Blow-up solutions of the self-dual Chern-Simons-Higgs vortex equation

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Blow-up solutions of the self-dual Chern–Simons–Higgs vortex equation

Solutions explosives de l’équation auto-duale de vortex de Chern–Simons–Higgs

Kwangseok Choe

, Namkwon Kim

aDepartment of Mathematics, Inha University, 253, Yonghyun-dong, Nam-ku, Incheon, 402-751, Republic of Korea bDepartment of Mathematics, Chosun University, Kwangju, 501-759, Republic of Korea

Received 6 April 2006; accepted 30 November 2006 Available online 13 June 2007

Abstract

We apply the variational method and the blow-up analysis to the self-dual Chern–Simons–Higgs vortex equation on a flat torus to obtain two solutions for certain values of the Chern–Simons constant. As the corresponding Chern–Simons constant tends to zero, one of corresponding solutions converges to zero and the other blows up at only one point in the sense of Brezis–Merle provided that the total number of vortex is greater than 2. Further, the below-up solution is of spike type and becomes a critical point ofJ⁺ when the total number of vortex is greater than 3. As a consequence, we show the existence of the third solution for some periodic configuration of vortices and some Chern–Simons constant.

©2007 Elsevier Masson SAS. All rights reserved.

Résumé

Nous nous appliquons la méthode variationnelle et l’analyse d’explosion à l’équation auto-duale de vortex de Chern–Simons–

Higgs sur un tore plat pour obtenir deux solutions pour certaines valeurs de la constante de Chern–Simons. Lorsque la constante correspondante de Chern–Simons tend vers zéro, une des solutions correspondantes converge vers zéro et l’autre solution explose en seulement un point dans le sens de Brezis–Merle à condition que le nombre de vortex total soit plus grand que 2. De plus, l’explosion est de type “pic” et, quand le nombre de vortex total est plus grand que 3, la solution est un point critique deJ⁺. Nous en déduisons l’existence d’une troisième solution pour une certaine configuration périodique des vortex et une certaine constante de Chern–Simons.

©2007 Elsevier Masson SAS. All rights reserved.

Keywords:Chern–Simons–Higgs vortex equation; Blow-up solutions

* Corresponding author.

E-mail addresses:kschoe@inha.ac.kr (K. Choe), kimnamkw@chosun.ac.kr (N. Kim).

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

doi:10.1016/j.anihpc.2006.11.012

1. Introduction

The Chern–Simons–Higgs model is a (2+1)dimensional gauge model and it was proposed in [19,20] in an attempt to explain the superconductivity of type II. Unlike the Abelian–Higgs (or Ginzburg–Landau) model, the Chern–Simons–Higgs model admits vortices which is charged both electrically and magnetically and is known to have two different type of solutions (see, for instance, [6,14,28] and references therein). Hence it has been studied actively in the mathematical literature(see [5,29,32] and references therein).

The self-dual Chern–Simons–Higgs vortex equation on a flat 2-torusΩcan be written as follows;

u= 1 ²e^u

e^u−1 +

k

j=1

4π mjδ_pj inΩ. (1.1)

Here, 2 >0 is the Chern–Simons constant,m_j∈N,p_j∈Ω, andj =1, . . . , k. The solutionuof (1.1) is often called a vortex solution and eachp_j (j=1, . . . , k) is called a vortex point andm_j the multiplicity ofp_j. The vortex points are related to the (local) maximum point of the magnetic flux in the Chern–Simons–Higgs model.

Meanwhile, (1.1) can be thought as a formal perturbation of the mean field equation. Indeed, if we letw=u−2 ln then (1.1) can be rewritten as

w= −e^w

1−²e^w +

k

j=1

4π mjδ_pj inΩ. (1.2)

If=0, (1.2) becomes the mean field equation. Indeed, whenk=m₁=1, it was proved by Tarantello [28] that (1.2) admits a family of solutions converging to a solution of the mean field equation astends to zero.

