Normalized Solutions of the Gross-Pitaevskii System with Inhomogeneous Interactions

Rui Sun

doi:10.4236/oalib.1114973

Open Access Library Journal > Vol.13 No.3, March 2026

Normalized Solutions of the Gross-Pitaevskii System with Inhomogeneous Interactions

Rui Sun
School of Mathematics, Liaoning Normal University, Dalian, China.
DOI: 10.4236/oalib.1114973 PDF HTML XML 50 Downloads 284 Views

Abstract

This paper is devoted to the normalized solutions of the two-dimensional Gross-Pitaevskii system with a microwave field and inhomogeneous interactions. By investigating the relevant $L^{2}$ -critical constrained variational problem, we get the existence and nonexistence of the normalized solutions under suitable assumptions about the interaction potentials. We establish the existence and nonexistence of minimizers of $e (γ, a)$ via a threshold $a^{*}$ , where $a^{*}$ is the square of $L^{2}$ -norm of the unique positive solution of $Δ u - u + u^{3} = 0$ in $ℝ^{2}$ . We also analyze the limiting behavior of the constraint minimizers as $a ↗ 2 a^{*}$ through overcoming the challenges associated with the sign-changing property of the logarithmic convolutions and the impact of inhomogeneous interactions.

Keywords

Gross-Pitaevskii System, -Critical Exponent, Logarithmic Convolution, Inhomogeneous Interaction

Share and Cite:

Sun, R. (2026) Normalized Solutions of the Gross-Pitaevskii System with Inhomogeneous Interactions. <i>Open Access Library Journal</i>, <b>13</b>, 1-25. doi: <a href='https://doi.org/10.4236/oalib.1114973' target='_blank' onclick='SetNum(149938)'>10.4236/oalib.1114973</a>.

1. Introduction

In this paper, we focus on the following two-component Gross-Pitaevskii system coupled with a microwave field and inhomogeneous interactions in $ℝ^{2}$ :

${\begin{array}{l} - Δ u_{1} + {| x |}^{2} u_{1} - ϕ u_{2} = λ u_{1} + a k (x) {| u_{2} |}^{2} u_{1} & in ℝ^{2}, \\ - Δ u_{2} + {| x |}^{2} u_{2} - ϕ u_{1} = λ u_{2} + a k (x) {| u_{1} |}^{2} u_{2} & in ℝ^{2}, \\ - Δ ϕ = γ u_{1} u_{2} & in ℝ^{2}, \end{array}$ (1.1)

where $λ \in ℝ$ is the chemical potential, $a \in ℝ$ denotes the contact interaction strength between two components, $k (x)$ stands for the inhomogeneous interactions, and $γ > 0$ (resp. < 0) represents that the magnetic field is attractive (resp. repulsive). The system (1.1) originates from Bose-Einstein Condensates (BECs) interacting with a microwave field at low temperature. The microwave field can influence BECs by the local field effect which plays an important role in the process of forming electromagnetic-matter waves. Due to the physical significance of this system, our goal is to research the existence, nonexistence and limiting behavior of the complex-valued states in system (1.1).

Another purpose of studying (1.1) derives from recent research [1]-[5]. In these references [1]-[4], the authors investigate various models, including single-component attractive BECs, two-component attractive BECs and rotating attractive BECs. They research a range of properties such as existence, nonexistence, uniqueness, quality concentration, symmetry breaking, and refined spike profiles of ground states. However, there is relatively little research on planar multi-component BECs interacting with electromagnetic fields. Recently, Wang, Cai and Wang [5] explored (1.1) in the complex-valued case. They obtained results about the existence and nonexistence of central vortex steady states. Now, we add inhomogeneous interactions to (1.1) for further research. For $0 < k (x) \leq 1$ , we give the existence of minimizers of $e (γ, a)$ in the case of $a \in (0, 2 a^{*})$ , where $a^{*}$ is the square of $L^{2}$ -norm of the unique positive solution of

$- Δ u + u = u^{3}, u \in H^{1} (ℝ^{2}) .$ (1.2)

In this paper, we are interested in the nontrivial solutions of (1.1) under the normalization constraint

$\int_{ℝ^{2}} (u_{1}^{2} + u_{2}^{2}) d x = 1.$ (1.3)

Towards this purpose, with the help of the fundamental solution of $- Δ$ , we define the energy functional

$\begin{array}{l} E_{γ, a} (u_{1}, u_{2}) : = \frac{1}{2} \int_{ℝ^{2}} [{| \nabla u_{1} |}^{2} + {| \nabla u_{2} |}^{2} + {| x |}^{2} (u_{1}^{2} + u_{2}^{2})] d x \\ + \frac{γ}{4 π} \int_{ℝ^{2}} \int_{ℝ^{2}} \ln | x - y | u_{1} (x) u_{2} (x) u_{1} (y) u_{2} (y) d x d y \\ - \frac{a}{2} \int_{ℝ^{2}} k (x) u_{1}^{2} u_{2}^{2} d x . \end{array}$ (1.4)

Due to the appearance of the harmonic potential term and the logarithmic convolution term, $E_{γ, a}$ is not well-defined on $H^{1} (ℝ^{2}) \times H^{1} (ℝ^{2})$ . Inspired by [3] [6] [7], we work in the subspace $X : = H \times H$ , where

$H : = {u \in H^{1} (ℝ^{2}) : {| u |}_{*} : = {(\int_{ℝ^{2}} {| x |}^{2} u^{2} (x) d x)}^{\frac{1}{2}} < \infty} .$

Define the norm on $X$ by

${‖ (u_{1}, u_{2}) ‖}_{X}^{2} = {‖ u_{1} ‖}_{H}^{2} + {‖ u_{2} ‖}_{H}^{2} = \int_{ℝ^{2}} [{| \nabla u_{1} |}^{2} + {| \nabla u_{2} |}^{2} + (1 + {| x |}^{2}) (u_{1}^{2} + u_{2}^{2})] d x .$

According to ([2] lemma 2.1), we have

$The embedding X = H \times H$ ↪ $L^{q} (ℝ^{2}) \times L^{q} (ℝ^{2}) is compact for all q \geq 2.$ (1.5)

We now define the constraint set

$M : = {(u_{1}, u_{2}) \in X : \int_{ℝ^{2}} (u_{1}^{2} + u_{2}^{2}) d x = 1},$

and focus on studying the following constrained variational problem:

$e (γ, a) : = \inf_{(u_{1}, u_{2}) \in M} E_{γ, a} (u_{1}, u_{2}) .$ (1.6)

Throughout the paper, we assume that the potential $- 1 \leq k (x) \leq 1$ satisfies the following assumptions:

$\lim_{x \to 0} k (x) = 1,$ (1.7)

$1 - \frac{a}{2 a^{*}} k (x) \leq C {| x |}^{b},$ (1.8)

where $C > 0$ and $b \geq 2$ are some constants, and $k (x) \in C^{0, α} (ℝ^{2})$ for some $α \in (0, 1)$ .

Stimulated by [6] [8], we perform that

$\begin{array}{l} \int_{ℝ^{2}} \int_{ℝ^{2}} \ln | x - y | u_{1} (x) u_{2} (x) u_{1} (y) u_{2} (y) d x d y \\ = \int_{ℝ^{2}} \int_{ℝ^{2}} \ln (1 + | x - y |) u_{1} (x) u_{2} (x) u_{1} (y) u_{2} (y) d x d y \\ - \int_{ℝ^{2}} \int_{ℝ^{2}} \ln (1 + {| x - y |}^{- 1}) u_{1} (x) u_{2} (x) u_{1} (y) u_{2} (y) d x d y \\ : = F_{1} (u_{1}, u_{2}) - F_{2} (u_{1}, u_{2}) . \end{array}$ (1.9)

For simplicity, we use ${| \cdot |}_{q}$ to denote the standard Lebesgue norm on $L^{q} (ℝ^{2})$ for $q \in [1, \infty]$ . Since

$ln (1 + | x - y |) \leq | x - y | \leq | x | + | y | for x, y \in ℝ^{2},$

We deduce by Hölder inequality that

$\begin{matrix} | F_{1} (u_{1}, u_{2}) | \leq \int_{ℝ^{2}} | x | | u_{1} (x) | | u_{2} (x) | d x \int_{ℝ^{2}} | u_{1} (y) | | u_{2} (y) | d y \\ + \int_{ℝ^{2}} | u_{1} (x) | | u_{2} (x) | d x \int_{ℝ^{2}} | y | | u_{1} (y) | | u_{2} (y) | d y \\ \leq 2 {| u_{1} |}_{*} {| u_{1} |}_{2} {| u_{2} |}_{2}^{2} for (u_{1}, u_{2}) \in X, \\ \leq {| u_{1} |}_{*} {| u_{2} |}_{2} for (u_{1}, u_{2}) \in M . \end{matrix}$ (1.10)

Applying Hardy-Littlewood-Sobolev inequality (cf. ([9], Theorem 4.3)), we then derive that there exists a constant $C > 0$ such that

$\begin{matrix} | F_{2} (u_{1}, u_{2}) | \leq \int_{ℝ^{2}} \int_{ℝ^{2}} \frac{| u_{1} (x) u_{2} (x) u_{1} (y) u_{2} (y) |}{| x - y |} d x d y \\ \leq C {| u_{1} u_{2} |}_{\frac{4}{3}}^{2} \leq C {| u_{1} |}_{\frac{8}{3}}^{2} {| u_{2} |}_{\frac{8}{3}}^{2} for (u_{1}, u_{2}) \in L^{\frac{8}{3}} (ℝ^{2}) \times L^{\frac{8}{3}} (ℝ^{2}) . \end{matrix}$ (1.11)

It follows from (1.9)-(1.11) that $\int_{ℝ^{2}} \int_{ℝ^{2}} \ln | x - y | u_{1} (x) u_{2} (x) u_{1} (y) u_{2} (y) d x d y$ is well-defined on $X$ . We can get that $E_{γ, a}$ is of class $C^{1}$ .

The existence and nonexistence of minimizers of the constrained variational problem (1.6) depend strongly on the following Gagliardo-Nirenberg type inequality (cf. ([2], lemma A. 2])):

$\int_{ℝ^{2}} {({| u_{1} |}^{2} + {| u_{2} |}^{2})}^{2} d x \leq \frac{2}{{| Q |}_{2}^{2}} \int_{ℝ^{2}} ({| \nabla u_{1} |}^{2} + {| \nabla u_{2} |}^{2}) d x \int_{ℝ^{2}} ({| u_{1} |}^{2} + {| u_{2} |}^{2}) d x,$ (1.12)

where $(u_{1}, u_{2}) \in H^{1} (ℝ^{2}) \times H^{1} (ℝ^{2})$ and $Q$ is the unique positive solution of (1.2). For any $θ \in (0, 2 π)$ , the equality in (1.12) can be attained at $(Q \sin θ, Q \cos θ)$ . Note from ([10], Proposition 4.1) that $Q$ decays exponentially as $| x | \to \infty$ in the sense that

$| Q (x) |, | \nabla Q (x) | = O ({| x |}^{- \frac{1}{2}} e^{- | x |}) as | x | \to \infty .$ (1.13)

Moreover, recall from [11] that $Q$ is the achievement function of the equality in the following classical Gagliardo-Nirenberg inequality:

$\int_{ℝ^{2}} {| u |}^{4} d x \leq \frac{2}{{| Q |}_{2}^{2}} \int_{ℝ^{2}} {| \nabla u |}^{2} d x \int_{ℝ^{2}} {| u |}^{2} d x, u \in H^{1} (ℝ^{2}) .$ (1.14)

We then conclude from (1.2) and (1.14) that

$\int_{ℝ^{2}} {| \nabla Q |}^{2} d x = \int_{ℝ^{2}} Q^{2} d x = \frac{1}{2} \int_{ℝ^{2}} Q^{4} d x .$ (1.15)

Based on the above results, we can establish the existence and nonexistence of minimizers for the constrained variational problem (1.6).

