Periodic Maxwell-Chern-Simons vortices with concentrating property

TL;DR

This paper analyzes the Maxwell-Chern-Simons model to establish the existence of periodic vortices with concentrating properties, providing new insights into the uniform Chern-Simons limit and addressing open problems in vortex theory.

Contribution

It derives the uniform Chern-Simons limit of the Maxwell-Chern-Simons model without restrictions and proves the existence of periodic vortices with concentrating density properties.

Findings

Established the uniform (CS) limit of the (MCS) model.

Proved existence of periodic Maxwell-Chern-Simons vortices with concentration properties.

Provided a key relation between Higgs and neutral scalar fields.

Abstract

In order to study electrically and magnetically charged vortices in fractional quantum Hall effect and anyonic superconductivity, the Maxwell-Chern-Simons (MCS) model was introduced by [Lee, Lee, Min (1990)] as a unified system of the classical Abelian-Higgs model (AH) and the Chern-Simons (CS) model. In this article, the first goal is to obtain the uniform (CS) limit result of (MCS) model with respect to the Chern-Simons parameter without any restriction on either a particular class of solutions or the number of vortex points. The most important step for this purpose is to derive the relation between the Higgs field and the neutral scalar field. Our (CS) limit result also provides the critical clue to answer the open problems raised by [Ricciardi,Tarantello (2000)] and [Tarantello (2004)], and we succeed to establish the existence of periodic Maxwell-Chern-Simons vortices satisfying the concentrating property of the density of superconductive electron pairs. Furthermore, we expect that the (CS) limit analysis in this paper would help to study the stability, multiplicity, and bubbling phenomena for solutions of the (MCS) model.

Equations

$${\mathcal{L}}^{MCS}(A,\varphi ,\mathfrak{n})$$

$$-|\varphi {|}^2{\left(\mathfrak{n}-\frac{q^2}{\mu }\right)}^2-\frac{q^2}{2}{\left(|\varphi {|}^2-\frac{\mu }{q^2}\mathfrak{n}\right)}^2,$$

$$V(|\varphi |,\mathfrak{n})=|\varphi {|}^2{\left(\mathfrak{n}-\frac{q^2}{\mu }\right)}^2+\frac{q^2}{2}{\left(|\varphi {|}^2-\frac{\mu }{q^2}\mathfrak{n}\right)}^2.$$

$${\mathcal{L}}^{AH}(A,\varphi )=-\frac{1}{4q^2}{F}_{\alpha \beta }{F}^{\alpha \beta }+D_{\alpha }\varphi \overline{\left(D^{\alpha }\varphi \right)}-\frac{q^2}{2}{\left(|\varphi {|}^2-1\right)}^2,$$

$${\mathcal{L}}^{CS}(A,\varphi )=-\frac{\mu }{4q^2}{\epsilon }^{\alpha \beta \gamma }{A}_{\alpha }{F}_{\beta \gamma }+D_{\alpha }\varphi \overline{\left(D^{\alpha }\varphi \right)}-\frac{q^4}{{\mu }^2}|\varphi {|}^2{\left(|\varphi {|}^2-1\right)}^2,$$

$$\left\{\begin{array}[]{l}D_{\alpha}D^{\alpha}\phi=-\frac{\partial V}{\partial\bar{\phi}},\\ \frac{1}{q^{2}}\partial_{\alpha}\partial^{\alpha}\mathfrak{n}=-\frac{\partial V}{\partial\mathfrak{n}},\\ \frac{1}{q^{2}}\partial_{\beta}F^{\alpha\beta}+\frac{\mu}{2q^{2}}\epsilon^{\alpha\beta\gamma}F_{\beta\gamma}=J^{\alpha},\end{array}\right.$$

$$\left(\psi ,B_{\alpha },{\mathfrak{n}}_2\right)=\left(e^{i\omega }\varphi ,A_{\alpha }+{\partial }_{\alpha }\omega ,{\mathfrak{n}}_1\right),\alpha =0,1,2.$$

$$\Omega =\left\{x\in {\mathbb{R}}^2|x=t_1{\mathbf{a}}_1+t_2{\mathbf{a}}_2,t_1,t_2\in (0,1)\right\},$$

$$\left\{\begin{array}[]{ll}{A_{j}\left(x+\mathbf{a}_{k}\right)=A_{j}(x)+\partial_{j}\omega_{k}(x),}\quad{j,k=1,2,}\\ {A_{0}\left(x+\mathbf{a}_{k}\right)=A_{0}(x)},\\ {\phi\left(x+\mathbf{a}_{k}\right)=e^{-i\omega_{k}(x)}\phi(x)},\\ {\mathfrak{n}\left(x+\mathbf{a}_{k}\right)=\mathfrak{n}(x),}\quad k=1,2,\end{array}\right.$$

$${\omega }_1\left(0,0^{+}\right)-{\omega }_1\left(0,1^{-}\right)+{\omega }_2\left(1^{-},0\right)-{\omega }_2\left(0^{+},0\right)=2\pi \mathfrak{M},$$

$$\left\{\begin{array}[]{l}{\left(D_{1}+iD_{2}\right)\phi=0}\\ {F_{12}=q^{2}|\phi|^{2}-\mu\mathfrak{n}}\\ {-A_{0}=\mathfrak{n}-\frac{q^{2}}{\mu}}\\ {-\Delta A_{0}+\mu F_{12}=-2q^{2}A_{0}|\phi|^{2}}\end{array}\right.$$

$$\left\{\begin{array}[]{l}\Delta u=\lambda\mu e^{u}-\mu N+4\pi\sum_{i=1}^{n}m_{i}\delta_{p_{i}},\\ \Delta N=\mu(\mu+\lambda e^{u})N-\lambda\mu(\mu+\lambda)e^{u},\end{array}\right.\textrm{in}\ \ \Omega.$$

$$\left(u_{\lambda ,\mu },\frac{N_{\lambda ,\mu }}{\lambda }\right)\to (0,1)\text{mbox}a.e.on\Omega \text{mbox}as\lambda \to \infty ,\mu \gg \lambda ,$$

$$\left(u_{\lambda ,\mu },\frac{N_{\lambda ,\mu }}{\lambda }\right)\to (-\infty ,0)\text{mbox}a.e.on\Omega \text{mbox}as\lambda \to \infty ,\mu \gg \lambda ,$$

$$\underset{\lambda ,\mu \to \infty ,\frac{\lambda }{\mu }\to 0}{\lim }{\Vert e^{u_{\lambda ,\mu }}-\frac{N_{\lambda ,\mu }}{\lambda }\Vert }_{L^{\infty }(\Omega )}=0.$$

$${\lambda }^2{e}^{u_{\lambda ,\mu }}\left(1-\frac{N_{\lambda ,\mu }}{\lambda }\right)\to \sum _j{\alpha }_j{\delta }_{{\hat{q}}_j},{\alpha }_j\ge 8\pi ,$$

$$\left\{\begin{array}[]{l}\Delta\left(u+\frac{N}{\mu}\right)=-\lambda^{2}e^{u}\left(1-\frac{N}{\lambda}\right)+4\pi\sum_{i=1}^{n}m_{i}\delta_{p_{i}},\\ \Delta N=\mu^{2}(1+\frac{\lambda}{\mu}e^{u})N-\lambda\mu^{2}\left(1+\frac{\lambda}{\mu}\right)e^{u}\end{array}\right.\mbox{ in }\Omega.$$

$$\Delta u=-{\lambda }^2{e}^u\left(1-e^u\right)+4\pi \sum _{i=1}^n{m}_i{\delta }_{p_i}\text{mbox}in\Omega .$$

$${\int }_{\Omega }{\lambda }^2{e}^u(1-\frac{N}{\lambda })=4\pi \mathfrak{M}.$$

$$\Delta w+e^w(1-e^w)=4\pi m{\delta }_0\text{in}{\mathbb{R}}^2.$$

$${\int }_{{\mathbb{R}}^2}{e}^{2w}dx=\pi ({\beta }^2-4\beta -4m^2-8m),\text{and}{\int }_{{\mathbb{R}}^2}{e}^{w}dx=\pi ({\beta }^2-2\beta -4m^2-4m).$$

$$\underset{r\to 0}{\lim }w(r;s)=s,\text{and}\underset{r\to 0}{\lim }{w}^{\prime }(r;s)=0,$$

$$\beta (s)\equiv \frac{1}{2\pi }{\int }_{{\mathbb{R}}^2}{e}^{w(r;s)}(1-e^{w(r;s)})dx={\int }_0^{\infty }{e}^{w(r;s)}(1-e^{w(r;s)})rdr.$$

$$\underset{s\to -\infty }{\lim }\beta (s)=4,\text{ and }\underset{s\to 0_{-}}{\lim }\beta (s)=+\infty .$$

$$u_{\lambda ,\mu }(x)<0,0<N_{\lambda ,\mu }(x)<\lambda \text{for any}x\in \Omega .$$

$${\int }_{\Omega }{\lambda }^2{e}^{u_{\lambda ,\mu }}\left(1-\frac{N_{\lambda ,\mu }}{\lambda }\right)dx={\int }_{\Omega }{\lambda }^2{e}^{u_{\lambda ,\mu }}\left|1-\frac{N_{\lambda ,\mu }}{\lambda }\right|dx=4\pi \mathfrak{M}.$$

$$\underset{\lambda \to \infty }{\lim }\left[{\int }_{\Omega }{e}^{u_{\lambda ,\mu }}\left|1-\frac{N_{\lambda ,\mu }}{\lambda }\right|dx\right]=0,$$

$$\text{either}{u}_{\lambda ,\mu }\to -\infty \text{or}\frac{N_{\lambda ,\mu }}{\lambda }\to 1\text{a.e. in}\Omega \text{as}\lambda \to \infty .$$

$${\int }_{\Omega }\left(1+\frac{\lambda }{\mu }{e}^{u_{\lambda ,\mu }}\right)\frac{N_{\lambda ,\mu }}{\lambda }dx={\int }_{\Omega }\left(1+\frac{\lambda }{\mu }\right)e^{u_{\lambda ,\mu }}dx.$$

$$-\Delta v=f\text{in}D,$$

$$(i)\text{if}\underset{\partial D}{\sup }v\le C,\text{then}\underset{D^{\prime }}{\sup }v\le \sigma \underset{D^{\prime }}{\inf }v+(1+\sigma )\tau \Vert f{\Vert }_{L^p}+(1-\sigma )C,$$

Full text

1. Introduction

As the pioneering work by Ginzburg and Landau, the classical Abelian-Higgs (AH) model (or, Maxwell-Higgs) was proposed in order to describe the superconductivity phenomena at low temperature (see [4, 35, 39, 47]). This model has been studied in [6, 35, 56, 59] for various domains. However, (AH) model can only describes electrically neutral vortices, which are static solutions of the corresponding Euler-Lagrange equation. In order to study the fractional quantum Hall effect and high temperature superconductivity, we should investigate electrically and magnetically ”charged” vortices. For this purpose, one might attempt to include Chern-Simons (CS) term into (AH) model. However, just adding (CS) term into (AH) model loses the self-dual structure, which is characterized by a special class of static solution corresponding to a constrained energy minimizer. The self-dual equation has a benefit in the gauge field theory since it is a reduced first-order equation, so called ”Bogomol’nyi equation”, for the more complicated second order equation of motion (see [5, 43]). In order to obtain a self-dual Chern-Simons theory, Hong-Kim-Pac in [33] and Jackiw-Weinberg in [34] independently proposed a model for charged vortices with electrodynamics governed only by the (CS) term without Maxwell term, which was included in the (AH) model. This pure (CS) model was suggested from the observation such that the (CS) term is dominant over the Maxwell term in the large scale. During the last few decades, the (CS) model has been extensively studied in [13, 14, 18, 51, 52, 58, 60] for entire solutions on a full space, in [8, 18, 19, 21, 22, 23, 27, 28, 29, 46, 53, 54] for the periodic case, and in [32] for bounded domains (see also [9, 10, 17, 30, 36, 37, 38, 50]).