DenotingN=k

j=1m_j and introducingv=u−u0, u₀=

k

j=1

4π mjδ_pj−4π N

|Ω| inΩ,

Ω

u₀dx=0, we can equivalently write (1.1) in a more favorable form as follows;

v= 1 ²e^u0+v

e^u0+v−1 +4π N

|Ω| inΩ. (1.3)

A solutionvof (1.3) is called of finite energy ifvbelongs toH¹. Indeed, it is well known that the corresponding physical energy of the solutionvis finite ifv∈H¹[5,29,32]. Thus, solutions of finite energy are indeed physically meaningful in (1.3) and has been sought in the literature. It was first proved in [5] that there is a critical number ₀=₀(m_j, p_j) >0 such that if < ₀ then (1.3) admits aH¹ solution, and if > ₀ then (1.3) admits no H¹ solution. This phenomenon is called a vortex confinement and it also appears in the Abelian–Higgs model [30]. Later, in [28], Tarantello showed that when < ₀, there exist at least twoH¹ solutions to (1.3). This multiplicity result was physically unexpected since the possibleH¹solutions of (1.3) have the same physical energy as well as the same distribution of vortex provided that the configurations ofm_j andp_j (j=1, . . . , k) are the same. We remind that such multiplicity does not happen in the Abelian–Higgs model by the uniqueness [30]. After that, naturally, the asymptotic behavior of the multiple solutions has been studied on a torus astends to 0 [24,25,28].

There are now many existence results forH¹-solutions of (1.3). By using the super-subsolution method, Caffarelli and Yang [5] constructed a maximal solutionv˜in the sense that ifvis another solution thenv <v. Asymptotics for˜ maximal solutions was obtained in [16–18]. It was also pointed out in [5,14,15,24,25,28] that (1.3) admits a variational structure: every solution of (1.3) is a critical point of the associated functional

F(v)=1

2∇v²_L2(Ω)+ 1 2²

Ω

e^u0+v−12

dx+4π N

|Ω|

Ω

v dx, v∈H¹(Ω).

Moreover, if we decompose in (1.3) v=w+c, c= 1

|Ω|

Ω

v dx,

we get the following quadratic equation e^2c

Ω

e^2u0+2wdx−e^c

Ω

e^u0+wdx+4π N ²=0, which implies that

w∈A≡

w∈H¹(Ω)

Ω

w dx=0,

Ω

e^u0+wdx 2

−16π N ²

Ω

e^2u0+2wdx0

.

Thus we may have two different variational formulations:

Forw∈A, define a constantc_±(w)by e^c±(w)=

Ωe^u0+wdx± (

Ωe^u0+wdx)²−16π N ²

Ωe^2u0+2wdx 2

Ωe^2u0+2wdx , (1.4)

so that F

w+c_±(w)

=J^±(w)+|Ω|

2²−2π N+4π Nln

8π N ² , where

J^±(w)=1

2∇w²2−4π Nln

Ω

e^u0+wdx− 4π N 1∓

1−²B(w)−4π Nln 1∓

1−²B(w) ,

B(w)=16π N

Ω

e^2u0+2wdx

Ω

e^u0+wdx 2

.

Once we find a critical pointw_±∈AofJ^±thenw_±+c_±(w_±)is a solution of (1.3). In particular, ifwis an interior infimum ofJ⁺thenw+c₊(w)is a local minimum ofF. Ifw is an interior infimum ofJ⁻, thenw+c₋(w) is a saddle point ofF. See [5,15,24,25,28] for details. The merit of this variational formulation is in analyzing the asymptotic behavior of solutions. In fact, in the case ofN=1, the Moser–Trudinger inequality enables us to find two interior infimumw_± ∈A for >0 sufficiently small [5,28]. Moreover, in this case,w₋ is uniformly bounded inH¹[28], and consequently, along a subsequence,u0+w₋ converges to a solution of the mean field equation as → +0. It is also proved in [15,24,25] that ifN=2, bothJ⁺andJ⁻attain global minimizers in the interior ofA. For this case, convergence to the solution of the mean field equation is not known [24,25].

For the caseN3, it was proved in [14] by the heat flow method that for >0 sufficiently small, (1.3) admits at least two solutionsv_1, andv_2, such thatv_1,→ −u₀ andu₀+v_2,→ −∞pointwisely almost everywhere as →0. However, asymptotics for solutions of (1.3) are not completely known forN2. We refer to [9,25] for this topic.