Theorem 1.1 Let $Q (x) = Q (| x |) > 0$ be the unique positive solution of (1.2) and $a^{*} : = {| Q |}_{2}^{2}$ , assume that $k (x)$ always satisfies (1.7) and (1.8).

1) If $- 1 \leq k (x) \leq 0$ , then there exists at least one minimizer of $e (γ, a)$ for $γ \neq 0$ and $a > 0$ .

2) If $0 < k (x) \leq 1$ , then there exists at least one minimizer of $e (γ, a)$ for $γ \neq 0$ and $0 < a < 2 a^{*}$ , and then there is no minimizer of $e (γ, a)$ for $γ \neq 0$ , $a > 2 a^{*}$ , or $γ > 0$ , $a = 2 a^{*}$ .

Theorem 1.1 establishes the existence and nonexistence of minimizers of $e (γ, a)$ . When $- 1 \leq k (x) \leq 0$ , we can get that $E_{γ, a}$ is bounded from below based on the non-positivity of $k (x)$ , then we can get the existence of minimizers of $e (γ, a)$ . As for $0 < k (x) \leq 1$ , then we can also get the existence of minimizers of $e (γ, a)$ by rewriting $E_{γ, a}$ into (2.8). Moreover, when $a \geq 2 a^{*}$ , we use appropriate test function to get that there is no minimizer of $e (γ, a)$ . If $(u_{1}^{a}, u_{2}^{a}) \in M$ is a minimizer of $e (γ, a)$ for $γ \neq 0$ and $0 < a < 2 a^{*}$ , then the variational theory shows that $(u_{1}^{a}, u_{2}^{a})$ solves

${\begin{array}{l} - Δ u_{1}^{a} + {| x |}^{2} u_{1}^{a} + \frac{γ}{2 π} \int_{ℝ^{2}} \ln | x - y | u_{1}^{a} (y) u_{2}^{a} (y) d y u_{2}^{a} = λ_{a} u_{1}^{a} + a k (x) {| u_{2}^{a} |}^{2} u_{1}^{a} in ℝ^{2}, \\ - Δ u_{2}^{a} + {| x |}^{2} u_{2}^{a} + \frac{γ}{2 π} \int_{ℝ^{2}} \ln | x - y | u_{1}^{a} (y) u_{2}^{a} (y) d y u_{1}^{a} = λ_{a} u_{2}^{a} + a k (x) {| u_{1}^{a} |}^{2} u_{2}^{a} in ℝ^{2}, \end{array}$ (1.16)

where $λ_{a} \in ℝ$ is the Lagrange multiplier associated with (1.3) and satisfies that

$\begin{array}{l} λ_{a} = 2 e (γ, a) - a \int_{ℝ^{2}} k (x) {| u_{1}^{a} |}^{2} {| u_{2}^{a} |}^{2} d x \\ + \frac{γ}{2 π} \int_{ℝ^{2}} \int_{ℝ^{2}} \ln | x - y | u_{1}^{a} (x) u_{2}^{a} (x) u_{1}^{a} (y) u_{2}^{a} (y) d x d y . \end{array}$ (1.17)

The second result in this paper deals with the limiting behavior of minimizers of $e (γ, a)$ as $a ↗ 2 a^{*}$ , where $γ > 0$ is fixed. We always denote by $\to$ and $⇀$ the strong convergence and the weak convergence, respectively.

Theorem 1.2 Let $(u_{1}^{a}, u_{2}^{a}) \in M$ be a minimizer of $e (γ, a)$ , where $γ > 0$ is fixed and $a \in (0, 2 a^{*})$ . Assume that $0 < k (x) \leq 1$ and $k (x) \in C^{0, α} (ℝ^{2})$ for some $α \in (0, 1)$ , then there exist a sequence ${y_{ε_{a}}} \subset ℝ^{2}$ and a point $x_{0} \in ℝ^{2}$ such that

$ε_{a} | u_{i}^{a} (ε_{a} x + ε_{a} y_{ε_{a}}) | \to \frac{Q (x - x_{0})}{\sqrt{2 a^{*}}}$ in $H^{1} (ℝ^{2})$ as $a ↗ 2 a^{*}$ ,

where $i = 1, 2$ and

$ε_{a} : = {[\int_{ℝ^{2}} ({| \nabla u_{1}^{a} |}^{2} + {| \nabla u_{2}^{a} |}^{2}) d x]}^{- \frac{1}{2}} \to 0$ as $a ↗ 2 a^{*}$ .

Moreover, $lim_{a ↗ 2 a^{*}} λ_{a} ε_{a}^{2} = - 1$ . Particularly, if $(u_{1}^{a}, u_{2}^{a}) \in M$ is non-negative, then $\sqrt{\frac{8 π (2 a^{*} - a)}{γ a^{*}}} u_{i}^{a} (\sqrt{\frac{8 π (2 a^{*} - a)}{γ a^{*}}} x + x_{i}^{a}) \to \frac{Q (x)}{\sqrt{2 a^{*}}}$ in $H \cap L^{\infty} (ℝ^{2})$ as $a ↗ 2 a^{*}$ , where $x_{i}^{a}$ is the unique global maximum point of $u_{i}^{a}$ and $x_{i}^{a} \to 0$ as $a ↗ 2 a^{*}$ .

For any minimizer $(u_{1}^{a}, u_{2}^{a}) \in M$ , Theorem 1.2 provides the preliminary limiting behavior as $a ↗ 2 a^{*}$ . Moreover, Theorem 1.2 also gives the more refined convergence information when the minimizer is non-negative. In order to prove Theorem 1.2, we shall show that

$e (γ, a) \approx \frac{γ}{32 π} \ln \frac{8 π (2 a^{*} - a)}{γ a^{*}} + C as a ↗ 2 a^{*} .$

The above Theorems show that the system (1.1) has a stable normalized state when $0 < a < 2 a^{*}$ , and the stable state disappears when $a \geq 2 a^{*}$ . The inhomogeneous interaction $k (x)$ only affects the range of stable states and does not alter the cole role of critical threshold. In short, this result reveals the stability conditions and critical behavior of two-component BECs under microwave field, providing a physical basis for the artificial control of BECs configuration by regulating the interaction strength.

The rest of this paper is organized as follows. In section 2, we prove Theorem 1.1 on the existence and nonexistence of minimizers. In section 3, we analyze the limiting behavior of minimizers and prove Theorem 1.2.

2. Existence and Nonexistence of Minimizers

In this section, we prove the existence and nonexistence of minimizers for $e (γ, a)$ , where $γ \neq 0$ and $a > 0$ . For the case that $0 < k (x) \leq 1$ , the crucial technique is that we rewrite the energy functional $E_{γ, a} (u_{1}, u_{2})$ into a new form (2.8) and we use an important estimate (2.6).

Proof of Theorem 1.1. 1) For the case that $- 1 \leq k (x) \leq 0$ , we prove the existence of minimizers of $e (γ, a)$ , where $γ \neq 0$ and $a > 0$ . By Young inequality, we derive from (1.10) that

$\begin{matrix} | F_{1} (u_{1}, u_{2}) | \leq \frac{π}{| γ |} {| u_{1} |}_{*}^{2} + \frac{| γ |}{4 π} {| u_{2} |}_{2}^{2} \\ \leq \frac{π}{| γ |} \int_{ℝ^{2}} {| x |}^{2} (u_{1}^{2} + u_{2}^{2}) d x + \frac{| γ |}{4 π} for (u_{1}, u_{2}) \in M . \end{matrix}$ (2.1)

Recall from [11] the general Gagliardo-Nirenberg inequality: for any $v \in H^{1} (ℝ^{2})$ and $q \geq 2$ ,

$\int_{ℝ^{2}} {| v |}^{q} d x \leq \frac{q}{2 {| Q_{q} |}_{2}^{q - 2}} {(\int_{ℝ^{2}} {| \nabla v |}^{2} d x)}^{\frac{q - 2}{2}} \int_{ℝ^{2}} {| v |}^{2} d x,$ (2.2)

where $Q_{q}$ is the unique positive solution of

$- \frac{q - 2}{2} Δ v + v = v^{q - 1}, v \in H^{1} (ℝ^{2}) .$

Combining (1.11) with (2.2) yields that there exists a constant $C > 0$ such that

$\begin{matrix} | F_{2} (u_{1}, u_{2}) | \leq C {(\int_{ℝ^{2}} {| \nabla u_{1} |}^{2} d x)}^{\frac{1}{4}} {(\int_{ℝ^{2}} {| \nabla u_{2} |}^{2} d x)}^{\frac{1}{4}} \\ \leq C {[\int_{ℝ^{2}} ({| \nabla u_{1} |}^{2} + {| \nabla u_{2} |}^{2}) d x]}^{\frac{1}{2}} for (u_{1}, u_{2}) \in M . \end{matrix}$ (2.3)

Following (1.9), (1.12), (2.1) and (2.3), we infer that

$\begin{array}{l} E_{γ, a} (u_{1}, u_{2}) \geq \frac{1}{2} \int_{ℝ^{2}} ({| \nabla u_{1} |}^{2} + {| \nabla u_{2} |}^{2}) d x + \frac{1}{4} \int_{ℝ^{2}} {| x |}^{2} (u_{1}^{2} + u_{2}^{2}) d x \\ - C {[\int_{ℝ^{2}} ({| \nabla u_{1} |}^{2} + {| \nabla u_{2} |}^{2}) d x]}^{\frac{1}{2}} - \frac{γ^{2}}{16 π^{2}} for (u_{1}, u_{2}) \in M, \end{array}$ (2.4)

which implies that $E_{γ, a}$ is bounded from below on $M$ . Let ${(u_{1, n}, u_{2, n})} \subset M$ be a minimizing sequence of $e (γ, a)$ . We get from (2.4) that $\int_{ℝ^{2}} ({| \nabla u_{1, n} |}^{2} + {| \nabla u_{2, n} |}^{2}) d x$ and $\int_{ℝ^{2}} {| x |}^{2} (u_{1, n}^{2} + u_{2, n}^{2}) d x$ are bounded uniformly with respect to $n$ . Combining with the fact ${| u_{1, n} |}_{2}^{2} + {| u_{2, n} |}_{2}^{2} = 1$ , we then obtain that ${(u_{1, n}, u_{2, n})}$ is bounded uniformly in $X$ . By (1.5), there is $(u_{1}, u_{2}) \in X$ such that

$(u_{1, n}, u_{2, n}) ⇀ (u_{1}, u_{2}) in X,$

$(u_{1, n}, u_{2, n}) \to (u_{1}, u_{2}) in L^{q} (ℝ^{2}) \times L^{q} (ℝ^{2}) for any q \in [2, \infty)$

as $n \to \infty$ , which indicates that $(u_{1}, u_{2}) \in M$ . By the weak lower semicontinuity, we have

$\begin{array}{l} \int_{ℝ^{2}} [{| \nabla u_{1} |}^{2} + {| \nabla u_{2} |}^{2} + {| x |}^{2} (u_{1}^{2} + u_{2}^{2})] d x \\ \leq \underset{n \to \infty}{\lim \inf} \int_{ℝ^{2}} [{| \nabla u_{1, n} |}^{2} + {| \nabla u_{2, n} |}^{2} + {| x |}^{2} (u_{1, n}^{2} + u_{2, n}^{2})] d x . \end{array}$

We then claim that

$\begin{array}{l} \int_{ℝ^{2}} \int_{ℝ^{2}} \ln | x - y | u_{1, n} (x) u_{2, n} (x) u_{1, n} (y) u_{2, n} (y) d x d y \\ \to \int_{ℝ^{2}} \int_{ℝ^{2}} \ln | x - y | u_{1} (x) u_{2} (x) u_{1} (y) u_{2} (y) d x d y as n \to \infty . \end{array}$ (2.5)