As stated above, a naive inclusion of both (AH) term and (CS) term in the Lagrangian fails to make the system self-dual. However, in [40], Lee, Lee, and Min succeeded in restoring the self-duality in Maxwell-Chern-Simons (MCS) model as a unified self-dual system of (AH) and (CS), by introducing a neutral scalar field. Moreover, the authors in [40] showed formally that the self-dual equation of (MCS) owns both (AH) model and (CS) model as limiting problems according to the limit behavior of the electric charge and the Chern-Simons mass scale (see also [24]). This formal argument in [40, 24] could be supported with mathematically rigorous proof in [11, 12, 48, 49]. In [11], Chae and Kim established the existence of topological multivortex solution for (MCS) model in a full space $\mathbb{R}^{2}$. Here, the topological entire solution in $\mathbb{R}^{2}$ satisfies the specific boundary condition such that its first component vanishes at infinity. Moreover, the authors in [11] showed the convergence of topological multivortex solutions to the (CS) model and (AH) model. The convergence depends on the asymptotic behavior of the electric charge and the Chern-Simons mass scale. In [12], they also obtained the corresponding result for topological solutions on a flat two torus (see (1.7) for the definition of topological solution on a flat two torus). In [49], Ricciardi and Tarantello showed that there exist at least two gauge distinct periodic multivortices (topological solution and mountain pass solution), and analyzed their asymptotic behavior in terms of the (CS) limit and the (AH) limit. Moreover, Ricciardi in [48] obtained the stronger convergence result for an arbitrary sequence of periodic multivortices while the Chern-Simons parameter, which is the ratio between electric charge and the Chern-Simons mass scale, is fixed.

In this article, one of main goals is to improve the (CS) limit analysis for (MCS) model without any restriction on either a particular class of solutions, the number of vortex points, or the Chern-Simons parameter. Moreover, in view of our first result, we could also obtain the affirmative answers for the open problems raised by Ricciardi and Tarantello in [49], and Tarantello in [55].

In order to introduce our results more precisely, let us recall the Lagrangian density $\mathcal{L}^{MCS}$ for the (MCS) model, which is defined in the $(2+1)$-dimensional Minkowski space $\mathbb{R}^{2,1}$ with the metric $\textrm{diag}(1,-1,-1)$:

[TABLE]

where the metric is used to raise or lower indices, all the Greek indices run over $0,1,2$, and $\epsilon^{\alpha\beta\gamma}$ is the totally skew-symmetric tensor fixed so that $\epsilon^{012}=1$. Here, $\phi:\mathbb{R}^{1+2}\to\mathbb{C}$ is the complex valued Higgs field, $\mathfrak{n}:\mathbb{R}^{1+2}\to\mathbb{R}$ is the neutral scalar field, $A_{\alpha}:\mathbb{R}^{1+2}\to\mathbb{R}$ is the gauge field, $D_{\alpha}=\partial_{\alpha}-iA_{\alpha}$ is the gauge covariant derivative associated with $A_{\alpha}$ where $i=\sqrt{-1}$, and $F_{\alpha\beta}=\partial_{\alpha}A_{\beta}-\partial_{\beta}A_{\alpha}$ is the field strength. The constant $q>0$ denotes the electric charge and $\mu>0$ is the Chern-Simons mass scale. The gauge potential field $\mathcal{A}$ with a 1-form (connection) is identified as $\mathcal{A}=-iA_{\alpha}dx^{\alpha}$, and the Maxwell gauge field $F_{\mathcal{A}}$ is expressed by $F_{\mathcal{A}}=\mathrm{d}\mathcal{A}=-\frac{i}{2}F_{\alpha\beta}\mathrm{d}x^{\alpha}\wedge\mathrm{d}x^{\beta}$ is expressed by the 2-form (curvature) . Let us denote the self-dual potential by

[TABLE]

Note that in $\mathcal{L}^{MCS}$, the Maxwell term for $\mathcal{A}$ is denoted by $F_{\alpha\beta}F^{\alpha\beta}$ and the Chern-Simons term is represented by the quantity $\frac{\mu}{4q^{2}}\epsilon^{\alpha\beta\gamma}A_{\alpha}F_{\beta\gamma}$. Indeed, the Lagrangian of the (AH) model and the (CS) model are given by

[TABLE]

and

[TABLE]

respectively. If we fix $q$, and assume the identity $\mathfrak{n}=\frac{q^{2}}{\mu}$ in (1.1), then as $\mu\to 0$, a limiting Lagrangian for $\mathcal{L}^{MCS}$ formally would be $\mathcal{L}^{{AH}}$. On the other hand, if we fix $\frac{q^{2}}{\mu}$, and insert the identity $\mathfrak{n}=\frac{q^{2}}{\mu}|\phi|^{2}$ into the potential of $\mathcal{L}^{MCS}$, then as $\mu\to\infty$, a limiting Lagrangian for $\mathcal{L}^{MCS}$ formally would be $\mathcal{L}^{{CS}}$.

The periodic patterns of vortex configurations have been predicted and founded in the experiment for the study of superconductivity (see [1]). Periodic vortices (or condensates) relative to (1.1) are defined as the static solutions, which is independent of the $x^{0}$-variable, for the following Euler-Lagrangian equations subject to the ’t Hooft type periodic boundary conditions (see [57]) :

[TABLE]

where $J^{\alpha}=i(\bar{\phi}D^{\alpha}\phi-\phi\overline{(D^{\alpha}\phi)})$ is the conserved current for the system. We say that $\left(\phi,A_{\alpha},\mathfrak{n}_{1}\right)$ is gauge equivalent to $\left(\psi,B_{\alpha},\mathfrak{n}_{2}\right),$ if there exists a smooth function $\omega$ satisfying

[TABLE]

The ’t Hooft type periodic boundary conditions are required for the invariance of (1.2) with respect to the gauge transformation. More precisely, the periodic cell domain is given by

[TABLE]

where $\mathbf{a}_{1}$ and $\mathbf{a}_{2}$ are linearly independent vectors in $\mathbb{R}^{2}.$ Let $\Gamma_{k}=\left\{x\in\mathbb{R}^{2}\ |\ x=t_{k}\mathbf{a}_{k},\ t_{k}\in(0,1)\right\},k=1,2,$ be a part of the boundary of $\Omega$. We assume that $(A,\phi,\mathfrak{n})$ is a static (that is, independent of the $x^{0}$-variable) solution of (1.2), and there exist smooth functions $\omega_{k}$, ($k=1,2$) in a neighborhood of $\Gamma_{1}\cup\Gamma_{2}\backslash\Gamma_{k}$, satisfying

[TABLE]

for $x\in\Gamma_{1}\cup\Gamma_{2}\backslash\Gamma_{k}$, $k=1,2$. We set $\omega_{k}\left(s^{1},s^{2}\right)=\omega_{k}\left(s^{1}\mathbf{a}_{1},s^{2}\mathbf{a}_{2}\right),k=1,2$ so that $\phi$ is single-valued in $\Omega.$ In view of the compatibility condition, we have

[TABLE]

where $\mathfrak{M}\in\mathbb{Z}_{+}$ is called the vortex number and coincides with the total number of zeroes of $\phi$ in $\Omega$ counted according to their multiplicities.

Since the Euler-Lagrangian equation (1.2) is very complicated to study even for stationary solution, we restrict to consider energy minimizers only. It is well known from the arguments in [5] that a global minimizer of static energy on suitable function spaces is achieved by the following self-dual equations:

[TABLE]

together with the boundary conditions (1.3). Due to Jaffe-Taubes argument in [35, 56], the self-dual equation (1.5) is reduced to the following elliptic system (see [11, 24, 31, 49, 54] for the detail):

[TABLE]

where $u=\ln|\phi|^{2}$, $\lambda=\frac{2q^{2}}{\mu}$, and $N=2\mathfrak{n}$. Here, $\delta_{p_{i}}\in\Omega$ stands for the Dirac measure concentrated at $p_{i}$, and $p_{i}\neq p_{j}$ if $i\neq j$. Each $p_{i}$ is called a vortex point and $m_{i}\in\mathbb{N}$ is the multiplicity of $p_{i}$.

In view of Remark 3 below, the equation (1.6) has two different kinds of periodic solutions satisfying one of the following asymptotic behaviors:

[TABLE]

Among the results obtained in [12, 49, 48] for (MCS) model, let us review the (CS) limit results for (1.6) on a flat two torus $\Omega$. In [12], Chae and Kim showed the existence of topological solution for (1.6), and its (CS) convergence whenever $\mu\to\infty$ and $\lambda$ is fixed (see [11] for the study in $\mathbb{R}^{2}$). In [49], Ricciardi and Tarantello extended the (CS) limit to other class of solutions. They showed that there exists $\lambda_{0}>0$ sufficiently large such that for any $\lambda>\lambda_{0}$, there is $\mu_{\lambda}>0$ satisfying that if $\mu>\mu_{\lambda}$ , then (1.6) has at least two distinct solutions, topological solution and mountain pass solution, which converge to (CS) multivortices as $\mu\to\infty$. Moreover, they derived the asymptotic behavior of these (CS) multivortices for not only topological solution but also mountain pass solution provided $\mathfrak{M}=\sum_{i=1}^{n}m_{i}=1$ as $\lambda\to\infty$. In [48], Ricciardi improved the results [12, 49] by obtaining the (CS) limit for arbitrary sequence of solutions in $C^{q}$ norm for any $q\geq 0$ whenever $\lambda=1$.

For given arbitrary configuration of vortex points, our first goal is to obtain the uniform (CS) limit result of (MCS) model for any class of solutions for (1.6) with large $\lambda,\mu>0$, and derive the following Brezis-Merle type alternatives for (MCS) model.

