follows that

0 r^2¯+1e^{V r}dr =₁ and lim_{1→−21 1}¹₁=

0 r²⁺¹e^{U r}dr < which satisfy 27 because Theorem 1.4 of (3). Therefore, we complete the proof of this result.

Proof of Theorem 1.2. Let _{1 2}∈ 2+2 ×2¯ +2 satisfies 116. From Remark 2.1, we see that _{11 1}¹₁=₁ and _{21 1}¹₁ is continuous on − ⁰₁. Since lim_1→⁰

1_{21 1}¹₁= and

₁ lim_{1→−21 1}¹₁

satisfies27by Lemma 2.3 and Lemma 2.4, the region of _{21 1}¹₁ is the interval

21+ ¯+ ^41+1+¯

1−2+1

from Lemma 2.2. Therefore, we complete the proof of this theorem by Proposition 2.1.

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3. UNIQUENESS OF TOPOLOGICAL SOLUTIONS FOR SMALL

In this section, we will prove Theorem 1.1 for sufficiently small from the following two a priori estimates lemmas for any topological solution of 11 when be small enough.

Lemma 3.1. Let >0 and u v be a topological solution of 11. Then for each 0<

d < ¹₄min p_i−p_j q_k−q 1≤i < j ≤N₁ 1≤k < ≤N₂

, there is a constant ₀ = ₀d >0 such that if0< < ₀, then

maxuC^2c_d vC^2c_d ≤c₀exp−c₁

(3.1)

for some positive constants c₀ c₁ depending only on d where _d=

i j B_dp_i∪B_dq_j .

Proof. We divide the proof into two steps.

Step 1. for each compact subsetK ⊂R²\, there are constants _∗ >0 and ₀K <0 such that₀K≤ux vx <0 inK for 0< < _∗ where =p₁ p_N

1 q₁ q_N

2. Let Ux =ux−

N₁

i=1

2_ilnx−p_i and Vx=vx−

N₂

j=1

2_jlnx−q_j

Thus it suffices to prove that if >0 is sufficiently small, then inf_B

ROUx inf_B

ROVx≥₀ for some ₀ =₀R <0 where R >max

i j p_i q_j and O =0 0. On the contrary, without loss of generality, we may assume that there exist a constant R₀>max_{i j}p_i q_j and two sequences _n x_n ⊂B_R0O such that

x_n →O _n →0 and U

nx_n= inf

B_R0OUx→ − as n→

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Accepted

Manuscript

For simplicity, we let U_n =U

n and V_n=V

n. Decompose U_n=U_1n+U_2n and V_n =V_1n+ V_2n where

⎧⎨

⎩

U_1n+ ¹2

ne^vn1−e^un=0in B_RO

V_1n+¹2

ne^un1−e^vn=0inB_RO

U_1nx=V_1nx=0 onB_RO and

⎧⎨

⎩

U_2n=0in B_RO V_2n =0inB_RO

U_2nx=U_nx V_1nx=V_nxonB_RO

for any R≥R₀. From the Harnack inequality, we see that U_2nx converges to − uniformly onB_RO.

On the other hand, following the argument of Lemma 3.1 of (20) and Poincar´e inequality, we can verify that U_1n is bounded in W₀^{1 q}B_RO for each 1< q <2. By passing to a subsequence, we may assume that

U_1n! U weakly inW₀^{1 q}B_RO

and strongly inL^pB_ROfor 1≤p < ₂^2q_−q. Consequently,U_n → −andV_n→ −almost everywhere onB_RO, otherwise, lim

n→e^vnx1−e^unx=0 almost everywhere onB_ROfrom 0≤

B_RO

nlim→e^vnx1−e^unxdx= lim

n→

B_RO

e^vnx1−e^unxdx < lim

n→4

N₁

i=1

_i²_n=0

Consider the equation 11 with no vortex and =1. Let M >8 ^N_i=¹_1i ^N_j=²_1j

and by Lemma 2.2(iii), there exists a point^∗₁ ^∗₂∈_NT such that^∗₁ < ⁰₁ and^∗₂ < ⁰₂ implies

_{11 2}−2 _{21 2}−2 > M for any _{1 2}∈ _NT ∩^∗₁ ⁰₁×^∗₂ ⁰₂ (3.2)

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Accepted

Manuscript

For each n, choose y_n y_n ∈R² such that u_ny_n=^∗₁ and v_ny_n=₂^∗, where y_n =supx u_nx=^∗₁ and y_n =supx v_nx=^∗₂

Since u_n v_n → − almost everywhere on each compact subset and lim

x→u_nx v_nx→ 0 0, y_n y_n → as n→ , otherwise, there exist a constant n_∗ >0 and a sequence x^u_n ⊆R² resp.,x_n^v⊆R² such that x^u_n>y_nresp.,x^v_n>y_n and u_nx^u_n≥ 0resp.,v_nx^v_n≥0 passing to a subsequence if necessary for all n≥n_∗ if y_nresp.,y_n is bounded. Without loss of generality, we let uˆ_nx=u_nnx+y_n, vˆ_nx=v_nnx+y_n and y_n ≥ y_n for n∈N, otherwise, we let uˆ_nx=u_nnx+y_n, vˆ_nx=v_nnx+y_n and y_n ≤ y_n forn∈N. Then uˆ_n and vˆ_n satisfy

⎧⎪

⎪⎨

⎪⎪

⎩

uˆ_n+e^vˆn1−e^uˆn=0in_n=

x< ^min₂^ynyn

n

vˆ_n+e^uˆn1−e^vˆn=0in _n=

x < ^min₂^ynyn

n

_ne^ˆvn1−e^uˆndx≤4N1

i=1_i

_ne^uˆn1−e^vˆndx ≤4N2

j=1_j To finish this step, we need the fact as follows.

Claim. uˆ_n and vˆ_n are bounded inC_locR².

Proof of the claim. uˆ_n is bounded in C_locR² from uˆ_nO=^∗₁ and Harnack inequality.

Consequently, vˆ_n is bounded in C_locR² by

_ne^uˆn1−e^vˆndx ≤4

N₂

j=1

_j for all n∈N. The proof of the claim is completed.

From claim, we see that uˆ_n vˆ_n converge in C_loc² _n×C_loc² _n to uˆ_∗ vˆ_∗ which is a solution of

⎧⎨

⎩

uˆ_∗+e^vˆ∗1−e^uˆ∗=0inR² vˆ_∗+e^uˆ∗1−e^vˆ∗=0inR² ˆ

u_∗x≤0 and vˆ_∗x ≤0in R²

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