Actually, note that

$\begin{array}{l} | \int_{ℝ^{2}} \int_{ℝ^{2}} \ln | x - y | u_{1, n} (x) u_{2, n} (x) u_{1, n} (y) u_{2, n} (y) d x d y \\ - \int_{ℝ^{2}} \int_{ℝ^{2}} \ln | x - y | u_{1} (x) u_{2} (x) u_{1} (y) u_{2} (y) d x d y | \\ \leq | \int_{ℝ^{2}} \int_{ℝ^{2}} \ln | x - y | (u_{1, n} (x) - u_{1} (x)) u_{2, n} (x) u_{1, n} (y) u_{2, n} (y) d x d y | \\ + | \int_{ℝ^{2}} \int_{ℝ^{2}} \ln | x - y | u_{1} (x) (u_{2, n} (x) - u_{2} (x)) u_{1, n} (y) u_{2, n} (y) d x d y | \\ + | \int_{ℝ^{2}} \int_{ℝ^{2}} \ln | x - y | u_{1} (x) u_{2} (x) (u_{1, n} (y) - u_{1} (y)) u_{2, n} (y) d x d y | \\ + | \int_{ℝ^{2}} \int_{ℝ^{2}} \ln | x - y | u_{1} (x) u_{2} (x) u_{1} (y) (u_{2, n} (y) - u_{2} (y)) d x d y | \\ : = L_{1, n} + L_{2, n} + L_{3, n} + L_{4, n} . \end{array}$

We denote the norm of any function $u$ in $H^{1} (ℝ^{2})$ by ${‖ u ‖}_{1} : = {[\int_{ℝ^{2}} ({| \nabla u |}^{2} + u^{2}) d x]}^{\frac{1}{2}}$ for convenience. In view of

$| \ln | x - y | | = | \ln (1 + | x - y |) - \ln (1 + {| x - y |}^{- 1}) | \leq | x | + | y | + {| x - y |}^{- 1},$

we then deduce that

$\begin{matrix} L_{1, n} \leq \int_{ℝ^{2}} | x | | u_{1, n} (x) - u_{1} (x) | | u_{2, n} (x) | d x \int_{ℝ^{2}} | u_{1, n} (y) | | u_{2, n} (y) | d y \\ + \int_{ℝ^{2}} | u_{1, n} (x) - u_{1} (x) | | u_{2, n} (x) | d x \int_{ℝ^{2}} | y | | u_{1, n} (y) | | u_{2, n} (y) | d y \\ + \int_{ℝ^{2}} | u_{1, n} (x) - u_{1} (x) | | u_{2, n} (x) | \int_{ℝ^{2}} \frac{| u_{1, n} (y) | | u_{2, n} (y) |}{| x - y |} d y d x \\ \leq {| u_{1, n} - u_{1} |}_{2} {| u_{2, n} |}_{*} {| u_{1, n} |}_{2} {| u_{2, n} |}_{2} + {| u_{1, n} - u_{1} |}_{2} {| u_{2, n} |}_{2}^{2} {| u_{1, n} |}_{*} \\ + C {‖ u_{1, n} ‖}_{1} {‖ u_{2, n} ‖}_{1} {| u_{1, n} - u_{1} |}_{2} {| u_{2, n} |}_{2} \\ \leq C {| u_{1, n} - u_{1} |}_{2} \to 0 as n \to \infty, \end{matrix}$

where we use the fact that there exists a constant $C > 0$ , independent of $n \in ℤ^{+}$ and $x \in ℝ^{2}$ , such that

$\begin{array}{l} \int_{ℝ^{2}} \frac{| u_{1, n} (y) | | u_{2, n} (y) |}{| x - y |} d y \\ = \int_{| x - y | < 1} \frac{| u_{1, n} (y) | | u_{2, n} (y) |}{| x - y |} d y + \int_{| x - y | \geq 1} \frac{| u_{1, n} (y) | | u_{2, n} (y) |}{| x - y |} d y \\ \leq {(\int_{| x - y | < 1} \frac{1}{{| x - y |}^{\frac{3}{2}}} d y)}^{\frac{2}{3}} {(\int_{| x - y | < 1} {| u_{1, n} (y) |}^{3} {| u_{2, n} (y) |}^{3} d y)}^{\frac{1}{3}} \\ + \int_{| x - y | \geq 1} | u_{1, n} (y) | | u_{2, n} (y) | d y \end{array}$

$\leq C {| u_{1, n} |}_{6} {| u_{2, n} |}_{6} + {| u_{1, n} |}_{2} {| u_{2, n} |}_{2} \leq C {‖ u_{1, n} ‖}_{1} {‖ u_{2, n} ‖}_{1} .$ (2.6)

Similarly, we also have

$L_{2, n} \leq C {| u_{2, n} - u_{2} |}_{2} \to 0 as n \to \infty,$

$L_{3, n} \leq C {| u_{1, n} - u_{1} |}_{2} \to 0 as n \to \infty,$

$L_{4, n} \leq C {| u_{2, n} - u_{2} |}_{2} \to 0 as n \to \infty .$

The claim (2.5) is thus proved.

It follows from Hölder inequality that

$\begin{array}{l} | \int_{ℝ^{2}} k (x) u_{1, n}^{2} u_{2, n}^{2} d x - \int_{ℝ^{2}} k (x) u_{1}^{2} u_{2}^{2} d x | \\ \leq \int_{ℝ^{2}} u_{1, n}^{2} | u_{2, n} + u_{2} | | u_{2, n} - u_{2} | d x + \int_{ℝ^{2}} u_{2}^{2} | u_{1, n} + u_{1} | | u_{1, n} - u_{1} | d x \\ \leq {| u_{2, n} + u_{2} |}_{3} {| u_{2, n} - u_{2} |}_{3} {| u_{1, n} |}_{6}^{2} + {| u_{1, n} + u_{1} |}_{3} {| u_{1, n} - u_{1} |}_{3} {| u_{2} |}_{6}^{2} \\ \leq C ({| u_{1, n} - u_{1} |}_{3} + {| u_{2, n} - u_{2} |}_{3}) \to 0 as n \to \infty \end{array}$

then we can get that

$\int_{ℝ^{2}} k (x) u_{1, n}^{2} u_{2, n}^{2} d x \to \int_{ℝ^{2}} k (x) u_{1}^{2} u_{2}^{2} d x as n \to \infty .$

We conclude from the above that

$e (γ, a) \leq E_{γ, a} (u_{1}, u_{2}) \leq \underset{n \to \infty}{\lim \inf} E_{γ, a} (u_{1, n}, u_{2, n}) = e (γ, a),$

which means that $e (γ, a) = E_{γ, a} (u_{1}, u_{2})$ , and hence $(u_{1}, u_{2})$ is a minimizer of $e (γ, a)$ .

2). For the case that $0 < k (x) \leq 1$ , we prove the existence and nonexistence of minimizers of $e (γ, a)$ .

Case 1: $γ \neq 0$ and $a \in (0, 2 a^{*})$ . For any $a \in (0, 2 a^{*})$ , there exists a constant $β > 0$ such that

$\frac{a}{2} \leq β < a^{*} .$ (2.7)

We rewrite the energy functional $E_{γ, a} (u_{1}, u_{2})$ as

$\begin{matrix} E_{γ, a} (u_{1}, u_{2}) = \frac{1}{2} \int_{ℝ^{2}} ({| \nabla u_{1} |}^{2} + {| \nabla u_{2} |}^{2}) d x - \frac{β}{4} \int_{ℝ^{2}} {(\sqrt{k (x)} u_{1}^{2} + \sqrt{k (x)} u_{2}^{2})}^{2} d x \\ + \frac{1}{2} \int_{ℝ^{2}} {| x |}^{2} (u_{1}^{2} + u_{2}^{2}) d x \\ + \frac{γ}{4 π} \int_{ℝ^{2}} \int_{ℝ^{2}} \ln | x - y | u_{1} (x) u_{2} (x) u_{1} (y) u_{2} (y) d x d y \\ + \frac{β}{4} \int_{ℝ^{2}} {(\sqrt{k (x)} u_{1}^{2} - \sqrt{k (x)} u_{2}^{2})}^{2} d x + \frac{2 β - a}{2} \int_{ℝ^{2}} k (x) u_{1}^{2} u_{2}^{2} d x . \end{matrix}$ (2.8)

Following (1.9), (1.12) (2.1), (2.3), (2.7) and (2.8), we infer that

$\begin{array}{l} E_{γ, a} (u_{1}, u_{2}) \geq \frac{1}{2} (1 - \frac{β}{a^{*}}) \int_{ℝ^{2}} ({| \nabla u_{1} |}^{2} + {| \nabla u_{2} |}^{2}) d x + \frac{1}{4} \int_{ℝ^{2}} {| x |}^{2} (u_{1}^{2} + u_{2}^{2}) d x \\ - C {[\int_{ℝ^{2}} ({| \nabla u_{1} |}^{2} + {| \nabla u_{2} |}^{2}) d x]}^{\frac{1}{2}} - \frac{γ^{2}}{16 π^{2}} for (u_{1}, u_{2}) \in M, \end{array}$ (2.9)

which indicates that $E_{γ, a}$ is bounded from below on $M$ . Using the similar argument as the case that $- 1 \leq k (x) \leq 0$ , one can obtain that $e (γ, a)$ admits at least one minimizer.

Case 2: $γ \neq 0$ and $a > 2 a^{*}$ . For any $a > 2 a^{*}$ , we next prove the nonexistence of minimizers of $e (γ, a)$ . Letting $0 \leq θ \leq 1$ , we consider the test function

$(u_{1, τ} (x), u_{2, τ} (x)) : = (\frac{\sqrt{θ} τ}{{| Q |}_{2}} Q (τ x), \frac{\sqrt{1 - θ} τ}{{| Q |}_{2}} Q (τ x)) for τ > 0.$

Clearly, $(u_{1, τ}, u_{2, τ}) \in M$ for all $τ > 0$ . It follows from (1.4) and (1.15) that

$\begin{matrix} E_{γ, a} (u_{1, τ}, u_{2, τ}) = [\frac{1}{2} - \frac{a θ (1 - θ)}{2 {(a^{*})}^{2}} \int_{ℝ^{2}} k (\frac{x}{τ}) Q^{4} (x) d x] τ^{2} + \frac{1}{2 a^{*} τ^{2}} \int_{ℝ^{2}} {| x |}^{2} Q^{2} (x) d x \\ - \frac{γ θ (1 - θ)}{4 π} ln τ + \frac{γ θ (1 - θ)}{4 π {(a^{*})}^{2}} \int_{ℝ^{2}} \int_{ℝ^{2}} \ln | x - y | Q^{2} (x) Q^{2} (y) d x d y . \end{matrix}$ (2.10)

Choose $θ = \frac{1}{2}$ , combining with Fatou’s lemma, then we obtain from (1.7), (1.13) and (2.10) that

$\begin{matrix} e (γ, a) \leq [\frac{1}{2} - \frac{a}{8 {(a^{*})}^{2}} \int_{ℝ^{2}} k (\frac{x}{τ}) Q^{4} (x) d x] τ^{2} + \frac{1}{2 a^{*} τ^{2}} \int_{ℝ^{2}} {| x |}^{2} Q^{2} (x) d x \\ - \frac{γ}{16 π} \ln τ + \frac{γ}{16 π {(a^{*})}^{2}} \int_{ℝ^{2}} \int_{ℝ^{2}} \ln | x - y | Q^{2} (x) Q^{2} (y) d x d y \\ \leq (\frac{1}{2} - \frac{a}{4 a^{*}}) τ^{2} + \frac{1}{2 a^{*} τ^{2}} \int_{ℝ^{2}} {| x |}^{2} Q^{2} (x) d x \\ - \frac{γ}{16 π} \ln τ + \frac{γ}{16 π {(a^{*})}^{2}} \int_{ℝ^{2}} \int_{ℝ^{2}} \ln | x - y | Q^{2} (x) Q^{2} (y) d x d y \\ \to - \infty as τ \to \infty, \end{matrix}$ (2.11)

which implies that there is no minimizers of $e (γ, a)$ for $γ \neq 0$ and $a > 2 a^{*}$ .