Theorem 1.1**.**

Let $Z\equiv\cup_{i}\{p_{i}\}$. We assume that $\{(u_{\lambda,\mu},N_{\lambda,\mu})\}$ is a sequence of solutions of (1.6). Then

[TABLE]

Moreover, as $\lambda,\mu\to\infty,\ \frac{\lambda}{\mu}\to 0$, up to subsequences, one of the following holds:

(i) $u_{\lambda,\mu}\to 0$ uniformly on any compact subset of $\Omega\setminus Z$;

(ii) $u_{\lambda,\mu}+2\ln\lambda-u_{0}\to\hat{w}$ in $C^{1}_{\textrm{loc}}(\Omega)$, where $\hat{w}$ satisfies $\Delta\hat{w}+e^{\hat{w}+u_{0}}=4\pi\mathfrak{M}$;

(iii) there exists a nonempty finite set $B=\{\hat{q}_{1},\cdots,\hat{q}_{k}\}\subset\Omega$ and $k$-number of sequences of points $q^{j}_{\lambda,\mu}\in\Omega$ such that $\lim_{\lambda,\mu\to\infty,\ \frac{\lambda}{\mu}\to 0}q^{j}_{\lambda,\mu}=\hat{q}_{j}$, $\left(u_{\lambda,\mu}+2\ln\lambda\right)(q^{j}_{\lambda,\mu})\to+\infty$, and $u_{\lambda,\mu}+2\ln\lambda\to-\infty$ uniformly on any compact subset of $\Omega\setminus B$. Moreover,

[TABLE]

in the sense of measure.

The most important step in the proof for Theorem 1.1 is to derive the relation (1.8) between $u_{\lambda,\mu}$ and $N_{\lambda,\mu}$. In order to achieve this purpose, we apply the Green’s representation formula for the gradient estimation of $u_{\lambda,\mu}$, and use the nondegeneracy of the operator $-\Delta+1$ in $\mathbb{R}^{2}$ after a suitable scaling.

We note that the elliptic system (1.6) is equivalent to

[TABLE]

To the best of our knowledge, the estimation (1.8) in Theorem 1.1 has been known for a fixed constant $\lambda>0$ as $\mu\to\infty$. We improve this result holds uniformly for large $\lambda>0$ satisfying $\lambda\ll\mu$. Due to the estimation (1.8), (1.9) would be regarded as a perturbation of the following equation arising from (CS) model:

[TABLE]

The corresponding result (i)-(iii) in Theorem 1.1 for (CS) equation (1.10) has been proved in [20] based on the arguments for Brezis-Merle type alternatives (see [2, 3, 7, 20, 45, 46]). However, since our case is the coupled system problem, a major obstacle arises from the interaction between two components $u_{\lambda,\mu}$ and $N_{\lambda,\mu}$. In order to overcome this difficulty, we should carry out a careful estimation for the gradient of $N_{\lambda,\mu}$ in the Pohozaev identity.

In [49], the authors made a conjecture such that the density of superconducting particles $e^{u_{\lambda,\mu}}$ of (1.6) converges to $e^{u_{\lambda}}$ of (1.10) as $\mu\to\infty$ without the restriction $\mathfrak{M}=1$, and it was proved in [48] for fixed $\lambda=1$. This result would be valid even uniformly for $\lambda>0$ since (1.9) and (1.10) share the similar asymptotic behavior in (i)-(iii) of Theorem 1.1 for any sequence of solutions to (1.6) including even mountain pass solution and for any $\mathfrak{M}>0$. Moreover, we can improve the (CS) convergence for blow up solutions, which are constructed below, in terms of not only $e^{u_{\lambda,\mu}}$ but also $u_{\lambda,\mu}$. We will continue to discuss the detail of uniform (CS) convergence for arbitrary solutions in forthcoming paper.

Now we consider the asymptotic behavior (iii) in Theorem 1.1. The case (iii) is called blow up phenomena. More precisely, we define the blow up solutions as follows:

Definition 1.1**.**

Let $B=\{\hat{q}_{j}\}_{j=1}^{k}\subset\Omega$ be a set of finite points. If $\{(u_{\lambda,\mu},N_{\lambda,\mu})\}$ is a family of solutions of (1.6) and there exist $k$-number of sequence of points $q^{j}_{\lambda,\mu}$, $j=1,\cdots,k$, satisfying

(i) $\lim_{\lambda,\mu\to\infty,\ \frac{\lambda}{\mu}\to 0}\left(u_{\lambda,\mu}+2\ln\lambda\right)(q^{j}_{\lambda,\mu})=+\infty,$ and

(ii) $\lim_{\lambda,\mu\to\infty,\ \frac{\lambda}{\mu}\to 0}q^{j}_{\lambda,\mu}=\hat{q}_{j}$,

then $B$ is called a blow-up set and $\{(u_{\lambda,\mu},N_{\lambda,\mu})\}$ is called a family of bubbling solutions (or blow up solutions) of (1.6) at $B$

In view of Theorem 1.1, we note that the blow up phenomena implies the concentration of density for the nonlinear terms in the first equation in (1.9). We emphasize that this observation provides the affirmative answer for the open problem raised in [55]. In other words, we would like to show the existence of blow up solutions with the concentrating property at the vortex points. It turns out that the construction of solutions blowing up outside vortex points, that is, at the regular points, is more difficult than at the vortex points since the limit problem for the first one has nontrivial kernel. We first construct solutions blowing up at a regular point, and continue to study solutions blowing up at a vortex point.

Theorem 1.2**.**

Assume $\mathfrak{M}>2$. Let $\hat{q}$ be a non-degenerate critical point of $u_{0}$ defined in (3.1). Assume that $\lambda,\ \mu$ are large enough and satisfy $(\ln\lambda)\lambda^{2}\ll\mu$. Then (1.6) has a solution $(u_{\lambda,\mu},N_{\lambda,\mu})$ satisfying

(i) $\lambda^{2}e^{u_{\lambda,\mu}}\left(1-\frac{N_{\lambda,\mu}}{\lambda}\right)\to 4\pi\mathfrak{M}\delta_{\hat{q}}$ in the sense of measure as $\lambda,\mu\to\infty,\frac{(\ln\lambda)\lambda^{2}}{\mu}\to 0$,

(ii) $\max_{y\in\Omega}u_{\lambda,\mu}(y)\geq c$ for some constant $c\in\mathbb{R}$, and

(iii) $\frac{N_{\lambda,\mu}}{\lambda}\to 0$ uniformly on any compact subset of $\Omega\setminus\{\hat{q}\}$ as $\lambda,\mu\to\infty,\frac{(\ln\lambda)\lambda^{2}}{\mu}\to 0$.

Remark 1**.**

By integrating the first equation of (1.9), we have

[TABLE]

Moreover, in view of Lemma 2.1 below, one knows that the local mass of the Chern-Simons equation without vortex points is strictly greater than $8\pi$. So necessarily one has $4\pi\mathfrak{M}>8\pi$, that is, $\mathfrak{M}>2$. This implies that when there is only one vortex point with multiplicity one, there should be no such kind of bubbling solutions considered in Theorem 1.2.

Motivated by Theorem 1.1 and Theorem 1.2, we also could solve the open problem raised in [55], and show the existence of blow up solutions with the concentrating property at the vortex point.

Theorem 1.3**.**

Assume $\mathfrak{M}>4$, $p_{1}\neq p_{j}$, $j=2,\cdots,n$, and $1\ll(\ln\lambda)^{5}\lambda^{5}\ll\mu$. Then (1.6) has a solution $(u_{\lambda,\mu},N_{\lambda,\mu})$ satisfying

(i) $\lambda^{2}e^{u_{\lambda,\mu}}\left(1-\frac{N_{\lambda,\mu}}{\lambda}\right)\to 4\pi\mathfrak{M}\delta_{p_{1}}$ in the sense of measure as $\lambda,\mu\to\infty,\frac{(\ln\lambda)^{5}\lambda^{5}}{\mu}\to 0$,

(ii) $\max_{y\in\Omega}u_{\lambda,\mu}(y)\geq c$ for some constant $c\in\mathbb{R}$,

(iii) $\frac{N_{\lambda,\mu}}{\lambda}\to 0$ uniformly on any compact subset of $\Omega\setminus\{p_{1}\}$ as $\lambda,\mu\to\infty,\frac{(\ln\lambda)^{5}\lambda^{5}}{\mu}\to 0$.

Remark 2**.**

If we consider the blow up solutions at the vortex point with the multiplicity one, and assume that the maximum of the first component has a finite lower bound, then the limit equation becomes the Chern-Simons equation containing the vortex point with the multiplicity one. In this case, the local mass should be greater than $16\pi$, necessarily $4\pi\mathfrak{M}>16\pi$, and thus we need the condition $\mathfrak{M}>4$ in Theorem 1.3.

We note that the conditions for $\lambda,$ $\mu>0$ in Theorem 1.2-1.3 is stronger than the condition $\lambda\ll\mu$ in Theorem 1.1 because of technical reason, which occurs from the lower bound of $u_{\lambda,\mu}$. The maximum of the first component for solutions in Theorem 1.2 and Theorem 1.3 has a finite lower bound since the profile of approximate solutions comes from the entire solution of (CS) model. In forthcoming paper, we will study the blow up solutions whose first component has no lower bound for the maximum value such that the limiting profile will be the Liouville equation.

The paper is organized as follows. In Section 2, we review some preliminaries in the gauge theory. In Section 3, we analyze the asymptotic behavior of solutions and prove Theorem 1.1. In Section 4-5, we study the existence of blow up solutions.

2. Preliminaries

In this section, we review some known results in the gauge theory. Firstly, we consider the following problem

[TABLE]

We recall the following results.

Lemma 2.1**.**

[7, 15*]** [20, Lemma 3.2] Let $m$ be a nonnegative integer, and $w$ be a solution of (2.1).

If $e^{w}(1-e^{w})\in L^{1}(\mathbb{R}^{2})$, then either*

(i) $w(x)\to 0$ as $|x|\to\infty$, or

(ii) $w(x)=-\beta\ln|x|+O(1)$ near $\infty$, where $\beta=-2m+{\frac{1}{2\pi}}\int_{\mathbb{R}^{2}}e^{w}(1-e^{w})dx.$

Assume that $w$ satisfies the boundary condition (ii). Then we have

[TABLE]

In particular, $\int_{\mathbb{R}^{2}}e^{w}(1-e^{w})dx>8\pi(1+m).$

Next we introduce the following result, which will help us to study the asymptotic behavior of solutions in $\Omega$.

Lemma 2.2**.**

[14, Theorem 2.1]** [16, Theorem 3.2] [51, Theorem 2.2] Let $m=0$, and $w$ be a solution of (2.1) with $e^{w}(1-e^{w})\in L^{1}(\mathbb{R}^{2})$. Then, $w(x)$ is smooth, radially symmetric with respect to some point $x_{0}$ in $\mathbb{R}^{2}$, and strictly decreasing function of $r=|x-x_{0}|$.