In this paper, we consider asymptotics of solutions of (1.1) whenN 3. We construct two kinds of solutions for (1.1) by the variational method for some values of Chern–Simons constant. One kind of solutions converges to 0 as tends to zero. This solutions become the maximal solutions whenis small enough. The other kind of solutions blows up at a single point in the sense of Brezis–Merle astends to zero. In particular, the blow-up solution we find is of spike type, that is, the maximum values of the exponential of the solutions remain bounded and the solutions converge to zero except the maximum point as the Chern–Simons constant tends to zero. Similar kind of spike solutions has been dealt with in the different area (see, for example [3,23,31] and references therein). Furthermore, whenN >3, it turns out that the blow-up solution is a critical point of the functionalJ⁺. It is well known [28] that, for >0 sufficiently small, the maximal solution is a critical point ofJ⁺. Therefore, it indicates that whenN >3,J⁺ may have more than one critical point and the structure of the solution space of (1.3) might be complicated. As a corollary of our main theorem, in the case that the distribution of the vortex points are periodic in a torus, we can show that there are solutions blowing up at several points in the sense of Brezis–Merle. Moreover, if the vortex points are distributed periodically with multiplicity 1 or 2, we show that there are at least three solutions for certain values of the Chern–Simons constant. In this respect, under the periodic distribution of single vortex, (1.1) shows all possibilities of Brezis–Merle type alternatives.

This paper is organized as follows. In Section 2, we find solutions for (1.3) for certain values ofby variational method. In Section 3, we present our main result, the asymptotics as→0 of the solutions we find using the results in Section 4. In Section 4, we develop typical blow-up alternatives for (1.2) following [1,2,4,24,25], which is used in Section 3.

2. Existence

Throughout this paper, we fix some notations and definitions. We let Z= {p₁, . . . , p_k} ⊂Ω the set of vortex points,m_j the multiplicities of the vortex pointsp_j,N=

jm_j1 as before. We also letGthe Green function for Ω satisfying

−_xG(x, y)=δ_y− 1

|Ω|, x, y∈Ω, and

Ω

G(x, y) dx=0

and γ (x, y)=G(x, y)+ _2π¹ ln|x −y| be the regular part of the Green function. It is obvious that u₀(x)=

−_k

j=14π mjG(x, p_j). Finally, we denote H_#¹=

v∈H¹(Ω)

Ω

v dx=0

,

J (v)=1

2∇v²2−4π Nln

Ω

e^u0+vdx forv∈H_#¹,

B(v)=16π N

Ωe^2u0+2vdx (

Ωe^u0+vdx)² forv∈H¹(Ω).

We also present the Green representation formula for a solutionvof (1.3) v= 1

|Ω|

Ω

v(y) dy+

Ω

⁻²G(x, y)

e^u0+v−e^2u0+2v

(y) dy. (2.1)

We note that for everyv∈H_#¹,B(v)16π N/|Ω|by the Hölder inequality. In fact, it is easy to show that, for any t >16π N/|Ω|, the set

S(t )=

v∈H_#¹|B(v)=t

is nonempty and thus weakly closed in H_#¹ by the Trudinger embedding theorem. Now, we borrow the following lemma from [24,25] to proceed.

Lemma 2.1.For everyv∈H_#¹and0< τ1,

Ω

e^u0+vdx B(v) 16π N

^1−τ

τ

Ω

e^{τ (u0+v)}dx ¹

τ

. (2.2)

This lemma could be shown by the Hölder inequality. For the sake of convenience, we denoteJ (t )≡infv∈S(t )J (v) from now on.

Lemma 2.2. For anyt >16π N/|Ω|,J (v)attains the infimum onS(t )and J (t )is continuous with respect tot >

16π N/|Ω|.

Proof. Letv∈S(t ). Takingτ=1/Nin (2.2) and using the Moser–Trudinger inequality, we have

Ω

e^u0+vdxCt^N−1

Ω

e^N1(u0+v)dx N

Ct^N−1exp 1

16π N∇v²2

.