Case 3: $γ > 0$ and $a = 2 a^{*}$ . Combining with (1.8), choosing $θ = \frac{1}{2}$ , we can obtain from (2.10) that

$\begin{matrix} e (γ, a) \leq \frac{τ^{2}}{4 a^{*}} \int_{ℝ^{2}} (1 - \frac{a}{2 a^{*}} k (\frac{x}{τ})) Q^{4} (x) d x + \frac{1}{2 a^{*} τ^{2}} \int_{ℝ^{2}} {| x |}^{2} Q^{2} (x) d x \\ - \frac{γ}{16 π} \ln τ + \frac{γ}{16 π {(a^{*})}^{2}} \int_{ℝ^{2}} \int_{ℝ^{2}} \ln | x - y | Q^{2} (x) Q^{2} (y) d x d y \\ \leq C \frac{τ^{2 - b}}{4 a^{*}} \int_{ℝ^{2}} {| x |}^{b} Q^{4} (x) d x + \frac{1}{2 a^{*} τ^{2}} \int_{ℝ^{2}} {| x |}^{2} Q^{2} (x) d x \\ - \frac{γ}{16 π} \ln τ + \frac{γ}{16 π {(a^{*})}^{2}} \int_{ℝ^{2}} \int_{ℝ^{2}} \ln | x - y | Q^{2} (x) Q^{2} (y) d x d y \\ \to - \infty as τ \to \infty . \end{matrix}$ (2.12)

which implies that there is no minimizers of $e (γ, a)$ for $γ > 0$ and $a = 2 a^{*}$ .

This completes the proof of Theorem 1.1.

3. Limiting Behavior of Minimizers

In this section, we prove Theorem 1.2 on the limiting behavior of minimizers $(u_{1}^{a}, u_{2}^{a})$ of $e (γ, a)$ as $a ↗ 2 a^{*}$ , where $γ > 0$ is fixed. We shall make full use of the rewritten form (3.7) of the energy functional (1.4). In order to analyze the blowing-up property of minimizers $(u_{1}^{a}, u_{2}^{a})$ , we define

$ε_{a} : = {[\int_{ℝ^{2}} ({| \nabla u_{1}^{a} |}^{2} + {| \nabla u_{2}^{a} |}^{2}) d x]}^{- \frac{1}{2}} .$ (3.1)

Lemma 3.1 Let $(u_{1}^{a}, u_{2}^{a}) \in M$ be a minimizer of $e (γ, a)$ , where $γ > 0$ is fixed and $a \in (0, 2 a^{*})$ , together with $0 < k (x) \leq 1$ satisfies (1.7) and (1.8). Then

1) $ε_{a} > 0$ satisfies that

$ε_{a} \to 0$ and $λ_{a} ε_{a}^{2} \to - 1$ as $a ↗ 2 a^{*}$ ;(3.2)

2) There exist a sequence ${y_{ε_{a}}} \subset ℝ^{2}$ , and constants $r_{0} > 0, α > 0$ independent of $a \in (0, 2 a^{*})$ , such that

$\underset{a ↗ 2 a^{*}}{\lim \inf} \int_{B_{r_{0}} (0)} \sqrt{k (ε_{a} x + ε_{a} y_{ε_{a}})} {| v_{i}^{a} |}^{2} d x \geq α > 0 for i = 1, 2,$ (3.3)

where

$v_{i}^{a} (x) : = ε_{a} u_{i}^{a} (ε_{a} x + ε_{a} y_{ε_{a}}) for i = 1, 2;$ (3.4)

3) There exists a point $x_{0} \in ℝ^{2}$ such that

$lim_{a ↗ 2 a^{*}} | v_{i}^{a} (x) | = \frac{Q (x - x_{0})}{\sqrt{2 a^{*}}} in H^{1} (ℝ^{2}) for i = 1, 2.$ (3.5)

Proof. 1) We first prove that

$ε_{a} \to 0 as a ↗ 2 a^{*} .$ (3.6)

Rewrite

$\begin{matrix} E_{γ, a} (u_{1}^{a}, u_{2}^{a}) = \frac{1}{2} \int_{ℝ^{2}} ({| \nabla u_{1}^{a} |}^{2} + {| \nabla u_{2}^{a} |}^{2}) d x - \frac{a^{*}}{4} \int_{ℝ^{2}} {(\sqrt{k (x)} {| u_{1}^{a} |}^{2} + \sqrt{k (x)} {| u_{2}^{a} |}^{2})}^{2} d x \\ + \frac{1}{2} \int_{ℝ^{2}} {| x |}^{2} ({| u_{1}^{a} |}^{2} + {| u_{2}^{a} |}^{2}) d x \\ + \frac{γ}{4 π} \int_{ℝ^{2}} \int_{ℝ^{2}} \ln | x - y | u_{1}^{a} (x) u_{2}^{a} (x) u_{1}^{a} (y) u_{2}^{a} (y) d x d y \\ + \frac{a^{*}}{4} \int_{ℝ^{2}} {(\sqrt{k (x)} {| u_{1}^{a} |}^{2} - \sqrt{k (x)} {| u_{2}^{a} |}^{2})}^{2} d x \\ + \frac{2 a^{*} - a}{2} \int_{ℝ^{2}} k (x) {| u_{1}^{a} |}^{2} {| u_{2}^{a} |}^{2} d x . \end{matrix}$ (3.7)

We derive from (1.9), (1.12), (2.1), (2.3) and (3.7) that

$e (γ, a) \geq - C ε_{a}^{- 1} - \frac{γ^{2}}{16 π^{2}} .$ (3.8)

Derive from (2.12), choosing $τ = {(2 a^{*} - a)}^{- \frac{1}{2}}$ , we can obtain that

$\begin{matrix} e (γ, a) \leq C \frac{{(2 a^{*} - a)}^{\frac{b - 2}{2}}}{4 a^{*}} \int_{ℝ^{2}} {| x |}^{b} Q^{4} (x) d x + \frac{2 a^{*} - a}{2 a^{*}} \int_{ℝ^{2}} {| x |}^{2} Q^{2} (x) d x \\ + \frac{γ}{32 π} ln (2 a^{*} - a) + \frac{γ}{16 π {(a^{*})}^{2}} \int_{ℝ^{2}} \int_{ℝ^{2}} ln | x - y | Q^{2} (x) Q^{2} (y) d x d y \\ \to - \infty as a ↗ 2 a^{*} . \end{matrix}$ (3.9)

It then yields from (3.8) and (3.9) that (3.6) holds true.

We then prove that $λ_{a} ε_{a}^{2} \to - 1$ as $a ↗ 2 a^{*}$ . It follows from (3.6) and (3.8) that

$\underset{a ↗ 2 a^{*}}{\lim \inf} ε_{a}^{2} e (γ, a) \geq \underset{a ↗ 2 a^{*}}{\lim \inf} (- \frac{γ^{2}}{16 π^{2}} ε_{a}^{2} - C ε_{a}) = 0.$ (3.10)

In addition, observe from (2.1), (2.3) and (3.7) that

$\begin{matrix} ε_{a}^{2} e (γ, a) \geq \frac{1}{2} - \frac{a^{*} ε_{a}^{2}}{4} \int_{ℝ^{2}} {(\sqrt{k (x)} {| u_{1}^{a} |}^{2} + \sqrt{k (x)} {| u_{2}^{a} |}^{2})}^{2} d x \\ + \frac{ε_{a}^{2}}{4} \int_{ℝ^{2}} {| x |}^{2} ({| u_{1}^{a} |}^{2} + {| u_{2}^{a} |}^{2}) d x \\ + \frac{a^{*} ε_{a}^{2}}{4} \int_{ℝ^{2}} {(\sqrt{k (x)} {| u_{1}^{a} |}^{2} - \sqrt{k (x)} {| u_{2}^{a} |}^{2})}^{2} d x \\ + \frac{(2 a^{*} - a) ε_{a}^{2}}{2} \int_{ℝ^{2}} k (x) {| u_{1}^{a} |}^{2} {| u_{2}^{a} |}^{2} d x - \frac{γ^{2}}{16 π^{2}} ε_{a}^{2} - C ε_{a} . \end{matrix}$ (3.11)

Using (1.12), we then obtain from (3.9) and (3.11) that

$\begin{matrix} 0 \geq \underset{a ↗ 2 a^{*}}{\lim \sup} ε_{a}^{2} e (γ, a) \\ \geq \underset{a ↗ 2 a^{*}}{\lim \sup} \frac{ε_{a}^{2}}{4} \int_{ℝ^{2}} {| x |}^{2} ({| u_{1}^{a} |}^{2} + {| u_{2}^{a} |}^{2}) d x + \lim_{a ↗ 2 a^{*}} (- \frac{γ^{2}}{16 π^{2}} ε_{a}^{2} - C ε_{a}) \\ \geq \underset{a ↗ 2 a^{*}}{\lim \inf} \frac{ε_{a}^{2}}{4} \int_{ℝ^{2}} {| x |}^{2} ({| u_{1}^{a} |}^{2} + {| u_{2}^{a} |}^{2}) d x \geq 0. \end{matrix}$ (3.12)

It follows from (3.10) and (3.12) that

$lim_{a ↗ 2 a^{*}} ε_{a}^{2} e (γ, a) = 0 and lim_{a ↗ 2 a^{*}} ε_{a}^{2} \int_{ℝ^{2}} {| x |}^{2} ({| u_{1}^{a} |}^{2} + {| u_{2}^{a} |}^{2}) d x = 0.$ (3.13)

Similarly, we also have

$lim_{a ↗ 2 a^{*}} ε_{a}^{2} \int_{ℝ^{2}} {(\sqrt{k (x)} {| u_{1}^{a} |}^{2} - \sqrt{k (x)} {| u_{2}^{a} |}^{2})}^{2} d x = 0,$

$lim_{a ↗ 2 a^{*}} ε_{a}^{2} \int_{ℝ^{2}} {(\sqrt{k (x)} {| u_{1}^{a} |}^{2} + \sqrt{k (x)} {| u_{2}^{a} |}^{2})}^{2} d x = \frac{2}{a^{*}},$ (3.14)

which imply that

$lim_{a ↗ 2 a^{*}} ε_{a}^{2} \int_{ℝ^{2}} k (x) {| u_{1}^{a} |}^{2} {| u_{2}^{a} |}^{2} d x = \frac{1}{2 a^{*}} .$ (3.15)

Due to (2.1), (2.3) and (3.13), we obtain that

$| ε_{a}^{2} F_{1} (u_{1}^{a}, u_{2}^{a}) | \leq \frac{π}{γ} ε_{a}^{2} \int_{ℝ^{2}} {| x |}^{2} ({| u_{1}^{a} |}^{2} + {| u_{2}^{a} |}^{2}) d x + \frac{γ}{4 π} ε_{a}^{2} \to 0 as a ↗ 2 a^{*},$

$| ε_{a}^{2} F_{2} (u_{1}^{a}, u_{2}^{a}) | \leq C ε_{a} \to 0 as a ↗ 2 a^{*},$

that is,

$lim_{a ↗ 2 a^{*}} ε_{a}^{2} F_{1} (u_{1}^{a}, u_{2}^{a}) = 0 and lim_{a ↗ 2 a^{*}} ε_{a}^{2} F_{2} (u_{1}^{a}, u_{2}^{a}) = 0.$ (3.16)

Hence we conclude from (1.17), (3.13), (3.15) and (3.16) that

$lim_{a ↗ 2 a^{*}} λ_{a} ε_{a}^{2} = - 1.$

2) Note from (3.4) that

$\int_{ℝ^{2}} ({| v_{1}^{a} |}^{2} + {| v_{2}^{a} |}^{2}) d x = 1 = \int_{ℝ^{2}} ({| \nabla v_{1}^{a} |}^{2} + {| \nabla v_{2}^{a} |}^{2}) d x .$ (3.17)

Thus ${v_{i}^{a}} (i = 1, 2)$ is bounded uniformly in $H^{1} (ℝ^{2})$ and in $L^{q} (ℝ^{2})$ for all $q \in [2, \infty)$ . Following (3.14) and (3.15), we then obtain from (3.4) that