Assume $w(r;s)$ be the radially symmetric solution with respect to [math] of (2.1) such that

[TABLE]

where $w^{\prime}$ denotes $\frac{dw}{dr}(r;s)$, and let us set

[TABLE]

Then one has

(i) $\beta(0)=0$ and $w(\cdot;0)\equiv 0$;

(ii) $\beta:(-\infty,0)\rightarrow(4,+\infty)$ is strictly increasing, bijective, and

[TABLE]

Lemma 2.3**.**

(Lemma 2.1, [49]) Let $(u_{\lambda,\mu},N_{\lambda,\mu})$ be solutions of (1.6) over $\Omega$. Then

[TABLE]

In view of Lemma 2.3, we can show that the nonlinear term of the first equation in (1.9) is uniformly bounded in $L^{1}(\Omega)$ with respect to $\lambda,\mu>0$ as in the following corollary.

Corollary 2.1**.**

Let $(u_{\lambda,\mu},N_{\lambda,\mu})$ satisfy (1.6) over $\Omega$. Then we have

[TABLE]

Proof.

By integrating (1.9) over $\Omega$ and using Lemma 2.3, we can obtain Corollary 2.1.

∎

Remark 3**.**

In view of Corollary 2.1, we obtain

[TABLE]

which implies

[TABLE]

Moreover, by integrating the second equation of (1.6) on $\Omega$, we also see that

[TABLE]

If $\mu>\lambda$, then it is reasonable to consider the class of solutions satisfying the asymptotic behavior in (1.7).

Let us also recall the following form of the Harnack inequality.

Lemma 2.4**.**

([3, 26]) Let $D\subseteq\mathbb{R}^{2}$ be a smooth bounded domain and $v$ satisfy:

[TABLE]

with $f\in L^{p}(D)$, $p>1$. For any subdomain $D^{\prime}\subset\subset D$, there exist two positive constants $\sigma\in(0,1)$ and $\tau>0$, depending on $D^{\prime}$ only such that:

[TABLE]

[TABLE]

3. Asymptotic behavior of solutions

In this section, we will study the asymptotic behavior of solutions to (1.9) and prove Theorem 1.1. We firstly introduce some notations. Let $G(x,y)$ be the Green’s function satisfying

[TABLE]

where $|\Omega|$ is the measure of $\Omega$, and we denote the regular part of $G(x,y)$ by

[TABLE]

Let $\mathfrak{M}=\sum_{i=1}^{n}m_{i}$, and

[TABLE]

We set $u=v+u_{0}$, and assume $|\Omega|=1$. Then (1.9) is equivalent to

[TABLE]

Lemma 3.1**.**

Let $(u_{\lambda,\mu},N_{\lambda,\mu})$ satisfy (1.9) over $\Omega$. Then there exists a constant $C>0$, independent of $\lambda>0$ and $\mu>0$, such that

[TABLE]

Proof.

By applying the Green’s representation formula for a solution $(u_{\lambda,\mu},N_{\lambda,\mu})$ of (1.9), we see

[TABLE]

Together with Lemma 2.3 and Corollary 2.1, we can obtain

[TABLE]

where $c_{0},\ c_{1},\ C>0$ are constants, independent of $\lambda,\mu>0$.

∎

Next we will have the key estimate which will reduce (1.9) to an almost decoupled system whose first equation is a perturbation of a single Chern-Simons equation.

Lemma 3.2**.**

Let $(u_{\lambda,\mu},N_{\lambda,\mu})$ satisfy (1.9) over $\Omega$. Then

[TABLE]

Proof.

Let $v_{\lambda,\mu}=u_{\lambda,\mu}-u_{0}$. Then $v_{\lambda,\mu}$ satisfies (3.2). We argue by contradiction and suppose that there exists $x_{\lambda,\mu}\in\Omega$ such that

[TABLE]

Let

[TABLE]

Then we see that

[TABLE]

Here, the last equality is obtained from Lemma 2.3.

Fix a constant $R>0$, independent of $\lambda,\mu>0$. The mean value theorem and $\left\|\nabla\left(v_{\lambda,\mu}+\frac{N_{\lambda,\mu}}{\mu}\right)\right\|_{L^{\infty}(\Omega)}=O(\lambda)$ in Lemma 3.1 yield some $\theta\in[0,1]$ satisfying

[TABLE]

We are going to consider the following cases according to the location of limit point for $x_{\lambda,\mu}$, up to subsequence.

Case 1. $\lim_{\lambda,\mu\to\infty,\frac{\lambda}{\mu}\to 0}x_{\lambda,\mu}=x_{0}\notin\cup^{n}_{i=1}\{p_{i}\}$.

Since $B_{d}(x_{0})\cap\cup^{n}_{i=1}\{p_{i}\}=\emptyset$ for a sufficiently small constant $d>0$, $u_{0}$ is smooth in $B_{d}(x_{0})$. Together with (LABEL:2.4), we see that

[TABLE]

here we used that if $|x|\leq R$, then $\mu^{-1}x+x_{\lambda,\mu}\in B_{d}(x_{0})$.

In view of Lemma 2.3, we have $|\tilde{N}_{\lambda,\mu}|\leq 1$, and thus there exists a function $N_{0}$ satisfying $\tilde{N}_{\lambda,\mu}\to N_{0}$ in $C^{1}_{\textrm{loc}}(\mathbb{R}^{2})$ as $\lambda,\mu\to\infty,\ \frac{\lambda}{\mu}\to 0$, where $N_{0}$ is a solution of

[TABLE]

and $c_{0}=\lim_{\lambda,\mu\rightarrow\infty}e^{(v_{\lambda,\mu}+u_{0}+\frac{{N}_{\lambda,\mu}}{\mu})(x_{\lambda,\mu})}$. Then we have $N_{0}\equiv c_{0}$ in $\mathbb{R}^{2}$ (for example, see [25, Proposition 2.3]).

We also note that

[TABLE]

and

[TABLE]

here, we used Lemma 2.3 and the assumption $1\ll\lambda\ll\mu$ in the third equality. However, (3.8) and (3.9) contradict the assumption (3.4).

Case 2. $x_{\lambda,\mu}\rightarrow x_{0}=p_{i}$ for some $i$.

Define $\hat{u}_{0}(x)\in C^{\infty}(B_{d}(p_{i}))$ such that

[TABLE]

Together with (LABEL:2.4), we have

[TABLE]

There are two cases according to the behavior of $|x_{\lambda,\mu}-p_{i}|\mu$.

Case 2-(1). $\lim_{\lambda,\mu\to\infty,\frac{\lambda}{\mu}\to 0}|x_{\lambda,\mu}-p_{i}|\mu=\infty$.

In this case, the equation (LABEL:eq_N) and (3.10) imply that if $|x|\leq R$, then

[TABLE]

Then the same arguments in Case 1 implies a contradiction again.

Case 2-(2). $\lim_{\lambda,\mu\to\infty,\frac{\lambda}{\mu}\to 0}|x_{\lambda,\mu}-p_{i}|\mu\leq c$ for some constant $c>0$.

In view of (3.2), Lemma 2.3, and the condition $\lambda,\mu\to\infty,\ \frac{\lambda}{\mu}\to 0$, we see that

[TABLE]

Let

[TABLE]

and $z=\mu^{-1}x+p_{i}$.

Then, we have

[TABLE]

In view of Lemma 2.3 and $\hat{u}_{0}\in C^{\infty}(B_{d}(p_{i}))$, we see that

[TABLE]

for some constant $c_{0}>0$, independent of $\lambda,\mu>0$.

By Lemma 3.1, we also see that

[TABLE]

From (3.14) and (3.15), we note that there are two possibilities as follows:

(i). $\sup_{\partial B_{1}(0)}|\hat{v}_{\lambda,\mu}|\leq C$.

In this case, (3.15) implies that $|\hat{v}_{\lambda,\mu}|$ is uniformly bounded in $C^{0}_{\textrm{loc}}\left(B_{d\mu}(0)\right)$ for $\lambda,\mu>0$. Then there exists a function $v_{0}$ such that $\hat{v}_{\lambda,\mu}\to v_{0}$ in $C^{1}_{\textrm{loc}}\left(B_{d\mu}(0)\right)$ and $\nabla v_{0}\equiv 0$ in $\mathbb{R}^{2}$. It implies that $v_{0}\equiv c$ for some constant $c\in{\mathbb{R}}$ and $\Delta v_{0}=0$ in $\mathbb{R}^{2}$. From the mean value theorem for harmonic function and (3.14), we see that for any constant $R>0$,

[TABLE]

We get a contradiction as $R\to\infty$ in (3.16).

(ii). $\sup_{\partial B_{1}(0)}\hat{v}_{\lambda,\mu}\rightarrow-\infty$.

In this case, (3.15) implies that $\hat{v}_{\lambda,\mu}\to-\infty$ is uniformly in $C^{0}_{\textrm{loc}}\left(B_{d\mu}(0)\right)$ for $\lambda,\mu>0$. By (LABEL:eq_N), we have

[TABLE]

Since $|\tilde{N}_{\lambda,\mu}(x)|\leq 1$ for all $x\in B_{{d\mu}}(0)$, there is a function $\tilde{N}_{0}$ satisfying $\tilde{N}_{\lambda,\mu}\to\tilde{N}_{0}$ in $C^{1}_{\textrm{loc}}(\mathbb{R}^{2})$, and

[TABLE]

which implies $\tilde{N}_{0}\equiv 0$ in $\mathbb{R}^{2}$ (for example, see [25, Proposition 2.3]). We note that

[TABLE]

and

[TABLE]

However, (3.18) and (3.19) contradict the assumption (3.4).

∎

In view of Lemma 3.2, the first equation of (1.9) can be regarded as a perturbation of a single Chern-Simons equation (1.10). By applying the arguments in [20, Lemma 4.1], we can obtain the following result.

Lemma 3.3**.**

Suppose that there exists a sequence of solutions $(u_{\lambda,\mu},N_{\lambda,\mu})$ of (1.9) such that

[TABLE]

Then, we have

[TABLE]

Proof.

Choose a sequence of points $\{x_{\lambda,\mu}\}\subseteq\Omega$ such that

[TABLE]

Passing to a subsequence (still denoted by $u_{\lambda,\mu}$), we may assume that $\lim_{\lambda,\mu\to\infty,\ \frac{\lambda}{\mu}\to 0}x_{\lambda,\mu}=x_{0}\in\Omega$. We consider the following two cases according to the location of $x_{0}$.

Case 1. $x_{0}\notin Z$.