This implies that J (v)1

4∇v²2−4π N (N−1)lnt−C. (2.3)

Thus,J is coercive onS(t )and attains the infimum onS(t ). Now let vt be a minimizer ofJ on S(t ). By direct calculation,

B(v_t)ϕ=2B(vt) ^Ωe^2u0+2vtϕ dx

Ωe^2u0+2vtdx −

Ωe^u0+vtϕ dx

Ωe^u0+vtdx

forϕ∈H¹. (2.4)

Hence,B(vt)=0. Chooseϕ∈H¹such thatB(vt)ϕ=1. Then, applying the implicit function theorem to the function a→B(v_t+aφ), we getε0>0 and

a:(−ε₀, ε₀)→R, da dε

ε=0

=1 (2.5)

such thatB(v_t+a(ε)ϕ)=t+εforε∈(−ε₀, ε₀). Thus, J (t+ε)J (v_t+a(ε)ϕ)→J (v_t)

as ε→0 by the continuity ofJ (v). That is, lim sup_ε→0J (t+ε)J (t ). Similar argument replacingv_t withv_t+ε givesJ (t )lim infε→0J (t+ε), which shows the continuity. 2

Lemma 2.3.J (t )= −2π N (N−2)lnt+O(1)forN2ast→ ∞.

Proof. Letv∈S(t ). As in Lemma 2.2, we plugτ=2/N1 into (2.2) to have

Ω

e^u0+vdxCt^N2−2

Ω

e^N2(u0+v)dx ^N

2

Ct^N−22 exp 1

8π N∇v²₂

. (2.6)

Then, (2.6) implies that

J (v)−2π N (N−2)lnt−C.

We show that the growth rate−2π N (N−2)is sharp in the above inequality. Without loss of generality, we may assume thatu₀attains a maximum at the origin. Fix a constantr >0 such that the ballB_2r(0)⊂Ω. Letχ∈C₀^∞(R²) be a cut-off function such thatχ≡1 onBr(0), andχ≡0 on[B2r(0)]^c. Consider the test function

ϕε(x)= −χ (x)ln

|x|²+ε²N

, ε >0. (2.7)

It is easily checked that asε→0,

∇ϕε²2=

|x|r

4N²|x|²

(|x|²+ε²)²dx+O(1)=8π N²ln1

ε+O(1),

Ω

e^u0+ϕεdx=

|x|r

e^u0(x)

(|x|²+ε²)^N dx+O(1)

=

|y|r/ε

ε^2−2Ne^u0(εy)

(|y|²+1)^N dy+O(1)=Cε^2−2N+O(1),

Ω

e^2u0+2ϕεdx=

|x|r

e^2u0(x)

(|x|²+ε²)^2N dx+O(1)

=

|y|r/ε

ε^2−4Ne^2u0(εy)

(|y|²+1)^2N dy+O(1)=Cε^2−4N+O(1), and

Ωϕ_εdx=O(1). Letϕ¯_ε=ϕ_ε−_|Ω¹_|

Ωϕ_εdx∈H_#¹. Since 0 is a maximum point ofu0, we have|e^u0(εy)−e^u0(0)| Cε²|y|²for|y|r/εand hence

B(ϕ¯_ε)=B(ϕ_ε)=16π N Cε^2−4N+O(1)

(Cε^2−2N+O(1))² =C₀ε⁻²+O(1).

Thus, fort sufficiently large, we can chooseε∼√

C₀/t >0 such thatt=B(ϕ_ε). Then inf

B(v)=tJ (v)J (ϕ¯ε)−4π N (N−2)ln1

ε+C−2π N (N−2)lnt+C ast→ ∞. 2

For a constantμ >0, we define a functionalI_μ:H_#¹→Rby I_μ(v)=J (v)+B(v)

μ . (2.8)

Lemma 2.4.For eachμ >0,I_μis coercive inH_#¹and there exists a global minimizer ofI_μ. Proof. As in Lemma 2.2, we setτ=1/N in (2.2) and repeat the calculation. Then, for allv∈H_#¹,