$\begin{array}{l} \int_{ℝ^{2}} k (ε_{a} x + ε_{a} y_{ε_{a}}) {| v_{1}^{a} |}^{2} {| v_{2}^{a} |}^{2} d x \\ = ε_{a}^{2} \int_{ℝ^{2}} k (x) {| u_{1}^{a} |}^{2} {| u_{2}^{a} |}^{2} d x \to \frac{1}{2 a^{*}} as a ↗ 2 a^{*}, \end{array}$ (3.18)

$\begin{array}{l} \int_{ℝ^{2}} {(\sqrt{k (ε_{a} x + ε_{a} y_{ε_{a}})} {| v_{1}^{a} |}^{2} - \sqrt{k (ε_{a} x + ε_{a} y_{ε_{a}})} {| v_{2}^{a} |}^{2})}^{2} d x \\ = ε_{a}^{2} \int_{ℝ^{2}} {(\sqrt{k (x)} {| u_{1}^{a} |}^{2} - \sqrt{k (x)} {| u_{2}^{a} |}^{2})}^{2} d x \to 0 as a ↗ 2 a^{*} . \end{array}$ (3.19)

For $i = 1, 2$ , we denote

${\bar{v}}_{i}^{a} (x) : = v_{i}^{a} (x - y_{ε_{a}}) = ε_{a} u_{i}^{a} (ε_{a} x) .$

To prove (3.3), it suffices to prove that there exist ${y_{ε_{a}}} \subset ℝ^{2}$ , $r_{0} > 0$ and $α > 0$ independent of $a \in (0, 2 a^{*}),$ such that

$\underset{a ↗ 2 a^{*}}{\lim \inf} \int_{B_{r_{0}} (y_{ε_{a}})} \sqrt{k (ε_{a} x)} {| {\bar{v}}_{i}^{a} |}^{2} d x \geq α > 0 for i = 1, 2.$ (3.20)

We first show (3.20) for $i = 1$ . Indeed, if it were false, then for any $r > 0$ , there exists a subsequence ${{\bar{v}}_{1}^{a_{k}}}$ , where $a_{k} ↗ 2 a^{*}$ as $k \to \infty$ , such that

$lim_{k \to \infty} sup_{y \in ℝ^{2}} \int_{B_{r} (y)} \sqrt{k (ε_{a} x)} {| {\bar{v}}_{1}^{a_{k}} |}^{2} d x = 0.$

By ([12] Lemma 1.21), we have $k {(ε_{a} x)}^{\frac{1}{4}} {\bar{v}}_{1}^{a_{k}} \to 0$ in $L^{q} (ℝ^{2})$ as $k \to \infty$ for any $q \in (2, \infty)$ . Hence

$\begin{array}{l} \int_{ℝ^{2}} k (ε_{a} x) {| {\bar{v}}_{1}^{a_{k}} |}^{2} {| {\bar{v}}_{2}^{a_{k}} |}^{2} d x \\ \leq {(\int_{ℝ^{2}} k (ε_{a} x) {| {\bar{v}}_{1}^{a_{k}} |}^{4} d x)}^{\frac{1}{2}} {(\int_{ℝ^{2}} k (ε_{a} x) {| {\bar{v}}_{2}^{a_{k}} |}^{4} d x)}^{\frac{1}{2}} \\ \leq C {(\int_{ℝ^{2}} k (ε_{a} x) {| {\bar{v}}_{1}^{a_{k}} |}^{4} d x)}^{\frac{1}{2}} \to 0 as k \to \infty, \end{array}$

which contradicts with (3.18).

We next show (3.20) for $i = 2$ . Let ${y_{ε_{a}}} \subset ℝ^{2}$ , $r_{0} > 0$ and $α > 0$ be obtained for ${\bar{v}}_{1}^{a}$ in (3.20). If (3.20) were false for ${\bar{v}}_{2}^{a}$ , then there is a subsequence ${{\bar{v}}_{2}^{a_{k}}}$ , where $a_{k} ↗ 2 a^{*}$ as $k \to \infty$ , such that

$\underset{k \to \infty}{\lim \sup} \int_{B_{r_{0}} (y_{ε_{k}})} \sqrt{k (ε_{a} x)} {| {\bar{v}}_{2}^{a_{k}} |}^{2} d x = 0,$

where $ε_{k} : = ε_{a_{k}}$ is defined by (3.1). Choose $q > 4$ and $0 < ζ < 1$ such that $\frac{1}{4} = \frac{1 - ζ}{q} + \frac{ζ}{2}$ . We then deduce that

$\begin{array}{l} \int_{B_{r_{0}} (y_{ε_{k}})} k (ε_{a} x) {| {\bar{v}}_{1}^{a_{k}} |}^{2} {| {\bar{v}}_{2}^{a_{k}} |}^{2} d x \\ \leq {(\int_{B_{r_{0}} (y_{ε_{k}})} k (ε_{a} x) {| {\bar{v}}_{1}^{a_{k}} |}^{4} d x)}^{\frac{1}{2}} {(\int_{B_{r_{0}} (y_{ε_{k}})} k (ε_{a} x) {| {\bar{v}}_{2}^{a_{k}} |}^{4} d x)}^{\frac{1}{2}} \\ \leq C {(\int_{B_{r_{0}} (y_{ε_{k}})} k (ε_{a} x) {| {\bar{v}}_{2}^{a_{k}} |}^{4} d x)}^{\frac{1}{2}} \\ \leq C {[\int_{B_{r_{0}} (y_{ε_{k}})} {(k {(ε_{a} x)}^{\frac{1}{4}} | {\bar{v}}_{2}^{a_{k}} |)}^{q} d x]}^{\frac{2 (1 - ζ)}{q}} {[\int_{B_{r_{0}} (y_{ε_{k}})} {(k {(ε_{a} x)}^{\frac{1}{4}} | {\bar{v}}_{2}^{a_{k}} |)}^{2} d x]}^{ζ} \\ \leq C {(\int_{B_{r_{0}} (y_{ε_{k}})} \sqrt{k (ε_{a} x)} {| {\bar{v}}_{2}^{a_{k}} |}^{2} d x)}^{ζ} \to 0 as k \to \infty . \end{array}$

Together with (3.20) for $i = 1$ , we then get that

$\begin{array}{l} \int_{B_{r_{0}} (y_{ε_{k}})} {(\sqrt{k (ε_{a} x)} {| {\bar{v}}_{1}^{a_{k}} |}^{2} - \sqrt{k (ε_{a} x)} {| {\bar{v}}_{2}^{a_{k}} |}^{2})}^{2} d x \\ \geq \frac{1}{2} \int_{B_{r_{0}} (y_{ε_{k}})} k (ε_{a} x) {| {\bar{v}}_{1}^{a_{k}} |}^{4} d x \geq \frac{1}{2 π r_{0}^{2}} {(\int_{B_{r_{0}} (y_{ε_{k}})} \sqrt{k (ε_{a} x)} {| {\bar{v}}_{1}^{a_{k}} |}^{2} d x)}^{2} \\ \geq \frac{α^{2}}{2 π r_{0}^{2}} > 0 as k \to \infty, \end{array}$

which contradicts with (3.19).

3) Since ${(v_{1}^{a}, v_{2}^{a})}$ is bounded uniformly in $H^{1} (ℝ^{2}) \times H^{1} (ℝ^{2})$ , up to a sub-sequence if necessary, there exists $(v_{1}^{*}, v_{2}^{*}) \in H^{1} (ℝ^{2}) \times H^{1} (ℝ^{2})$ such that as $a ↗ 2 a^{*}$ ,

$(v_{1}^{a}, v_{2}^{a}) ⇀ (v_{1}^{*}, v_{2}^{*}) in H^{1} (ℝ^{2}) \times H^{1} (ℝ^{2}),$

$(v_{1}^{a}, v_{2}^{a}) \to (v_{1}^{*}, v_{2}^{*}) in L_{l o c}^{q} (ℝ^{2}) \times L_{l o c}^{q} (ℝ^{2}) for q \in [2, \infty),$

$(v_{1}^{a}, v_{2}^{a}) \to (v_{1}^{*}, v_{2}^{*}) a . e . on ℝ^{2} .$

It follows from Fatou’s lemma and (3.17) that $\int_{ℝ^{2}} ({| v_{1}^{*} |}^{2} + {| v_{2}^{*} |}^{2}) d x \leq 1$ . Moreover, we obtain from (3.3) that $v_{1}^{*} \equiv 0$ and $v_{2}^{*} \equiv 0$ . According to Brézis-Lieb lemma and (3.17), we derive that

$\begin{matrix} 1 = \int_{ℝ^{2}} ({| v_{1}^{a} |}^{2} + {| v_{2}^{a} |}^{2}) d x \\ = \int_{ℝ^{2}} ({| v_{1}^{*} |}^{2} + {| v_{2}^{*} |}^{2}) d x + \int_{ℝ^{2}} ({| v_{1}^{a} - v_{1}^{*} |}^{2} + {| v_{2}^{a} - v_{2}^{*} |}^{2}) d x + o (1), \end{matrix}$ (3.21)

$\begin{matrix} 1 = \int_{ℝ^{2}} ({| \nabla v_{1}^{a} |}^{2} + {| \nabla v_{2}^{a} |}^{2}) d x \\ = \int_{ℝ^{2}} ({| \nabla v_{1}^{*} |}^{2} + {| \nabla v_{2}^{*} |}^{2}) d x + \int_{ℝ^{2}} ({| \nabla v_{1}^{a} - \nabla v_{1}^{*} |}^{2} + {| \nabla v_{2}^{a} - \nabla v_{2}^{*} |}^{2}) d x + o (1), \end{matrix}$ (3.22)

where and below $o (1)$ represents the quantities tending to 0 as $a ↗ 2 a^{*}$ . Additionally, there holds

$\begin{array}{l} \int_{ℝ^{2}} k (ε_{a} x + ε_{a} y_{ε_{a}}) {| v_{1}^{a} v_{2}^{a} - v_{1}^{*} v_{2}^{*} |}^{2} d x \\ = \int_{ℝ^{2}} k (ε_{a} x + ε_{a} y_{ε_{a}}) {| v_{1}^{a} - v_{1}^{*} |}^{2} {| v_{2}^{a} - v_{2}^{*} |}^{2} d x + o (1), \end{array}$ (3.23)

where we use the facts that $a ↗ 2 a^{*}$ ,

$\int_{ℝ^{2}} (k (ε_{a} x + ε_{a} y_{ε_{a}}) {| v_{1}^{*} |}^{2} {| v_{2}^{a} |}^{2} - k (ε_{a} x + ε_{a} y_{ε_{a}}) v_{1}^{a} v_{1}^{*} {| v_{2}^{a} |}^{2}) d x \to 0,$

$\int_{ℝ^{2}} (k (ε_{a} x + ε_{a} y_{ε_{a}}) {| v_{1}^{a} |}^{2} {| v_{2}^{*} |}^{2} - k (ε_{a} x + ε_{a} y_{ε_{a}}) {| v_{1}^{a} |}^{2} v_{2}^{a} v_{2}^{*}) d x \to 0,$

$\int_{ℝ^{2}} (k (ε_{a} x + ε_{a} y_{ε_{a}}) v_{1}^{a} v_{1}^{*} v_{2}^{a} v_{2}^{*} - k (ε_{a} x + ε_{a} y_{ε_{a}}) v_{1}^{a} v_{1}^{*} {| v_{2}^{a} |}^{2}) d x \to 0,$

$\int_{ℝ^{2}} (k (ε_{a} x + ε_{a} y_{ε_{a}}) v_{1}^{a} v_{1}^{*} v_{2}^{a} v_{2}^{*} - k (ε_{a} x + ε_{a} y_{ε_{a}}) {| v_{1}^{a} |}^{2} v_{2}^{a} v_{2}^{*}) d x \to 0,$

$\int_{ℝ^{2}} (k (ε_{a} x + ε_{a} y_{ε_{a}}) v_{1}^{a} v_{1}^{*} v_{2}^{a} v_{2}^{*} - k (ε_{a} x + ε_{a} y_{ε_{a}}) v_{1}^{a} v_{1}^{*} {| v_{2}^{*} |}^{2}) d x \to 0,$