Let $d>0$ be a small constant satisfying $B_{d}(x_{0})\cap Z=\emptyset$. We argue by contradiction and suppose that there exist a compact set $K\subset\Omega\setminus Z$, a positive constant $c_{K}>0$, and a sequence $\{z_{\lambda,\mu}\}\subset K$ such that $\sup_{K}|u_{\lambda,\mu}|=|u_{\lambda,\mu}(z_{\lambda,\mu})|\geq c_{K}>0$ for large $\lambda,\mu>0$. We choose a connected compact set $K_{1}\subset\Omega\setminus Z$ satisfying $B_{d}(x_{0})\cup K\subset K_{1}$. Since $u_{\lambda,\mu}(z_{\lambda,\mu})\leq-c_{K}<0$, Lemma 2.2 implies there is a constant $s_{1}<0$ such that

[TABLE]

We can also choose $y_{\lambda,\mu}\in K_{1}$ such that $u_{\lambda,\mu}(y_{\lambda,\mu})=s_{1}$ by the intermediate value theorem.

Let $\bar{u}_{\lambda,\mu}(x)=\left(u_{\lambda,\mu}+\frac{N_{\lambda,\mu}}{\mu}\right)(\lambda^{-1}x+y_{\lambda,\mu})$ for $x\in\Omega_{y_{\lambda,\mu}}\equiv\{\ x\in\mathbb{R}^{2}\ |\ \lambda^{-1}x+y_{\lambda,\mu}\in K_{1}\ \}$.

By Corollary 2.1 and $u_{\lambda,\mu}(y_{\lambda,\mu})=s_{1}$, we see that $\bar{u}_{\lambda,\mu}$ satisfies

[TABLE]

By using Lemma 3.1 and $W^{2,p}$ estimation, we see that $\bar{u}_{\lambda,\mu}$ is bounded in $C^{1,\sigma}_{\textrm{loc}}(\Omega_{y_{\lambda,\mu}})$ for some $\alpha\in(0,1)$. In view of Lemma 2.3 and Lemma 3.2, we see that if $0<\lambda\ll\mu$, then

[TABLE]

and $\bar{u}_{\lambda,\mu}(0)=s_{1}+o(1).$ Passing to a subsequence, $\bar{u}_{\lambda,\mu}$ converges in $C^{1}_{\textrm{loc}}(\mathbb{R}^{2})$ to a function $u_{*}$, which is a solution of

[TABLE]

By using Lemma 2.2, we see that $u_{*}$ is radially symmetric with respect to a point $\bar{p}$ in $\mathbb{R}^{2}$.

In view of Lemma 2.2, we have

[TABLE]

which implies a contradiction. Thus (3.20) holds true in Case 1.

Case 2. $x_{0}=p_{i}\in Z$ for some $i$.

Fix a small constant $r_{0}>0$ such that $B_{r_{0}}(x_{0})\cap Z=\{x_{0}\}$. For simplicity, we assume that $x_{0}=0$. We are going to show that

[TABLE]

Once we have (3.26), the argument in Case 1 implies (3.20). In order to prove (3.26), we argue by contradiction again and suppose that, up to a subsequence, $\lim_{\lambda,\mu\to\infty,\ \frac{\lambda}{\mu}\to 0}\left(\inf_{|x|=r_{0}}|u_{\lambda,\mu}(x)|\right)\geq\tau_{0}$ for some constant $\tau_{0}>0$. Since $u_{\lambda,\mu}<0$, we have

[TABLE]

We divide our discussion into the following two cases.

(i). $\lim_{\lambda,\mu\to\infty,\ \frac{\lambda}{\mu}\to 0}\left(\lambda|x_{\lambda,\mu}|\right)<+\infty$.

Note that $u_{\lambda,\mu}(x)=2m_{i}\ln|x|+v_{\lambda,\mu}(x)$ near $x=0$ for some $1\leq i\leq n$, where $v_{\lambda,\mu}$ is a smooth function in $B_{d}(0)$. Let

[TABLE]

Then $\hat{v}_{\lambda,\mu}$ satisfies

[TABLE]

By (3.21), we note that

[TABLE]

Together with Lemma 3.1 and the assumption $\liminf_{\lambda,\mu\to\infty,\ \frac{\lambda}{\mu}\to 0}\left(\lambda|x_{\lambda,\mu}|\right)<+\infty$, we see that $\hat{v}_{\lambda,\mu}$ is bounded in $C^{0}_{\textrm{loc}}\left(B_{r_{1}}(0)\right)$. Passing to a subsequence, we may assume that

[TABLE]

for some function $\hat{v}$. By using Lemma 2.3 and Lemma 3.2 as in (LABEL:ee), we see that $\hat{u}(x)=\hat{v}(x)+2m_{i}\ln|x|$ satisfies

[TABLE]

In view of Lemma 2.3, we have $0\leq\frac{N_{\lambda,\mu}(|x_{\lambda,\mu}|x)}{\lambda}\leq 1$. Moreover, Lemma 3.2 implies $\lim_{\lambda,\mu\to\infty,\ \frac{\lambda}{\mu}\to 0}\frac{N_{\lambda,\mu}(|x_{\lambda,\mu}|x)}{\lambda}=e^{\hat{u}(x)}\in[0,1]$. Since $\hat{u}\leq 0$ in $\mathbb{R}^{2}$ and $\hat{u}(y_{0})=0$, we have $\hat{u}\equiv 0$ by Hopf Lemma, which implies a contradiction.

(ii). $\lim_{\lambda,\mu\to\infty,\ \frac{\lambda}{\mu}\to 0}\left(\lambda|x_{\lambda,\mu}|\right)=+\infty$.

Lemma 2.2 implies there is a constant $s_{2}<0$ such that

[TABLE]

where $\tau_{0}$ is the constant in (3.27). We can also choose $\hat{y}_{\lambda,\mu}$ on the line segment joining $x_{\lambda,\mu}$ and $\frac{r_{0}x_{\lambda,\mu}}{|x_{\lambda,\mu}|}$ such that $u_{\lambda,\mu}(\hat{y}_{\lambda,\mu})=s_{2}$ and $|\hat{y}_{\lambda,\mu}|\geq|x_{\lambda,\mu}|$ by the intermediate value theorem. Let $\hat{u}_{\lambda,\mu}(x)=\left(u_{\lambda,\mu}+\frac{N_{\lambda,\mu}}{\mu}\right)(\lambda^{-1}x+\hat{y}_{\lambda,\mu})$ for $x\in\hat{\Omega}_{\hat{y}_{\lambda,\mu}}\equiv\{\ x\in\mathbb{R}^{2}\ |\ \lambda^{-1}x+\hat{y}_{\lambda,\mu}\in B_{\frac{|x_{\lambda,\mu}|}{2}}(\hat{y}_{\lambda,\mu})\ \}$. Here we note that $0\notin B_{\frac{|x_{\lambda,\mu}|}{2}}(\hat{y}_{\lambda,\mu})$. Then $\hat{u}_{\lambda,\mu}$ satisfies

[TABLE]

Using the same argument as in Case 1, we get a contradiction by comparing the upper/lower bound of $L^{1}$ norm of the nonlinear term in (3.30). Thus the claim (3.26) holds true. Then we can again apply the arguments in Case 1 and prove (3.20) holds true.

∎

As a corollary of Lemma 3.3, we get the following proposition.

Proposition 3.1**.**

Let $(u_{\lambda,\mu},N_{\lambda,\mu})$ be a sequence of solutions of (1.9). Then, up to subsequences, one of the following holds true:

(i) $u_{\lambda,\mu}\to 0$ uniformly on any compact subset of $\Omega\setminus Z$ as $\lambda,\mu\to\infty,\ \frac{\lambda}{\mu}\to 0$, or

(ii) there exists a constant $\nu_{0}>0$ such that \sup_{\lambda,\mu\to\infty,\ \frac{\lambda}{\mu}\to 0}\Big{(}\sup_{\Omega}u_{\lambda,\mu}\Big{)}\leq-\nu_{0}.

Completion of the proof of Theorem 1.1. Note that Proposition 3.1-(i) corresponds to Theorem 1.1-(i). In order to complete the proof of Theorem 1.1, from now on, we will study the asymptotic behavior for the solution $(u_{\lambda,\mu},N_{\lambda,\mu})$ of (1.9) satisfying Proposition 3.1-(ii). Let us denote

[TABLE]

Clearly $w_{\lambda,\mu}$ satisfies

[TABLE]

By Lemma 2.3 and Lemma 3.2, as $\lambda,\mu\to\infty,\ \frac{\lambda}{\mu}\to 0$, we have (for example, see (LABEL:ee))

[TABLE]

and

[TABLE]

Then we have

[TABLE]

We consider the following two cases:

Case 1. $\sup_{\Omega}w_{\lambda,\mu}\leq C$ for some constant $C>0$.

In this case, we note that the Harnack inequality (i.e. Lemma 2.4) and (3.33) imply $w_{\lambda,\mu}-u_{0}$ is uniformly bounded in $L^{\infty}(\Omega)$. Moreover, in view of $W^{2,2}$ estimation, $w_{\lambda,\mu}-u_{0}$ is uniformly bounded in $C^{1,\alpha}(\Omega)$ for some $\alpha\in(0,1)$. Then $w_{\lambda,\mu}-u_{0}\to\hat{w}$ in $C^{1}_{\textrm{loc}}(\Omega)$, where $\hat{w}$ satisfies

[TABLE]

We note that Case 1 implies Theorem 1.1-(ii).

Case 2. $\lim_{\lambda,\mu\to\infty,\ \frac{\lambda}{\mu}\to 0}\sup_{\Omega}w_{\lambda,\mu}=+\infty.$

Following [7], we say that a point $q\in\Omega$ is a blow-up point for $\{w_{\lambda,\mu}\}$ if there exists a sequence $\{x_{\lambda,\mu,q}\}$ such that

[TABLE]

The set $S\subset\Omega$, which consists of blow-up points for $\{w_{\lambda,\mu}\}$, is called the blow-up set for $\{w_{\lambda,\mu}\}$.

Step 1. Let $p\in\Omega$ be a blow-up point for $\{w_{\lambda,\mu}\}$. Then we have the following “minimal mass” result.

[TABLE]

Indeed, we note that the equation (3.32) is a perturbation of

[TABLE]

and the minimal mass result for (3.35) was obtained in [20, Lemma 4.2]. By the similar arguments in [20], we can also get (3.34) for the solution $w_{\lambda,\mu}$ of (3.32). Here we skip the detail and refer to [20].