I_μ(v)1

4∇v²₂−4π N (N−1)lnB(v)+B(v) μ −C

1

4∇v²2+ inf

t >16π N/|Ω|

(t /μ)−4π N (N−1)lnt

−C

1

4∇v²₂−4π N (N−1)lnμ−C,

whereC depends only onΩ andZ. Thus Iμis bounded from below and coercive inH_#¹. SinceIμis lower semi- continuous, there exists a minimizer for eachμ >0. 2

For eachμ >0, letv_μ∈H_#¹be a minimizer ofIμ. By the Lagrange multiplier theorem, the variational equation forIμis given by

v_μ= 2

μB( v_μ) e^2u0+2vμ

Ωe^2u0+2vμdx − 4π N+ 2 μB( v_μ)

e^u0+vμ

Ωe^u0+vμdx+4π N

|Ω| onΩ. (2.9)

Lemma 2.5. B( v_μ)is strictly increasing with respect to μ. Furthermore, when N 3, there exist two constants C₁, C₂>0depending only onΩ andZsuch thatC₁μB( v_μ)C₂μforμsufficiently large.

Proof. First, givenμ₁> μ₂>0, I_μ1( v_μ

1)I_μ1( v_μ

2)=I_μ2( v_μ

2)+ 1

μ1 − 1 μ2

B( v_μ

2) Iμ₂( v_μ1)+ 1

μ₁− 1 μ₂

B( v_μ2)

by the minimizing property ofv_μ. However, I_μ1( v_μ

1)−I_μ2( v_μ

1)= 1

μ₁− 1 μ₂

B( v_μ

1).

From this, we deduce thatB( v_μ)is monotonically increasing with respect toμ. Since the equality holds only when I_μ1( v_μ

1)=I_μ1( v_μ

2), the equality implies thatv_μ

2is a minimizer of bothI_μ1 andI_μ2. But then, for anyφ∈H_#¹, B( v_μ

2)φ= μ₁μ₂ μ₂−μ₁(I_μ

1−I_μ

2)( v_μ

2)φ=0,

which is a contradiction by (2.4). Therefore,B( v_μ)is strictly increasing andv_μ

1=v_μ

2 ifμ₁=μ₂. Next, whenN3,

I_μ( v_μ) inf

B(v)=μI_μ(v)= inf

B(v)=μJ (v)+1−2π N (N−2)lnμ+C forμsufficiently large by Lemma 2.3. On the other hand, (2.6) implies that

I_μ( v_μ)=J ( v_μ)+B( v_μ)

μ −2π N (N−2)ln B( v_μ) μ

+B( v_μ)

μ −2π N (N−2)lnμ−C.

Consequently, it follows that B( v_μ)

μ −2π N (N−2)ln B( v_μ) μ

C

forμsufficiently large. Then, we haveC₁< B( v_μ)/μ < C₂for someC₁,C₂>0 from the asymptotics of the function t→t−2π N (N−2)lnt. 2

Theorem 2.1.LetN3,v_μbe a minimizer ofIμas before, and =_μ≡

8π N

μ 2π N+B( v_μ) μ

₋1

(2.10) for someμ >0. Then, there exist a solutionu_μ∈H¹for(1.3)with=μ. Furthermore,B( u_μ)→ ∞asμ→ ∞ and

μlim→0_μ= lim

μ→∞_μ=0.

Proof. By Lemma 2.4 and 2.5, for anyμ >0, there existsv_μsatisfying (2.9). Let us definec_μ∈Randu_μ∈H¹(Ω) by

c_μ=ln

Ωe^u0+vμdx

Ωe^2u0+2vμdx +ln B( v_μ) 2π N μ+B( v_μ)

, (2.11)

u_μ=v_μ+cμ. (2.12)

Then, by direct calculation,u_μis a solution of (1.3) with=_μ. SinceC₁μ < B( v_μ) < C₂μfor large enoughμand B( u_μ)=B( v_μ), it follows from (2.10) that

μlim→0μ= lim

μ→∞μ=0, lim

μ→∞B( u_μ)= ∞. 2

It is easily checked thatu_μ∈H¹is a critical point ofF with=_μ, andc_μ=(1/|Ω|)

Ωu_μdx. Thenc_μ= c₊( v_μ)orc_μ=c₋( v_μ), wherec_±is defined in (1.4). We will prove in Section 3 thatc_μ=c₊( v_μ)forμsufficiently large andN >3, and consequently,v_μis a critical point ofJ⁺with=_μ.