$\int_{ℝ^{2}} (k (ε_{a} x + ε_{a} y_{ε_{a}}) v_{1}^{a} v_{1}^{*} v_{2}^{a} v_{2}^{*} - k (ε_{a} x + ε_{a} y_{ε_{a}}) {| v_{1}^{*} |}^{2} v_{2}^{a} v_{2}^{*}) d x \to 0.$

It follows from Brézis-Lieb lemma and (3.23) that

$\begin{array}{l} \int_{ℝ^{2}} k (ε_{a} x + ε_{a} y_{ε_{a}}) {| v_{1}^{a} |}^{2} {| v_{2}^{a} |}^{2} d x \\ = \int_{ℝ^{2}} k (ε_{a} x + ε_{a} y_{ε_{a}}) {| v_{1}^{*} |}^{2} {| v_{2}^{*} |}^{2} d x \\ + \int_{ℝ^{2}} k (ε_{a} x + ε_{a} y_{ε_{a}}) {| v_{1}^{a} - v_{1}^{*} |}^{2} {| v_{2}^{a} - v_{2}^{*} |}^{2} d x + o (1), \end{array}$

which jointly with

$\begin{array}{l} \int_{ℝ^{2}} (k (ε_{a} x + ε_{a} y_{ε_{a}}) {| v_{1}^{a} |}^{4} + k (ε_{a} x + ε_{a} y_{ε_{a}}) {| v_{2}^{a} |}^{4}) d x \\ = \int_{ℝ^{2}} (k (ε_{a} x + ε_{a} y_{ε_{a}}) {| v_{1}^{*} |}^{4} + k (ε_{a} x + ε_{a} y_{ε_{a}}) {| v_{2}^{*} |}^{4}) d x \\ + \int_{ℝ^{2}} (k (ε_{a} x + ε_{a} y_{ε_{a}}) {| v_{1}^{a} - v_{1}^{*} |}^{4} + k (ε_{a} x + ε_{a} y_{ε_{a}}) {| v_{2}^{a} - v_{2}^{*} |}^{4}) d x + o (1), \end{array}$

yields that

$\begin{array}{l} \int_{ℝ^{2}} {(\sqrt{k (ε_{a} x + ε_{a} y_{ε_{a}})} {| v_{1}^{a} |}^{2} + \sqrt{k (ε_{a} x + ε_{a} y_{ε_{a}})} {| v_{2}^{a} |}^{2})}^{2} d x \\ = \int_{ℝ^{2}} {(\sqrt{k (ε_{a} x + ε_{a} y_{ε_{a}})} {| v_{1}^{*} |}^{2} + \sqrt{k (ε_{a} x + ε_{a} y_{ε_{a}})} {| v_{2}^{*} |}^{2})}^{2} d x \\ + \int_{ℝ^{2}} {(\sqrt{k (ε_{a} x + ε_{a} y_{ε_{a}})} {| v_{1}^{a} - v_{1}^{*} |}^{2} + \sqrt{k (ε_{a} x + ε_{a} y_{ε_{a}})} {| v_{2}^{a} - v_{2}^{*} |}^{2})}^{2} d x + o (1) . \end{array}$ (3.24)

Due to (3.14), we have

$\begin{array}{l} \int_{ℝ^{2}} {(\sqrt{k (ε_{a} x + ε_{a} y_{ε_{a}})} {| v_{1}^{a} |}^{2} + \sqrt{k (ε_{a} x + ε_{a} y_{ε_{a}})} {| v_{2}^{a} |}^{2})}^{2} d x \\ = ε_{a}^{2} \int_{ℝ^{2}} {(\sqrt{k (x)} {| u_{1}^{a} |}^{2} + \sqrt{k (x)} {| u_{2}^{a} |}^{2})}^{2} d x \to \frac{2}{a^{*}} as a ↗ 2 a^{*} . \end{array}$ (3.25)

Then we deduce from (1.12), (3.21), (3.22), (3.24) and (3.25) that

$\begin{matrix} 0 = lim_{a ↗ 2 a^{*}} [\int_{ℝ^{2}} ({| \nabla v_{1}^{a} |}^{2} + {| \nabla v_{2}^{a} |}^{2}) d x - \frac{a}{4} \int_{ℝ^{2}} {(\sqrt{k (ε_{a} x + ε_{a} y_{ε_{a}})} ({| v_{1}^{a} |}^{2} + {| v_{2}^{a} |}^{2}))}^{2} d x] \\ = lim_{a ↗ 2 a^{*}} [\int_{ℝ^{2}} ({| \nabla v_{1}^{*} |}^{2} + {| \nabla v_{2}^{*} |}^{2}) d x + \int_{ℝ^{2}} ({| \nabla v_{1}^{a} - \nabla v_{1}^{*} |}^{2} + {| \nabla v_{2}^{a} - \nabla v_{2}^{*} |}^{2}) d x \\ - \frac{a}{4} \int_{ℝ^{2}} {(\sqrt{k (ε_{a} x + ε_{a} y_{ε_{a}})} {| v_{1}^{*} |}^{2} + \sqrt{k (ε_{a} x + ε_{a} y_{ε_{a}})} {| v_{2}^{*} |}^{2})}^{2} d x \\ - \frac{a}{4} \int_{ℝ^{2}} {(\sqrt{k (ε_{a} x + ε_{a} y_{ε_{a}})} {| v_{1}^{a} - v_{1}^{*} |}^{2} + \sqrt{k (ε_{a} x + ε_{a} y_{ε_{a}})} {| v_{2}^{a} - v_{2}^{*} |}^{2})}^{2} d x] \\ \geq \frac{a^{*}}{2} [{(\int_{ℝ^{2}} ({| v_{1}^{*} |}^{2} + {| v_{2}^{*} |}^{2}) d x)}^{- 1} - 1] \int_{ℝ^{2}} {({| v_{1}^{*} |}^{2} + {| v_{2}^{*} |}^{2})}^{2} d x \\ + lim_{a ↗ 2 a^{*}} [1 - \int_{ℝ^{2}} ({| v_{1}^{a} - v_{1}^{*} |}^{2} + {| v_{2}^{a} - v_{2}^{*} |}^{2}) d x] \\ \times \int_{ℝ^{2}} ({| \nabla v_{1}^{a} - \nabla v_{1}^{*} |}^{2} + {| \nabla v_{2}^{a} - \nabla v_{2}^{*} |}^{2}) d x \\ \geq 0, \end{matrix}$ (3.26)

which implies that

$\int_{ℝ^{2}} ({| v_{1}^{*} |}^{2} + {| v_{2}^{*} |}^{2}) d x = 1,$ (3.27)

$lim_{a ↗ 2 a^{*}} \int_{ℝ^{2}} ({| \nabla v_{1}^{a} - \nabla v_{1}^{*} |}^{2} + {| \nabla v_{2}^{a} - \nabla v_{2}^{*} |}^{2}) d x = 0.$

Therefore, we conclude that

$(v_{1}^{a}, v_{2}^{a}) \to (v_{1}^{*}, v_{2}^{*}) in H^{1} (ℝ^{2}) \times H^{1} (ℝ^{2}) as a ↗ 2 a^{*} .$ (3.28)

We obtain from (3.17) and (3.28) that

$\int_{ℝ^{2}} ({| \nabla v_{1}^{*} |}^{2} + {| \nabla v_{2}^{*} |}^{2}) d x = 1.$ (3.29)

Additionally, we also get from (3.26) that

$\int_{ℝ^{2}} ({| \nabla v_{1}^{*} |}^{2} + {| \nabla v_{2}^{*} |}^{2}) d x = \frac{a^{*}}{2} \int_{ℝ^{2}} {({| v_{1}^{*} |}^{2} + {| v_{2}^{*} |}^{2})}^{2} d x .$ (3.30)

Applying (3.19) and (3.28) we have $\int_{ℝ^{2}} {(\sqrt{k (ε_{a} x + ε_{a} y_{ε_{a}})} ({| v_{1}^{*} |}^{2} - {| v_{2}^{*} |}^{2}))}^{2} d x = 0$ . This further implies that $| v_{1}^{*} | = | v_{2}^{*} |$ a.e. on $ℝ^{2}$ . It then follows from (3.27), (3.29) and (3.30) that for $i = 1, 2$ ,

$\int_{ℝ^{2}} {| v_{i}^{*} |}^{2} d x = \frac{1}{2} and \frac{1}{2} = \int_{ℝ^{2}} {| \nabla v_{i}^{*} |}^{2} d x = a^{*} \int_{ℝ^{2}} {| v_{i}^{*} |}^{4} d x .$

Obviously, the identity in (1.14) is attained at $| v_{i}^{*} |$ for $i = 1, 2$ . We can derive from the Lagrange multiplier rule that for $i = 1, 2$ ,

$- Δ | v_{i}^{*} | + | v_{i}^{*} | = 2 a^{*} {| v_{i}^{*} |}^{3} in ℝ^{2} .$

The uniqueness (up to translations) of positive solutions of (1.2) yields that

$| v_{1}^{*} (x) | = | v_{2}^{*} (x) | = \frac{Q (x - x_{0})}{\sqrt{2 a^{*}}} for some x_{0} \in ℝ^{2} .$

This completes the proof of Lemma 3.1.

In what follows, we assume that the minimizer $(u_{1}^{a}, u_{2}^{a})$ of $e (γ, a)$ is non-negative. Following Lemma 3.1, we continue to analyze the refined limiting behavior of non-negative $(u_{1}^{a}, u_{2}^{a})$ as $a ↗ 2 a^{*}$ . The exponential decay of non-negative minimizers $(u_{1}^{a}, u_{2}^{a})$ at infinity needs to be proved first.

Lemma 3.2 Let $(u_{1}^{a}, u_{2}^{a}) \in M$ be a non-negative minimizer of $e (γ, a)$ , where $γ > 0$ is fixed and $a \in (0, 2 a^{*})$ , together with $0 < k (x) \leq 1$ satisfies (1.7) and (1.8). Then

1) There exists a large constant $R > 0$ , independent of $a \in (0, 2 a^{*})$ , such that

$| v_{i}^{a} (x) | \leq C e^{- \frac{2}{3} | x |} uniformly for | x | \geq R as a ↗ 2 a^{*},$ (3.31)

where $v_{i}^{a}$ is defined by (3.4) and $i = 1, 2$ ;

2) There results

$ε_{a} y_{ε_{a}} \to 0 as a ↗ 2 a^{*} .$ (3.32)

Proof. 1) Note from (1.16) that $(v_{1}^{a}, v_{2}^{a})$ solves

${\begin{cases} - Δ v_{1}^{a} + ε_{a}^{2} {| ε_{a} x + ε_{a} y_{ε_{a}} |}^{2} v_{1}^{a} + \frac{γ ε_{a}^{2}}{2 π} \int_{ℝ^{2}} \ln | x - y | v_{1}^{a} (y) v_{2}^{a} (y) d y v_{2}^{a} \\ = λ_{a} ε_{a}^{2} v_{1}^{a} + a k (ε_{a} x + ε_{a} y_{ε_{a}}) {| v_{2}^{a} |}^{2} v_{1}^{a} - \frac{γ ε_{a}^{2}}{2 π} \ln ε_{a} \int_{ℝ^{2}} v_{1}^{a} (y) v_{2}^{a} (y) d y v_{2}^{a} in ℝ^{2}, \\ - Δ v_{2}^{a} + ε_{a}^{2} {| ε_{a} x + ε_{a} y_{ε_{a}} |}^{2} v_{2}^{a} + \frac{γ ε_{a}^{2}}{2 π} \int_{ℝ^{2}} \ln | x - y | v_{1}^{a} (y) v_{2}^{a} (y) d y v_{1}^{a} \\ = λ_{a} ε_{a}^{2} v_{2}^{a} + a k (ε_{a} x + ε_{a} y_{ε_{a}}) {| v_{1}^{a} |}^{2} v_{2}^{a} - \frac{γ ε_{a}^{2}}{2 π} \ln ε_{a} \int_{ℝ^{2}} v_{1}^{a} (y) v_{2}^{a} (y) d y v_{1}^{a} in ℝ^{2} . \end{cases}$ (3.33)