The estimation (3.34) shows that if Case 2 happens, then $\{w_{\lambda,\mu}\}$ has a nonempty finite blow-up set $B\subset\Omega$, and $|B|\leq\frac{{4\pi\mathfrak{M}}}{8\pi}=\frac{\mathfrak{M}}{2}$. We also see that for any compact set $K\subseteq\Omega\setminus B$, there exists a constant $C_{K}>0$ such that

[TABLE]

Step 2. In this step, we are going to prove that the blow up phenomena implies the concentration of mass as in [3, 7, 20]. However, our case is a coupled system problem, we should carry out a delicate analysis in order to prove the concentration of mass. Firstly, we claim that

[TABLE]

Choose a small constant $d>0$ satisfying for any $q\in B$, $B_{2d}(q)\cap\left[B\cup Z\right]=\{q\}$. For each $q\in B$, we let $\{x_{\lambda,\mu,q}\}$ be a sequence of points such that

[TABLE]

We shall prove that

[TABLE]

for any $r\in(0,d]$ and all $q\in B$. Then by (3.36) and Harnack’s inequality, we get that

[TABLE]

To prove (3.38), we argue by contradiction and suppose that there exist $r\in(0,d]$ and $q\in B$ such that

[TABLE]

for some constant $c\in\mathbb{R}$. For simplicity, we assume that $q=0$. By using (3.36) and Harnack’s inequality, we can verify that $\{w_{\lambda,\mu}-u_{0}\}$ is bounded in $C_{loc}^{0}(B_{2d}(0)\setminus\{0\})$. Then elliptic estimates imply that there exists a function $\xi\in C^{1,\sigma}_{loc}(B_{2d}(0)\setminus\{0\})$ such that along a subsequence $w_{\lambda,\mu}-u_{0}\to\xi$ in $C^{1}_{loc}(B_{2d}(0)\setminus\{0\})$. Let

[TABLE]

In view of Lemma 2.3 and Lemma 3.2 as in (LABEL:ee), we see that

[TABLE]

in the sense of measure on $B_{2d}(0)$. By Corollary 2.1 and Fatou’s lemma, we have that $e^{\xi+u_{0}}\in L^{1}(B_{2d}(0))$. Moreover, Green’s representation formula implies that

[TABLE]

where $\eta\in C^{1}(B_{r}(0))$ for every $r\in(0,d)$, and we let

[TABLE]

We note that

[TABLE]

Then we see that $e^{\xi(x)}=|x|^{-\alpha_{q}/2\pi}e^{\phi+\eta}\geq c|x|^{-\alpha_{q}/2\pi}$ for $0<|x|\leq d$ and some constant $c>0$. Then the integrability of $e^{\xi+u_{0}}$ implies that

[TABLE]

where $m=m_{i}$ if $q=p_{i}\in B\cap{Z}$, and $m=0$ if $q\in B\setminus{Z}$.

Let $\phi_{\lambda,\mu}(x)=w_{\lambda,\mu}(x)-2m\ln|x|$. Then $\phi_{\lambda,\mu}$ satisfies

[TABLE]

Multiplying (3.41) by $x\cdot\nabla\phi_{\lambda,\mu}$ and integrating over $B_{r}(0)$ for $r\in(0,d)$, we get that

[TABLE]

We recall the second equation in (1.9), which can be written into

[TABLE]

Multiplying (3.43) by $x\cdot\nabla N_{\lambda,\mu}$ and integrating over $B_{r}(0)$ for $r\in(0,d)$, we have

[TABLE]

here we used $\nabla N_{\lambda,\mu}\cdot x=\frac{\partial N_{\lambda,\mu}}{\partial r}r$ and $|\nabla N_{\lambda,\mu}|^{2}=\left(\frac{\partial N_{\lambda,\mu}}{\partial r}\right)^{2}+\frac{\left(\frac{\partial N_{\lambda,\mu}}{\partial\theta}\right)^{2}}{r^{2}}$.

We claim that there is a sequence $\{r_{l}\}_{l\in\mathbb{N}}$ satisfying

[TABLE]

In order to prove the claim (3.45), we multiply the second equation in (1.9) by $N_{\lambda,\mu}$ and integrate over $\Omega$. Then we have

[TABLE]

Together with Lemma 2.3 and (3.33), we see that there is a constant $C>0$ such that

[TABLE]

In view of (3.46), we note that there exists a sequence $r_{l}$ satisfying (3.45). Otherwise, there exist constants $\varepsilon>0$ and $\bar{r}>0$ satisfying

[TABLE]

From (3.46), we have

[TABLE]

which is a contradiction.

At this point, in view of (LABEL:eqp1), (LABEL:eqp2), and (3.45), we see that for any $\varepsilon>0$, there is $l_{\varepsilon}$ such that if $l\geq l_{\varepsilon}$, then

[TABLE]

Let $\varphi_{0}(x)=(\xi+u_{0})(x)-2m\ln|x|$. Letting $\lambda,\mu\to\infty,\ \frac{\lambda}{\mu}\to 0$ in (3.47), Lemma 2.3 and Lemma 3.2 as in (LABEL:ee) imply

[TABLE]

There exists a constant $c>0$ such that $|x|^{2m}e^{\varphi_{0}}=e^{\xi+u_{0}}\leq c|x|^{-\tau}e^{\phi}$ in $B_{r}(0)$ for any $r\in(0,d)$, where $\tau\equiv\max\{0,\frac{\alpha_{q}}{2\pi}-2m\}$. We note that $\tau\in[0,2)$ from (3.40). In view of (3.39) and Corollary 1 in [7], we see that $e^{|\phi|}\in L_{loc}^{k}(B_{d}(0))$ for any $k\in[1,\infty)$. Since $\xi\in C^{2}_{loc}(B_{2d}(0)\setminus\{0\})$, we have $|x|^{2m}e^{\varphi_{0}}\in L^{t}(B_{d}(0))$ for any $t\in(1,\frac{2}{\tau})$. Then Hölder’s inequality implies that $\phi\in L^{\infty}(B_{d}(0))$ and

[TABLE]

We note that for $|x|=r<d$,

[TABLE]

Fix $t\in(1,\frac{2}{\tau})$ and choose a constant $a\in(0,\min\{1,2-\tau\})$ such that $\frac{at}{t-1}<2$. Hölder’s inequality implies that

[TABLE]

Since $|x|=r$, we have $B_{r/2}(x)\subseteq\Omega\setminus B_{r/2}(0)$. It follows from (3.49) that

[TABLE]

Since $a\in(0,2-\tau)$, we see that $|\nabla\phi(x)|=O(|x|^{a-1}+1)$ as $|x|\to 0$. Consequently $\nabla\varphi_{0}(x)=-\frac{\alpha_{q}x}{2\pi|x|^{2}}+\nabla h(x)$ with $|\nabla h(x)|=O(|x|^{a-1}+1)$ as $|x|\to 0$. Letting $\varepsilon\to 0$ and $r_{l}\to 0$ in (3.48), we obtain that $(2m+2)\alpha_{q}\leq\frac{\alpha_{q}^{2}}{4\pi}$, which contradicts (3.40).

Therefore, it follows from Harnack’s inequality that $w_{\lambda,\mu}-u_{0}\to-\infty$ and $w_{\lambda,\mu}\to-\infty$ uniformly on any compact subset of $\Omega\setminus B$.

Step 3. In view of Lemma 2.3 and Corollary 2.1, along a subsequence, ${e}^{w_{\lambda,\mu}-\frac{N_{\lambda,\mu}}{\mu}}\left(1-\frac{N_{\lambda,\mu}}{\lambda}\right)$ converges to a nonnegative measure. However, this measure must be supported in $B$ since $w_{\lambda,\mu}\to-\infty$ uniformly on any compact set $K\subset\Omega\setminus B$. Then we see that as $\lambda,\mu\to\infty,\ \frac{\lambda}{\mu}\to 0$,

[TABLE]

in the sense of measure. In view of the above arguments, we conclude that Case 2 implies Theorem 1.1-(iii). Now we complete the proof of Theorem 1.1.

∎

4. Proof of Theorem 1.2

In this section, we are going to construct blow up solutions of (3.2) such that

[TABLE]

Based on Theorem 1.1 above, our construction in this section was inspired by the construction in [42] where the authors constructed blow up solutions for the $SU(3)$ Chern-Simons system on torus using an entire regular solution for the single Chern-Simons equation as the building blocks.

4.1. The approximate solution and the reduction

Without loss of generality, we assume $|\Omega|=1$. We recall the equation (3.2) as follows:

[TABLE]

We are going to define the approximate solutions for (3.2). Let $w$ be the radially symmetric solution of

[TABLE]

where $a_{1}$ and $I_{1}$ are constants (see [14, Theorem 2.1, Lemma 2.6] for the existence of $w$ satisfying (4.2)). We set

[TABLE]

where

[TABLE]

which makes $U_{\lambda,q}\in C^{1}(\Omega)$, We would find a solution of (3.2) with the following form:

[TABLE]

here $(\varphi,S)$ would be regard as an error term. For the convenience, we also denote

[TABLE]

The notation $1_{B_{2d}(q)}$ means that $1_{B_{2d}(q)}(x)=1$ if $x\in B_{2d}(q)$ and $1_{B_{2d}(q)}(x)=0$ if $x\notin{B_{2d}(q)}$.

Then the equation (3.2) is reduced to a system for $(\varphi,S)$:

[TABLE]

where

[TABLE]

4.2. The linear and nonlinear problem

In this subsection, we are going to study the linear and nonlinear problem. First, let us introduce the space we are going to work in. Fix a small constant $0<\alpha<\frac{1}{2}$. Let us introduce two function spaces $X_{\alpha,q}$ and $Y_{\alpha,q}$. Define

[TABLE]

We say that $\psi\in X_{\alpha,q}$ if

[TABLE]

where $\tilde{\psi}(z)=\psi(\lambda^{-1}z+q)$. We say $\psi\in Y_{\alpha,q}$ if

[TABLE]

Let $\chi(y)$ be a smooth cut off function such that $\chi=1$ in $B_{d}(0)$, $\chi=0$ in $\Omega/B_{2d}(0)$ and $0\leq\chi\leq 1$. Define

[TABLE]

Let

[TABLE]

We define two subspace of $X_{\alpha,q}$ and $Y_{\alpha}$ as

[TABLE]

and

[TABLE]

Define the projection operator to $F_{q}$ as

[TABLE]

where $c_{j}$ are chosen so that $Q_{q}u\in F_{q}$. We have the following estimates:

Lemma 4.1**.**

There holds $\|Q_{q}u\|_{Y_{\alpha}}\leq C\|u\|_{Y_{\alpha}}$ for some positive constant $C$ independent of $q$.

First we need the preliminary results for the linear operators $L_{1,q},\ L_{2}$ in [25, 42], where

[TABLE]

Theorem 4.1** (Theorem B.3 in [42]).**

The operator $Q_{q}L_{1,q}$ is an isomorphism from $E_{q}$ to $F_{q}$. Moreover if $\varphi\in E_{q}$ and $g_{1}\in F_{q}$ satisfies $Q_{q}L_{1,q}(\varphi)=g_{1}$, then there exists a constant $C>0$ independent of $q$ such that

[TABLE]

Theorem 4.2**.**

The operator

[TABLE]

is an isomorphism. Moreover, for any $S\in W^{2,2}(\Omega)$ and $g_{2}\in L^{2}(\Omega)$ satisfies $L_{2}(S)=g_{2}$, there exists a positive constant $C$ independent of $\mu$ such that

[TABLE]

Proof.