3. Asymptotics of the solutions

In this section, we study the asymptotic behavior ofu_μ=v_μ+c_μasμ→0 andμ→ ∞. We first present some preliminary facts.

Lemma 3.1.Letu∈H¹be a solution of (1.1). Then,u=v+u₀0and

Ω

1 ²e^u

1−e^u

=4π N.

The lemma is well known (see, for example, [5,29]) and can be shown simply by the maximum principle. Now, we consider (1.1) on the whole ofR²when the distribution of vortex points,Z= {0}.

Lemma 3.2.Letmbe a nonnegative integer, andube a(smooth)solution of the following equation u=e^u

e^u−1

+4π mδp=0 inR². (3.1)

Ife^u(e^u−1)∈L¹(R²), either (i) u(x)→0as|x| → ∞, or

(ii) u(x)= −βln|x| +O(1)near∞, where β= −2m+ 1

2π

R²

e^u 1−e^u

dx.

Assume thatusatisfies the boundary condition(ii). Then we have

R²

e^2udx=π

β²−4β−4m²−8m and

R²

e^udx=π

β²−2β−4m²−4m

. (3.2)

In particular,

R²e^u(1−e^u) dx >8π(1+m).

Proof. This lemma might be well-known. But since we cannot find its’ proof in the literature, we present the sketch of the proof here following the argument in [12]. Sincee^u(e^u−1)∈L¹(R²), the argument of [4] implies thatuis bounded from above andu∈C_loc(R²\{0}). Moreover, by [12],u(x)= −βln|x| +O(1)near∞for some constant β∈Randu=2mln|x|+O(1)near the origin. Then it follows from theL¹-condition and elliptic estimates that either β =0 orβ >2. In the case thatβ=0, we arrive at (i) by theL¹-condition. In the case thatβ >2, we further have

∇u(x)= −β ^x

|x|² +o(|x|⁻¹)near∞by [12]. Multiplying (3.1) byx· ∇uand integrating on the domainΣ= {x|r <

|x|< R}, we obtain

∂Σ

1

2(x·ν)|∇u|²−(x· ∇u)(ν· ∇u)+(x·ν) 1

2e^2u−e^u

dσ=

Σ

e^2u−2e^u dx.

Letting r→0 andR→ ∞, we obtain

R²(2e^u−e^2u) dx=π(β²−4m²). Meanwhile, integrating (3.1) onΣ and lettingr→0 andR→ ∞, we have

R²(e^u−e^2u) dx=π(4m+2β). Thus, (3.2) immediately follows. Then the first identity in (3.2) implies thatβ >2m+4, which in turn implies thate^u(e^u−1)_L¹_(R²₎>8π(1+m). 2

If u is a solution of (3.1) with m=0 and e^u(e^u−1)∈L¹(R²), we further have the following lemma due to [7,10,27].

Lemma 3.3.Letube a solution of (3.1)withm=0 ande^u(e^u−1)∈L¹(R²). Then,uis radially symmetric and smooth. Letu(r;s)be the radial solution of (3.1)such thatlimr→0u(r;s)=s and limr→0u_r(r;s)=0. Then we further obtain

(a) u(· ;0)=0,

(b) Ifs <0,u(r;s)→ −∞asr→ ∞,

(c) Ifs >0,u(r;s)blows up at somer=r(s) >0.

Moreover, if we define a functionξ₀:(−∞,0)→R₊=(0,∞)by ξ₀(s)=

∞

0

e^u(r;s)

1−e^u(r;s)

r dr (3.3)

thenlims→0⁻ξ0(s)= ∞,lims→−∞ξ0(s)=4, and ξ0 is continuously differentiable and strictly increasing on the interval(−∞,0).

The following is an analogy of the Brezis–Merle type alternatives [1,2,4,24,25] for (1.3). It is not only interesting in itself but also will be used frequently in this section.