Using the same argument as (2.6), we then obtain from (3.17) that

$\int_{ℝ^{2}} \frac{v_{1}^{a} (y) v_{2}^{a} (y)}{| x - y |} d y \leq C uniformly for any x \in ℝ^{2} and a \in (0, 2 a^{*}) .$

This further implies that

$\begin{array}{l} \int_{ℝ^{2}} \ln | x - y | v_{1}^{a} (y) v_{2}^{a} (y) d y \\ = \int_{ℝ^{2}} \ln (1 + | x - y |) v_{1}^{a} (y) v_{2}^{a} (y) d y - \int_{ℝ^{2}} \ln (1 + {| x - y |}^{- 1}) v_{1}^{a} (y) v_{2}^{a} (y) d y \\ \geq - C for any x \in ℝ^{2} and a \in (0, 2 a^{*}) . \end{array}$

Hence we derive from (3.2) and (3.33) that as $a ↗ 2 a^{*}$ ,

${\begin{array}{l} - Δ v_{1}^{a} + \frac{2}{3} v_{1}^{a} \leq 2 a^{*} {| v_{2}^{a} |}^{2} v_{1}^{a} + o (1) v_{2}^{a} in ℝ^{2}, \\ - Δ v_{2}^{a} + \frac{2}{3} v_{2}^{a} \leq 2 a^{*} {| v_{1}^{a} |}^{2} v_{2}^{a} + o (1) v_{1}^{a} in ℝ^{2} . \end{array}$ (3.34)

Using the De Giorgi-Nash-Moser theory (cf. [13] Theorem 4.1), we then deduce from (3.17) that as $a ↗ 2 a^{*}$ ,

$max_{B_{1} (ξ)} v_{i}^{a} \leq C [{(\int_{B_{2} (ξ)} {| v_{1}^{a} |}^{2} d x)}^{\frac{1}{2}} + {(\int_{B_{2} (ξ)} {| v_{2}^{a} |}^{2} d x)}^{\frac{1}{2}}] for i = 1, 2,$

where $ξ$ is an arbitrary point in $ℝ^{2}$ and $C > 0$ is a constant independent of $a \in (0, 2 a^{*})$ . We then obtain from (3.5) that

$v_{i}^{a} \in L^{\infty} (ℝ^{2}) and v_{i}^{a} (x) \to 0 as | x | \to \infty uniformly for a ↗ 2 a^{*} with i = 1, 2.$ (3.35)

Therefore, we infer from (3.34) that there exists a large constant $R > 0$ independent of $a \in (0, 2 a^{*})$ such that

$- Δ (v_{1}^{a} + v_{2}^{a}) + \frac{4}{9} (v_{1}^{a} + v_{2}^{a}) \leq 0 uniformly for | x | \geq R as a ↗ 2 a^{*} .$

Applying the comparision principle, it implies that there is a constant $C > 0$ independent of $a \in (0, 2 a^{*})$ such that

$v_{1}^{a} (x) + v_{2}^{a} (x) \leq C e^{- \frac{2}{3} | x |} uniformly for | x | \geq R as a ↗ 2 a^{*} .$

Together with the non-negativity of $v_{i}^{a}$ for $i = 1, 2$ , we conclude that (3.31) holds true.

3) We now prove that

$lim_{a ↗ 2 a^{*}} ε_{a} y_{ε_{a}} = 0.$

Otherwise, there is a subsequence of $ε_{a} y_{ε_{a}}$ , still denoted by itself, such that

$ε_{a} y_{ε_{a}} \to z_{0} \neq 0 as a ↗ 2 a^{*} .$

Set $x_{a} : = \int_{ℝ^{2}} x ({| u_{1}^{a} |}^{2} + {| u_{2}^{a} |}^{2}) d x$ . It follows from (3.4) and (3.5) that as $a ↗ 2 a^{*}$ ,

$x_{a} = \int_{ℝ^{2}} (ε_{a} x + ε_{a} y_{ε_{a}}) ({| v_{1}^{a} |}^{2} + {| v_{2}^{a} |}^{2}) d x = ε_{a} y_{ε_{a}} + ε_{a} (x_{0} + o (1)) \to z_{0} .$

Define ${\bar{u}}_{i}^{a} (x) : = u_{i}^{a} (x + x_{a})$ for $i = 1, 2$ . Then $({\bar{u}}_{1}^{a}, {\bar{u}}_{2}^{a}) \in M$ and

$\int_{ℝ^{2}} ({| \nabla {\bar{u}}_{1}^{a} |}^{2} + {| \nabla {\bar{u}}_{2}^{a} |}^{2}) d x = \int_{ℝ^{2}} ({| \nabla u_{1}^{a} |}^{2} + {| \nabla u_{2}^{a} |}^{2}) d x,$

$\begin{array}{l} \int_{ℝ^{2}} \int_{ℝ^{2}} \ln | x - y | {\bar{u}}_{1} (x) {\bar{u}}_{2} (x) {\bar{u}}_{1} (y) {\bar{u}}_{2} (y) d x d y \\ = \int_{ℝ^{2}} \int_{ℝ^{2}} \ln | x - y | u_{1} (x) u_{2} (x) u_{1} (y) u_{2} (y) d x d y . \end{array}$

In addition, we deduce from (3.4) and (3.31) that as $a ↗ 2 a^{*}$ ,

$\begin{matrix} \int_{ℝ^{2}} {| x |}^{2} ({| {\bar{u}}_{1}^{a} |}^{2} + {| {\bar{u}}_{2}^{a} |}^{2}) d x = \int_{ℝ^{2}} {| x - x_{a} |}^{2} ({| u_{1}^{a} |}^{2} + {| u_{2}^{a} |}^{2}) d x \\ = ε_{a}^{2} \int_{ℝ^{2}} {| x - (x_{0} + o (1)) |}^{2} ({| v_{1}^{a} |}^{2} + {| v_{2}^{a} |}^{2}) d x \\ = O (ε_{a}^{2}), \end{matrix}$

$\int_{ℝ^{2}} {| x |}^{2} ({| u_{1}^{a} |}^{2} + {| u_{2}^{a} |}^{2}) d x = \int_{ℝ^{2}} {| ε_{a} x + ε_{a} y_{ε_{a}} |}^{2} ({| v_{1}^{a} |}^{2} + {| v_{2}^{a} |}^{2}) d x \to {| z_{0} |}^{2} .$

Using (1.7), (3.4) and (3.31), we derive that

$\begin{matrix} a \int_{ℝ^{2}} k (x) {| {\bar{u}}_{1} |}^{2} {| {\bar{u}}_{2} |}^{2} d x = a \int_{ℝ^{2}} k (x - x_{a}) {| u_{1}^{a} |}^{2} {| u_{2}^{a} |}^{2} d x \\ = a ε_{a}^{- 2} \int_{ℝ^{2}} k (ε_{a} x - ε_{a} (x_{0} + o (1))) {| v_{1} |}^{2} {| v_{2} |}^{2} d x \\ = ε_{a}^{- 2} [1 + o (1)] as a ↗ 2 a^{*}, \end{matrix}$

$\begin{matrix} a \int_{ℝ^{2}} k (x) {| u_{1} |}^{2} {| u_{2} |}^{2} d x = a ε_{a}^{- 2} \int_{ℝ^{2}} k (ε_{a} x + ε_{a} y_{ε_{a}}) {| v_{1} |}^{2} {| v_{2} |}^{2} d x \\ = ε_{a}^{- 2} [k (z_{0}) + o (1)] as a ↗ 2 a^{*} . \end{matrix}$

Note that $k (z_{0}) \leq 1$ , then $E_{γ, a} ({\bar{u}}_{1}^{a}, {\bar{u}}_{2}^{a}) < E_{γ, a} (u_{1}^{a}, u_{2}^{a})$ as $a ↗ 2 a^{*}$ , which contradicts with the fact that $(u_{1}^{a}, u_{2}^{a})$ is the minimizer of $e (γ, a)$ , which ensures that (3.32) is true. The proof of Lemma 3.2 is thus complete.

Next, we shall apply Lemma 3.2 to prove the convergence behavior of non-negative minimizers $(u_{1}^{a}, u_{2}^{a})$ as $a ↗ 2 a^{*}$ for given $γ > 0$ .

Lemma 3.3 Let $(u_{1}^{a}, u_{2}^{a}) \in M$ be a non-negative minimizer of $e (γ, a)$ , where $γ > 0$ is fixed and $a \in (0, 2 a^{*})$ , together with $0 < k (x) \leq 1$ satisfies (1.7) and (1.8) and $k (x) \in C^{0, α} (ℝ^{2})$ for some $α \in (0, 1)$ . Define

${\tilde{v}}_{i}^{a} (x) : = ε_{a} u_{i}^{a} (ε_{a} x + x_{i}^{a}), i = 1, 2,$ (3.36)

where $ε_{a}$ is given by (3.1) and $x_{i}^{a}$ is a global maximum point of $u_{i}^{a}$ . Then for $i = 1, 2$ , $x_{i}^{a}$ is unique and satisfies

$x_{i}^{a} \to 0$ and $\frac{| x_{1}^{a} - x_{2}^{a} |}{ε_{a}} \to 0$ as $a ↗ 2 a^{*}$ .(3.37)

Moreover, there results

${\tilde{v}}_{i}^{a} (x) \to \frac{Q (x)}{\sqrt{2 a^{*}}}$ in $H \cap L^{\infty} (ℝ^{2})$ as $a ↗ 2 a^{*}$ with $i = 1, 2$ .(3.38)

Proof. We first prove (3.37). For any given $a \in (0, 2 a^{*})$ , each $u_{i}^{a}$ has a global maximum point $x_{i}^{a}$ for $i = 1, 2$ by means of (3.3) and (3.35). Then $v_{i}^{a}$ achieves its global maximum point at $ε_{a}^{- 1} (x_{i}^{a} - ε_{a} y_{ε_{a}})$ for $i = 1, 2$ . Using (3.3) and (3.35) again, we obtain that $ε_{a}^{- 1} (x_{i}^{a} - ε_{a} y_{ε_{a}})$ is bounded uniformly as $a ↗ 2 a^{*}$ for $i = 1, 2$ . Together with (3.2) and (3.32), we have

$x_{i}^{a} \to 0 as a ↗ 2 a^{*} for i = 1, 2.$ (3.39)

Observe from (3.4) and (3.36) that

${\tilde{v}}_{i}^{a} (x) : = v_{i}^{a} (x + \frac{x_{i}^{a} - ε_{a} y_{ε_{a}}}{ε_{a}}) in ℝ^{2} for i = 1, 2.$ (3.40)

We deduce from (3.33) that $({\tilde{v}}_{1}^{a}, {\tilde{v}}_{2}^{a})$ solves

${\begin{cases} - Δ {\tilde{v}}_{1}^{a} = G_{1}^{a} in ℝ^{2}, \\ - Δ {\tilde{v}}_{2}^{a} = G_{2}^{a} in ℝ^{2}, \end{cases}$ (3.41)

where

$\begin{array}{l} G_{1}^{a} : = - ε_{a}^{2} {| ε_{a} x + x_{1}^{a} |}^{2} {\tilde{v}}_{1}^{a} - \frac{γ ε_{a}^{2}}{2 π} \int_{ℝ^{2}} \ln | x - y | {\tilde{v}}_{1}^{a} (y) {\tilde{v}}_{2}^{a} (y + \frac{x_{1}^{a} - x_{2}^{a}}{ε_{a}}) d y {\tilde{v}}_{2}^{a} (\cdot + \frac{x_{1}^{a} - x_{2}^{a}}{ε_{a}}) \\ + λ_{a} ε_{a}^{2} {\tilde{v}}_{1}^{a} + a k (x + \frac{x_{1}^{a} - ε_{a} y_{ε_{a}}}{ε_{a}}) {| {\tilde{v}}_{2}^{a} (\cdot + \frac{x_{1}^{a} - x_{2}^{a}}{ε_{a}}) |}^{2} {\tilde{v}}_{1}^{a} \\ - \frac{γ ε_{a}^{2}}{2 π} \ln ε_{a} {| {\tilde{v}}_{1}^{a} {\tilde{v}}_{2}^{a} |}_{1} {\tilde{v}}_{2}^{a} (\cdot + \frac{x_{1}^{a} - x_{2}^{a}}{ε_{a}}), \end{array}$