It has been shown in [25, Theorem 2.4 ] that $L_{2}$ is an isomorphism, $\mu\|S\|_{L^{\infty}(\Omega)}\leq C\|g_{2}\|_{L^{2}(\Omega)}$, and $\mu^{2}\|S\|_{L^{\infty}(\Omega)}\leq C\|g_{2}\|_{L^{\infty}(\Omega)}$ if $g_{2}\in L^{\infty}(\Omega)$. In order to complete the proof of Theorem 4.1, if is enough to prove

[TABLE]

Here we prove the estimate (4.16).

Multiply the equation $L_{2}(S)=g_{2}$ by $S$ and integrate over $\Omega$, one has

[TABLE]

By Holder’s inequality,

[TABLE]

which implies that

[TABLE]

Since $\Delta S-\mu^{2}S=g_{2}$, one can get that

[TABLE]

so

[TABLE]

Now, we complete the proof of Theorem 4.2.

∎

Next, let us consider the corresponding nonlinear problem. We define an operator $\Psi$ by

[TABLE]

where $\hat{S}=L^{-1}_{2}(g_{2,\lambda,\mu}(\varphi,S))$, and a subset $M_{\lambda,\mu}$ of $E_{q}\times W^{2,2}(\Omega)$ by

[TABLE]

where

[TABLE]

We note that if $(\varphi,S)\in M_{\lambda,\mu}$, then

[TABLE]

To apply contraction argument we need some estimations for the right hand side of (4.6).

Lemma 4.2**.**

There exists a constant $C$ such that

[TABLE]

for any $(\varphi,S)\in M_{\lambda,\mu}$.

Proof.

We have

[TABLE]

First, we consider the $L^{2}$ norm of $\Delta\{e^{U_{\lambda,q}+u_{0}}(1+\varphi)\}$ in $B_{d}(q)$.

In $B_{d}(q)$, we get that

[TABLE]

and

[TABLE]

Moreover, since $\gamma(y,q)$ and $u_{0}$ are smooth functions in $B_{d}(q)$, we get that

[TABLE]

and

[TABLE]

By the definitions of $U_{\lambda,q}$ and $u_{0}$, we obtain that

[TABLE]

Then,

[TABLE]

Let

[TABLE]

for any function $u$. Then,

[TABLE]

where $\tilde{\varphi}(z):=\varphi(\lambda^{-1}z+q)$ as defined in (4.22). In the last inequality in (4.23), we used the decay rate of $w(z)$ in (4.2). From (4.2), we also have

[TABLE]

We can rewrite $III$ by

[TABLE]

Since $|e^{w(z)}w^{\prime}(z)|\leq C(1+|z|)^{-2\mathfrak{M}-1}$,

[TABLE]

There exist finite number of points $\{z_{i}\}$ such that $B_{d\lambda}(0)\subseteq\cup_{i}B_{1}(z_{i})\subseteq\cup_{i}B_{2}(z_{i})\subseteq B_{2d\lambda}(0)$ and for any index $i$, $|\{z_{j}\mid B_{1}(z_{i})\cap B_{1}(z_{j})\neq\emptyset\}|\leq c_{1},$ where a positive constant $c_{1}$ is independent of $\lambda$. Then, we have

[TABLE]

Moreover, there exists some constant $c_{2}$ from $W^{2,2}$ estimate satisfying

[TABLE]

Therefore, it follows that

[TABLE]

Similarly, we have

[TABLE]

From the decay rate of $e^{w(z)}w^{\prime}(z)$, we have

[TABLE]

Next, we consider $\|\Delta\{e^{U_{\lambda,q}+u_{0}}(1+\varphi)\}\|_{L^{2}(\Omega\setminus B_{d}(q))}$. In $\Omega\setminus B_{d}(q)$, we have

[TABLE]

and

[TABLE]

We see that $G(y,q)$ is a smooth function in $\Omega\setminus B_{d}(q)$ and thus

[TABLE]

We also see that

[TABLE]

where $m_{i}$ is the multiplicity of $p_{i}$. Since $\gamma(y,p_{i})$ and $G(y,p_{j})$, $j\neq i$, are smooth functions in $B_{d}(p_{i})$, we have $e^{u_{0}}(y)=O(|y-p_{i}|^{2m_{i}})\in C^{2}(B_{d}(p_{i}))$. Obviously, we also have $e^{u_{0}}\in C^{2}(\Omega\setminus\cup_{i=1}^{n}B_{d}(p_{i}))$. From this observation, we see that

[TABLE]

We also see that

[TABLE]

We are going to estimate $\|\Delta\{e^{U_{\lambda,q}+u_{0}}(1+\varphi)\}\|_{L^{2}(\Omega\setminus B_{d}(q))}$ by dividing the region $\Omega\setminus B_{d}(q)$ into $\Omega\setminus B_{\frac{3}{2}d}(q)$ and $B_{\frac{3}{2}d}(q)\setminus B_{d}(q)$.

Firstly, the estimations (LABEL:Bd1)-(LABEL:detail_exp) and $W^{2,2}$-estimation imply

[TABLE]

Secondly, in $B_{\frac{3}{2}d}(q)\setminus B_{d}(q)$, the estimations (LABEL:Bd1)-(LABEL:detail_exp) again imply

[TABLE]

In order to estimate $\|e^{U_{\lambda,q}+u_{0}}\nabla\left(U_{\lambda,q}+{u_{0}}\right)\cdot\nabla\varphi\|_{L^{2}(B_{\frac{3}{2}d}(q)\setminus B_{d}(q))}$, we note that the estimation (4.2) yields

[TABLE]

From the definition of $U_{\lambda,q}$ and $u_{0}$, it is clear that $\nabla\left(U_{\lambda,q}+{u_{0}}\right)$ are uniformly bounded in $B_{\frac{3}{2}d}(q)\setminus B_{d}(q)$. Therefore, we obtain that

[TABLE]

where $\{B_{1}(z_{i})\}$ is finite covering of $B_{\frac{3}{2}d\lambda}(0)\setminus B_{d\lambda}(0)$ as in the calculus in $III$. Since

[TABLE]

and

[TABLE]

we obtain

[TABLE]

Therefore, the proof is complete.

∎

Proposition 4.1**.**

There exists a fixed point $({\varphi}_{q},{S}_{q})\in M_{\lambda,\mu}$ of the operator $\Psi$.

Proof.

In order to prove Proposition 4.1, it is enough to show that $\Psi$ is a contraction map from $M_{\lambda,\mu}$ to $M_{\lambda,\mu}$ due to the contraction mapping theorem. We are going to prove that $\Psi$ is a contraction map from $M_{\lambda,\mu}$ to $M_{\lambda,\mu}$ with the following two steps.

Step 1. We claim that $\Psi(\varphi,S)\in M_{\lambda,\mu}$ for any $(\varphi,S)\in M_{\lambda,\mu}$. First, we consider $\|g_{2,\lambda,\mu}(\varphi,S)\|_{L^{2}(\Omega)}$.

From the definition of (4.5), we note that

[TABLE]

This implies that for any $(\varphi,S)\in M_{\lambda,\mu}$, $\|\frac{\lambda}{\mu}h(\varphi,S)\|_{L^{\infty}(\Omega)}=O(1)$.

From the definition of $U_{\lambda,q}$ and $u_{0}$, $e^{U_{\lambda,q}}=e^{w_{\lambda,q}(y)+O(1)}$ and $u_{0}=O(1)$ in $B_{d}(q)$. This implies that

[TABLE]

In $\Omega\setminus B_{d}(q)$, $e^{U_{\lambda,q}}=e^{-2\mathfrak{M}\ln(d\lambda)+O(1)}$ and $e^{u_{0}}=O(1)$. This implies that

[TABLE]

It follows that $\|e^{U_{\lambda,q}+u_{0}}\|_{L^{2}(\Omega)}\leq C\lambda^{-1}$ and $\|e^{U_{\lambda,q}+u_{0}}\|_{L^{\infty}(\Omega)}=O(1)$.

From (4.7), we get that

[TABLE]

Using Taylor’s Theorem, we see that for some $0\leq\sigma\leq 1$,

[TABLE]

We also obtain

[TABLE]

From Lemma 4.2, we obtain that

[TABLE]

Therefore,

[TABLE]

Next, we consider $\|g_{1,\lambda,\mu}(\varphi,\hat{S})\|_{Y_{\alpha}}$, where $\hat{S}=L^{-1}_{2}(g_{2,\lambda,\mu}(\varphi,S))$.

From (4.20), we can rewrite (4.7) in $B_{2d}(q)$ as

[TABLE]

From the decay of $w$ and mean value theorem, there exists some $0\leq\sigma,\sigma^{\prime}\leq 1$ such that

[TABLE]

This implies that

[TABLE]

We again apply mean value theorem to $u_{0}(\lambda^{-1}z+q)-u_{0}(q)$ and $\gamma(\lambda^{-1}z+q,q)-\gamma(q,q)$, then for some $0\leq\sigma,\sigma^{\prime}\leq 1$ we get that

[TABLE]

and

[TABLE]

Since $u_{0}$ and $\gamma$ are regular and $e^{w(z)}\leq C(1+|z|)^{-2\mathfrak{M}}$ in $B_{2d\lambda}(0)$, we get that

[TABLE]

From Taylor’s Theorem, there exists some $0\leq\sigma\leq 1$ such that

[TABLE]

We have

[TABLE]

where $y:=\lambda^{-1}z+q$. From Theorem 4.2 and $\|g_{2,\lambda,\mu}(\varphi,\hat{S})\|_{L^{2}(\Omega)}$, we note that

[TABLE]

This implies that

[TABLE]

From the definition of $\theta$, it follows that

[TABLE]

Finally,

[TABLE]

Since

[TABLE]

this implies that

[TABLE]

In $\Omega\setminus B_{d}(q)$, we have

[TABLE]

Since $e^{U_{\lambda,q}}\leq c\lambda^{-2\mathfrak{M}}$ in $\Omega\setminus B_{d}(q)$, it follows that

[TABLE]

Therefore, Theorem 4.2 and the assumption $\lambda^{2}\ln\lambda<\mu$ yield for any $(\varphi,S)\in M_{\lambda,\mu}$

[TABLE]

From Theorem 4.1 and 4.2, the inequalities (4.40) and (4.52) yield that $\Psi(\varphi,S)\in M_{\lambda,\mu}$ for any $(\varphi,S)\in M_{\lambda,\mu}$.