Theorem 3.1.Letv,→0be a sequence of solutions of (1.3). Then, up to subsequences, one of the following holds true:

(i) v→ −u₀inC_loc(Ω\Z), or

(ii) v−2 lnis bounded uniformly inC⁰(Ω), or

(iii) lim supsup_Ω(u₀+v) <0and there exist a nonempty finite setS= {q₁, . . . , q_l} ⊂Ωandlnumber of sequences of pointsx_j,→q_j,j=1, . . . , lsuch that

(v−2 ln)(xj,)→ ∞

for anyj=1, . . . , landv−2 ln→ −∞uniformly on any compact subset ofΩ\S. Moreover, 1

²e^u0+v

1−e^u0+v

→

αjδq_j, αj8π in the sense of measure.

The proof of the above theorem is a bit technical, so we postpone it to Section 4.

In view of the above theorem, we define the blow-up solutions for (1.2) as follows.

For a sequence of solutions{w}of (1.2), if there existq∈Ω andx∈Ωsatisfying w(x)→ ∞, x→q

as→0, we call{w}blow-up solutions of (1.2) following Brezis–Merle [4]. Also, we callq a blow-up point and call the collection of all blow-up points of{w}the blow-up set for{w}.

Now, we consider the asymptotics whenμ→0.

Lemma 3.4.Letv_μbe as in Section2.B( v_μ)converges to16π N/|Ω|asμ→0.

Proof. Given anyδ >0, letϕbe a smooth function such thatB(ϕ) <16π N/|Ω| +δ. Then (2.3) implies that B( v_μ)

μ −4π N (N−1)lnB( v_μ)−CI_μ( v_μ)I_μ(ϕ)=J (ϕ)+B(ϕ) μ , which in turn implies that

B( v_μ)−4π N (N−1)μlnB( v_μ)B(ϕ)+Cμ.

SinceB( v_μ)is monotone by Lemma 2.5, lettingμ→0 in the above inequality, we get lim sup

μ→0

B( v_μ)B(ϕ) <16π N/|Ω| +δ.

However,B( v_μ)16π N/|Ω|andδ >0 is arbitrary, Lemma 3.4 immediately follows. 2

The following theorem tells that{u0+u_μ}satisfies the first alternative in Theorem 3.1 asμ→0.

Theorem 3.2.u₀+u_μL^∞(K)→0asμ→0for any compact subsetKofΩ\Z.

Proof. We argue by contradiction, and suppose that there exists a sequence of μ’s (still denoted by μ) such that μ→0 and{u₀+u_μ}does not satisfy the alternative (i) in Theorem 3.1. Let_μas in (2.10). Thenu_μ−2 ln_μis a solution of the following equation.

v= −e^u0+v

1−²e^u0+v +4π N

|Ω| .

If (ii) of Theorem 3.1 is the case, RHS of the above equation is uniformly bounded. Thus, by the elliptic theory, we arrive that u_μ−2 ln_μconverges uniformly to a smooth functionφ up to subsequences. Then,B( v_μ)=B( u_μ− 2 ln_μ)→B(φ). However,B(φ) >16π N/|Ω|for any smoothφ, which contradicts Lemma 3.4.

If (iii) of Theorem 3.1 is the case, there exists a blow-up setS= {q1, . . . , ql}ande^{uμ−2 lnμ}→0 inC_loc⁰ (Ω\S)up to subsequences. Thus, denotingwμ=u0+u_μ−2 lnμ, for anyr >0 small enough,

Ω

e^wμ=

B_r(q_i)

e^wμ+o(1)Cr

B_r(q_i)

e^2wμ 1/2

+o(1)Cr

Ω

e^2wμ 1/2

+o(1).

But then, sincee^wμL¹(Ω)4π Nby Lemma 3.1, B( v_μ)=16π N

Ωe^2wμ (

Ωe^wμ)² C/r²

asμ→0. Takingrsmall enough, we are led to a contradiction. Theorem 3.2 is proved. 2

The following theorem follows from the uniqueness of the solution of (1.1) near the maximal solutions in [13].

Theorem 3.3.Forμ >0sufficiently small, the functionμ→B( v_μ)is continuous and{u_μ}becomes the continuous family of maximal solutions. Thus, there exists a constantμ₀>0such that if=_μfor someμ > μ₀, we have two solutions for(1.1).

Proof. By Lemma 3.2 and [13], whenμ >0 is small enough,u_μmust be the maximal solution of (1.3) for=μ. Therefore, there exists a constantμ1such that the mappingsμ→B( v_μ)andμ→μare (single-valued) continuous forμ < μ1. Then, by Theorem 2.1, there always existsμ < μ1for any < μ₁ such thatμ=. Meanwhile, there existsμ0μ1such thatμ₀ < μ₁ by Theorem 2.1. Consequently, if=μwithμ > μ0, we have two solutions for (1.1), one withμ < μ1and the other withμ > μ0. 2

We now concentrate on the other situation, μ→ +∞. In this case, Lemma 2.5 imply that either (ii) or (iii) of Theorem 3.1 is the case and thus there is a constantν=ν(Ω,Z) >0 such that

sup

μ>1

sup

Ω

(u₀+u_μ)−ν. (3.4)

It will turn out that (iii) of Theorem 3.1 holds in this case. Moreover, the blow-up set consists of a single pointq, which should be a maximum point ofu0. To prove it, we need the following lemma dealing with a special case of (iii) of Theorem 3.1, blow-up away from the vortex points.

Lemma 3.5.Letw=v−2 lnbe the blow-up sequence in(iii)of Theorem3.1andq_j,α_jas in(iii)of Theorem3.1.

Assume thatq_j ∈/Z. Then, givenr >0small enough, there exist a constantC >0and a sequence of points{x} ⊂ B_r(q_j)with the property that

w(x)= max

|x−qj|rw(x)→ ∞ as→0 (3.5)

and

|x−maxqj|r

w(x)+2 ln|x−x|

C. (3.6)

Moreover, for any sequence{R}such thatR→ ∞,

lim→0

|y−x|Rs

e^w

1−²e^w

(y) dy=α_j (3.7)

wheres=exp[−¹₂w(x)].

Proof. See Section 4. 2

Now, we are ready to show our main result.

Theorem 3.4.Assume thatN3andu_μ,v_μas before.

(i) Asμ→ ∞, along a subsequence, u_μ−2 ln_μ→ −∞ uniformly on any compact set K⊂Ω\{q}for some q∈Ω, and

1

_μ²e^u0+uμ

1−e^u0+uμ

→4π N δq in the sense of measure.

Furthermore,u₀(q)=maxΩu₀. (ii) limμ→∞B( v_μ)

μ =2π N (N−2).

(iii) v_μis a critical point of the functionalJ⁺with=_μprovided thatN >3andμis sufficiently large.

Proof. We first show (i). We break it into several steps.

Step1. maxΩ( u_μ−2 ln_μ)→ ∞, and hence∇u_μ2→ ∞.

If not, there would be a sequence(still denoted byμ) such thatμ→ ∞and maxΩ( u_μ−2 ln_μ)C for some constantC >0. Then case (ii) of Theorem 3.1 must hold true. That is, along a subsequence,{u_μ−2 ln_μ}is bounded inC⁰(Ω). It follows that B( v_μ)=B( u_μ−2 ln_μ)C, which contradicts Lemma 2.5 and shows Step 1. Step 1 implies that case (iii) of Theorem 3.1 holds true foru_μ. In particular, we obtain that∇u_μ2→ ∞.

Step2.|S| =1.

We argue by contradiction, and suppose that, there is a sequence still denoted by u_μ which blows up at more than two points. LetS= {q₁, . . . , q_l}be the blow-up set foru_μwithl2. We take a small constantr >0 such that B_2r(q_i)’s are mutually disjoint. It follows from Theorem 3.1 and Green’s representation formula (2.1) that

v_μ=

Ω

1

_μ²G(x, y)

e^u0+uμ−e^2u0+2uμ (y) dy

and

v_μ→ l

i=1

α_iG(x, q_i), α_i8π (3.8)

inC_loc¹ (Ω\S). In particular, v_μ is bounded in C¹(Ω\_l

i=1B_r(q_i)). Moreover, Theorem 3.1 imply that there is a positive constantc₀independent ofμsuch that

c0

B_r(q_i)

1

_μ²e^u0+uμdx < 1 c₀.