Step 2. We claim that for any $(\varphi_{1},S_{1})$ and $(\varphi_{2},S_{2})$ in $M_{\lambda,\mu}$, there exists some constant $0<\tau<1$ such that

[TABLE]

Firstly, we see that

[TABLE]

By the similar way in Step 1, we can get that

[TABLE]

Next, we consider $\|g_{1,\lambda,\mu}(\varphi_{1},\hat{S}_{1})-g_{1,\lambda,\mu}(\varphi_{2},\hat{S}_{2})\|_{Y_{\alpha}}$, where $\hat{S}_{i}=L^{-1}_{2}(g_{2,\lambda,\mu}(\varphi_{i},S_{i}))$, $i=1,2$. We see that

[TABLE]

By the similar way in Step 1, we can get that

[TABLE]

In view of (LABEL:diffg2)-(4.56), Theorem 4.1, and Theorem 4.2, we can prove the claim (4.53).

∎

Completion of the proof of Theorem 1.2. By Proposition 4.1, we get that for any large $\lambda,\mu>0$ and any $q$ close to $\hat{q}$, where ${\hat{q}}$ is a non-degenerate critical point of $u_{0}$, there are $(\varphi_{q},S_{q})\in M_{\lambda,\mu}$ and constants $c_{q,j}$ such that

[TABLE]

In the following, we will choose $q$ suitably (depending on $\lambda,\mu>0$) such that the corresponding constants $c_{q,j}$ are zero and thus $(u_{\lambda,\mu},N_{\lambda,\mu})$ is a solution to (1.9), where

[TABLE]

It is standard to prove the following.

Lemma 4.3**.**

If

[TABLE]

then $c_{q,j}=0$ for $j=1,2$.

Next we have the following reduced problem:

Lemma 4.4**.**

[TABLE]

for some $a_{0}\neq 0$.

Proof.

Since W_{q,j}=\chi(y-q)\frac{\partial w(\lambda|y-q|)}{\partial q_{j}}=-\lambda\chi(y-q)\frac{\partial w(z)}{\partial z_{j}}\Big{|}_{z=\lambda(y-q)}, we see that

[TABLE]

We will estimate the above term by term.

Step 1. We claim that $I=o(1)$.

Note that

[TABLE]

Together with the integration by parts, we have

[TABLE]

Step 2. We claim that $II=a_{0}D_{j}u_{0}(q)+o(1)$ for some $a_{0}\neq 0$.

Recall the definition of $U_{\lambda,q}$ from (4.3), and let $\Gamma(y)=u_{0}(y)+4\pi\mathfrak{M}(1-\theta)\gamma(y,q)$. From the radial symmetry and decay rate of $w(z)$, we see that for some $0\leq\tau\leq 1$

[TABLE]

It has been known that $D_{j}\gamma(q,q)=0$ for any $q\in\Omega$ and $j=1,2$, which implies $D_{j}\Gamma(q)=D_{j}u_{0}(q)$. Together with $F(w(x))>0$ for all $x\in\mathbb{R}^{2}$, we prove the claim.

**Step 3.**We claim that $III+IV+V+VI=o(1)$. For some $0\leq\tau\leq 1$ we note that

[TABLE]

Moreover, since $(\varphi_{q},S_{q})$ is a fixed point of $\Psi$, we have

[TABLE]

From the proof of Proposition 4.1, we know that $\|g_{2,\lambda,\mu}(\varphi_{q},S_{q})\|_{L^{2}(\Omega)}=O(\mu+\frac{\mu^{2}(\ln\lambda)^{4}}{\lambda^{3}})$ since $(\varphi_{q},S_{q})\in M_{\lambda,\mu}$. From Theorem 4.2, it follows $\|S_{q}\|_{L^{2}(\Omega)}\leq O(\frac{1}{\mu}+\frac{(\ln\lambda)^{4}}{\lambda^{3}})$. Then, by the assumption $\lambda^{2}\ln\lambda<\mu$ and the similar way in (4.44), we get that

[TABLE]

We recall $h(\varphi_{q},S_{q})=e^{U_{\lambda,q}+u_{0}}(1+\varphi_{q})+S_{q}$. In the estimation in (4.49), the assumption $\lambda^{2}\ln\lambda<\mu$ yields that

[TABLE]

From the above estimates, we can derive (LABEL:finalclaim). ∎

From Lemma 4.4, we can derive (LABEL:finalclaim). Since we assume that $Du_{0}(\hat{q})=0$ and $D^{2}(u_{0})(\hat{q})$ is nondegenerate, from Lemma 4.4, we can find a point $q$ near $\hat{q}$ such that the right hand side of (LABEL:finalclaim) is equal to zero. Together with Lemma 4.3, we can find $q$ satisfying $c_{q,j}=0$ for $j=1,2$. At this point, we complete the proof of Theorem 1.2.

∎

5. Proof of Theorem 1.3

In this section, we are going to construct blow up solutions of (3.2) at the vortex point satisfying $\sup u_{\lambda,\mu}\geq-c_{0}>-\infty.$ Based on Theorem 1.1, our construction in this section was inspired by the arguments in section 4 and the construction in [41] where the authors construct blow up solutions at the vortex point using an entire solution for the Chern-Simons equation, which has a singularity, as the building blocks.

We recall the equation (3.2) as follows:

[TABLE]

Throughout this section, we assume that $\mathfrak{M}>4$. First of all, we are going to define the approximate solutions for (3.2). Let $V$ be the radially symmetric solution of

[TABLE]

where $a_{1}$ and $I_{1}$ are constants (see [14, Theorem 2.1, Lemma 2.6] for the existence of $V$ satisfying (5.1)). We set

[TABLE]

where $C_{\lambda}=2\ln\lambda+4\pi\left(\gamma(p_{1},p_{1})+\sum_{j=2}^{N}m_{j}G(p_{1},p_{j})\right)$ and

[TABLE]

which makes $U_{\lambda}\in C^{1}(\Omega)$, We would find a solution of (3.2) with the following form:

[TABLE]

here $(\varphi,S)$ would be regard as an error term.

We note that $e^{U_{\lambda}}=O(\lambda^{2})$ in (5.2), but $e^{U_{\lambda,q}}=O(1)$ in the section 4. In order to control the difficulties arising from the error parts related to $\varphi^{2}$ term we need to make the difference between (4.4) and (5.3).

For the convenience, we also denote

[TABLE]

The equation (3.2) is reduced to a system for $(\varphi,S)$:

[TABLE]

where

[TABLE]

For a small constant $0<\alpha<\frac{1}{2}$, recall that

[TABLE]

We say that $\psi\in X_{\alpha}$ if

[TABLE]

where $\tilde{\psi}(z)=\psi(\lambda^{-1}z+p_{1})$, and that $\psi\in Y_{\alpha}$ if

[TABLE]

We note that $\|\cdot\|_{X_{\alpha}}$ and $\|\cdot\|_{Y_{\alpha}}$ are similar to the norms $\|\cdot\|_{X_{\alpha,q}}$ and $\|\cdot\|_{Y_{\alpha,q}}$ in section 4, but the scaled area is different. We recall the preliminary results for the linear operator $L_{1}$ in [41], where

[TABLE]

Theorem 5.1** (Theorem B.1 in [41]).**

$L_{1}$* is an isomorphism from $X_{\alpha}$ to $Y_{\alpha}$. Moreover, if $w\in X_{\alpha}$ and $h\in Y_{\alpha}$ satisfy $L_{1}w=h$, then there is a constant $C>0$, independent of $\lambda>0$, such that*

[TABLE]

Next, let us consider the corresponding nonlinear problem. We define an operator $\Psi$ by

[TABLE]

where $\hat{S}=L^{-1}_{2}(h_{2,\lambda,\mu}(\varphi,S))$, and a subset $C_{\lambda,\mu}$ of $X_{\alpha}\times W^{2,2}(\Omega)$ by

[TABLE]

where

[TABLE]

We note that if $(\varphi,S)\in C_{\lambda,\mu}$, then

[TABLE]

The following estimation would be important for the contraction argument.

Lemma 5.1**.**

There exists a constant $C$ such that

[TABLE]

for any $(\varphi,S)\in C_{\lambda,\mu}$.

Proof.

Although we have $e^{U_{\lambda,q}}=O(1)$ from (4.3) in the section 4, we note that $e^{U_{\lambda}}=O(\lambda^{2})$ from (5.2). Except this observation, we can follow the arguments in the proof of Lemma 4.2, and obtain Lemma 5.1. We skip the detail. ∎

Completion of the proof of Theorem 1.3. First of all, we claim that there exists a fixed point $(\bar{\varphi},\bar{S})\in C_{\lambda,\mu}$ of the operator $\Psi$.

As in the proof of Proposition 4.1, Lemma 5.1 and (5.2) imply that there is a constant $C>0$ satisfying

[TABLE]

and

[TABLE]

We remark that the difference between (4.40)-(4.52) and (5.10)-(5.11) comes from the setting of solution in (4.4) and (5.3) in addition to Lemma 4.2 and Lemma 5.1. From Theorem 5.1 and 4.2, the inequalities (5.10) -(5.11) and the assumption $1\ll(\ln\lambda)^{5}\lambda^{5}\ll\mu$ yield that $\Psi(\varphi,S)\in C_{\lambda,\mu}$ for any $(\varphi,S)\in C_{\lambda,\mu}$.

Similarly, we can also get that if $1\ll(\ln\lambda)^{5}\lambda^{5}\ll\mu$ and $(\varphi_{1},S_{1}),(\varphi_{2},S_{2})\in C_{\lambda,\mu}$, then

[TABLE]

and

[TABLE]

The estimations (LABEL:5diffg2)-(5.13), Theorem 5.1, and Theorem 4.2 imply that if $(\varphi_{1},S_{1}),\ (\varphi_{2},S_{2})\in C_{\lambda,\mu}$, then there exists a constant $0<\tau<1$ satisfying

[TABLE]

In view of the contraction mapping theorem, we get that if $1\ll(\ln\lambda)^{5}\lambda^{5}\ll\mu$, there exists $(\bar{\varphi},\bar{S})\in C_{\lambda,\mu}$ satisfying

[TABLE]

We note that $\left(u_{\lambda,\mu},N_{\lambda,\mu}\right):=\left(U_{\lambda}+\bar{\varphi}-\frac{\lambda}{\mu}\left(e^{U_{\lambda}+u_{0}+\bar{\varphi}}+\bar{S}\right),\lambda\left(e^{U_{\lambda}+u_{0}+\bar{\varphi}}+\bar{S}\right)\right)$ satisfies the system (3.2), and thus complete the proof of Theorem 1.3. ∎

**Acknowledgement

**W. Ao was supported by NSFC (No. 11801421 and No. 11631011). O. Kwon was supported by Young Researcher Program through the National Research Foundation of Korea (NRF) (No. NRF-2016R1C1B2014942). Y. Lee was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. NRF-2018R1C1B6003403).