$\begin{array}{l} G_{2}^{a} : = - ε_{a}^{2} {| ε_{a} x + x_{2}^{a} |}^{2} {\tilde{v}}_{2}^{a} - \frac{γ ε_{a}^{2}}{2 π} \int_{ℝ^{2}} \ln | x - y | {\tilde{v}}_{1}^{a} (y + \frac{x_{2}^{a} - x_{1}^{a}}{ε_{a}}) {\tilde{v}}_{2}^{a} (y) d y {\tilde{v}}_{1}^{a} (\cdot + \frac{x_{2}^{a} - x_{1}^{a}}{ε_{a}}) \\ + λ_{a} ε_{a}^{2} {\tilde{v}}_{2}^{a} + a k (x + \frac{x_{2}^{a} - ε_{a} y_{ε_{a}}}{ε_{a}}) {| {\tilde{v}}_{1}^{a} (\cdot + \frac{x_{2}^{a} - x_{1}^{a}}{ε_{a}}) |}^{2} {\tilde{v}}_{2}^{a} \\ - \frac{γ ε_{a}^{2}}{2 π} \ln ε_{a} {| {\tilde{v}}_{1}^{a} {\tilde{v}}_{2}^{a} |}_{1} {\tilde{v}}_{1}^{a} (\cdot + \frac{x_{2}^{a} - x_{1}^{a}}{ε_{a}}) . \end{array}$

It follows from (3.5) and (3.40) that

${\tilde{v}}_{i}^{a} (x) \to \frac{1}{\sqrt{2 a^{*}}} Q (x + y_{i} - x_{0}) in H^{1} (ℝ^{2}) as a ↗ 2 a^{*},$ (3.42)

where $y_{i} : = lim_{a ↗ 2 a^{*}} ε_{a}^{- 1} (x_{i}^{a} - ε_{a} y_{ε_{a}})$ for $i = 1, 2$ . Obviously, ${{\tilde{v}}_{i}^{a}}$ is bounded uniformly as $a ↗ 2 a^{*}$ in $H^{1} (ℝ^{2})$ and in $L^{q} (ℝ^{2})$ for all $q \in [2, \infty)$ with $i = 1, 2$ . Notice that

$| \int_{ℝ^{2}} \ln | x - y | {\tilde{v}}_{1}^{a} (y) {\tilde{v}}_{2}^{a} (y + \frac{x_{1}^{a} - x_{2}^{a}}{ε_{a}}) d y | \leq | x | + C,$ (3.43)

$| \int_{ℝ^{2}} \ln | x - y | {\tilde{v}}_{1}^{a} (y + \frac{x_{2}^{a} - x_{1}^{a}}{ε_{a}}) {\tilde{v}}_{2}^{a} (y) d y | \leq | x | + C,$ (3.44)

where $C > 0$ is a constant independent of $a \in (0, 2 a^{*})$ . Applying the $L^{p}$ theory (cf. ([14] Theorem 9.11)) to (3.41), we then deduce from (3.31) and (3.39) that ${{\tilde{v}}_{i}^{a}}$ is bounded uniformly as $a ↗ 2 a^{*}$ in $W_{l o c}^{2, q} (ℝ^{2})$ for all $q \in [2, \infty)$ with $i = 1, 2$ . The standard Sobolev embedding theorem implies that ${{\tilde{v}}_{i}^{a}}$ is bounded uniformly as $a ↗ 2 a^{*}$ in $C_{l o c}^{1, μ} (ℝ^{2})$ for some $μ \in (0, 1)$ with $i = 1, 2$ . Similar to the proof of [8, Proposition 2.3], we get that as $a ↗ 2 a^{*}$ ,

$\int_{ℝ^{2}} \ln | x - y | {\tilde{v}}_{1}^{a} (y) {\tilde{v}}_{2}^{a} (y + \frac{x_{1}^{a} - x_{2}^{a}}{ε_{a}}) d y \in C_{l o c}^{3, μ} (ℝ^{2}),$

$\int_{ℝ^{2}} \ln | x - y | {\tilde{v}}_{1}^{a} (y + \frac{x_{2}^{a} - x_{1}^{a}}{ε_{a}}) {\tilde{v}}_{2}^{a} (y) d y \in C_{l o c}^{3, μ} (ℝ^{2}) .$

Note that $ε_{a}^{2} {| ε_{a} x + x_{i}^{a} |}^{2}$ is locally Lipschitz continuous in $ℝ^{2}$ for $i = 1, 2$ . Using the Schauder estimate (cf. ([14] Theorem 6.2)) to (3.41), we further obtain that ${{\tilde{v}}_{i}^{a}}$ is bounded uniformly as $a ↗ 2 a^{*}$ in $C_{l o c}^{2, μ} (ℝ^{2})$ for some $μ \in (0, 1)$ with $i = 1, 2$ . Passing to a subsequence, there is a function ${\tilde{v}}_{i}^{*} \in C_{l o c}^{2} (ℝ^{2})$ such that

${\tilde{v}}_{i}^{a} \to {\tilde{v}}_{i}^{*} in C_{l o c}^{2} (ℝ^{2}) as a ↗ 2 a^{*} for i = 1, 2.$

Then we have ${\tilde{v}}_{i}^{*} = \frac{1}{\sqrt{2 a^{*}}} Q (. + y_{i} - x_{0})$ for $i = 1, 2$ in view of (3.42). Since the origin is a global maximum point of ${\tilde{v}}_{i}^{a}$ , it is also a global maximum point of ${\tilde{v}}_{i}^{*}$ for $i = 1, 2$ . The radially symmetric and decreasing property of $Q$ yields that

$y_{1} = y_{2} = x_{0} .$ (3.45)

Hence we derive that

$\frac{| x_{1}^{a} - x_{2}^{a} |}{ε_{a}} \leq \frac{| x_{1}^{a} - ε_{a} y_{ε_{a}} |}{ε_{a}} + \frac{| x_{2}^{a} - ε_{a} y_{ε_{a}} |}{ε_{a}} \to | y_{1} | - | y_{2} | = 0 as a ↗ 2 a^{*} .$ (3.46)

We next prove (3.38). By (3.42) and (3.45), we get that

${\tilde{v}}_{i}^{a} (x) \to \frac{Q (x)}{\sqrt{2 a^{*}}} in H^{1} (ℝ^{2}) as a ↗ 2 a^{*} with i = 1, 2.$ (3.47)

It follows from (3.31) and (3.40) that there is a large constant $\tilde{R} \geq max {R + M, 2 M}$ , where $M > 0$ is the uniform upper bound of $ε_{a}^{- 1} | x_{i}^{a} - ε_{a} y_{ε_{a}} |$ as $a ↗ 2 a^{*}$ for $i = 1, 2$ , such that

$| {\tilde{v}}_{i}^{a} (x) | \leq C e^{- \frac{1}{3} | x |} uniformly for | x | \geq \tilde{R} as a ↗ 2 a^{*} with i = 1, 2.$ (3.48)

Combining (1.13) with (3.47) and (3.48), we deduce that

$\int_{ℝ^{2}} {| x |}^{2} | {\tilde{v}}_{i}^{a} (x) - \frac{Q (x)}{\sqrt{2 a^{*}}} | d x \to 0 as a ↗ 2 a^{*} with i = 1, 2.$ (3.49)

We then obtain from (3.47) and (3.49) that

${\tilde{v}}_{i}^{a} (x) \to \frac{Q (x)}{\sqrt{2 a^{*}}} in H as a ↗ 2 a^{*} for i = 1, 2.$

Now we show that

${\tilde{v}}_{i}^{a} (x) \to \frac{Q (x)}{\sqrt{2 a^{*}}} in L^{\infty} (ℝ^{2}) as a ↗ 2 a^{*} for i = 1, 2.$

By means of (1.13) and (3.48), we only need to prove the $L^{\infty}$ -uniform convergence of ${{\tilde{v}}_{i}^{a}}$ as $a ↗ 2 a^{*}$ for $i = 1, 2$ on any compact subset of $ℝ^{2}$ . Using (3.17), (3.39), (3.43), (3.44) and (3.48), we see that ${G_{i}^{a}}$ is bounded uniformly as $a ↗ 2 a^{*}$ in $L^{2} (ℝ^{2})$ for $i = 1, 2$ . For any $r > 0$ , it follows from [14], Theorem 8.8] that there is a constant $C > 0$ independent of $a \in (0, 2 a^{*})$ and $r > 0$ such that for $i = 1, 2$ ,

${‖ {\tilde{v}}_{i}^{a} ‖}_{H^{2} (B_{r} (0))} \leq C ({‖ {\tilde{v}}_{i}^{a} ‖}_{H^{1} (B_{r + 1} (0))} + {‖ G_{i}^{a} ‖}_{L^{2} (B_{r + 1} (0))}) as a ↗ 2 a^{*} .$

Hence ${v_{i}^{a}}$ is bounded uniformly as $a ↗ 2 a^{*}$ in $H_{l o c}^{2} (ℝ^{2})$ for $i = 1, 2$ . Then we derive from the compact embedding $H^{2} (B_{r} (0))$ ↪ $L^{\infty} (B_{r} (0))$ (cf. ([14] Theorem 7.26)) that there is a subsequence of ${{\tilde{v}}_{i}^{a}}$ , still denoted by itself, such that

${\tilde{v}}_{i}^{a} (x) \to \frac{Q (x)}{\sqrt{2 a^{*}}} in L^{\infty} (B_{r} (0)) as a ↗ 2 a^{*} for i = 1, 2.$

Note that the above convergence is independent of the choice of the subsequence and $r > 0$ is arbitrary. We conclude that the convergence holds for the whole sequence in $L_{l o c}^{\infty} (ℝ^{2})$ as $a ↗ 2 a^{*}$ for $i = 1, 2$ .

Finally, we prove the uniqueness of the global maximum point $x_{i}^{a}$ of $u_{i}^{a}$ for $i = 1, 2$ . Since

${\tilde{v}}_{i}^{a} (x) \to \frac{Q (x)}{\sqrt{2 a^{*}}} in C_{l o c}^{2} (ℝ^{2}) as a ↗ 2 a^{*} for i = 1, 2,$

and the origin is the unique global maximum point of $Q$ , we see that all local maximum points of ${\tilde{v}}_{i}^{a}$ must approach the origin and thus stay in a small ball $B_{ϵ} (0)$ as $a ↗ 2 a^{*}$ for some small constant $ϵ > 0$ with $i = 1, 2$ . Due to $Q^{″} (r) < 0$ , we can take $ϵ > 0$ small enough such that $Q^{″} (0) < 0$ for $r \in [0, ϵ]$ . It follows from ([15] Lemma 4.2) that ${\tilde{v}}_{i}^{a}$ has no local maximum points other than the origin as $a ↗ 2 a^{*}$ for $i = 1, 2$ . Hence the global maximum point of $u_{i}^{a}$ is unique as $a ↗ 2 a^{*}$ for $i = 1, 2$ . This completes the proof of Lemma 3.3.

In view of above lemmas, we are now ready to complete the proof of Theorem 1.2.

Proof of Theorem 1.2. According to Lemma 3.1 and Lemma 3.3, it suffices to prove that

$ε_{a} = {[\frac{8 π (2 a^{*} - a)}{γ a^{*}}]}^{\frac{1}{2}} (1 + o (1)) and lim_{a ↗ 2 a^{*}} \frac{x_{i}^{a}}{ε_{a}} = 0 for i = 1, 2.$

We first estimate $e (γ, a)$ to get the explicit blowing-up rate of $(u_{1}^{a}, u_{2}^{a})$ as $a ↗ 2 a^{*}$ . Taking $θ = \frac{1}{2} (1 + o (1))$ and $τ = {[\frac{γ a^{*}}{8 π (2 a^{*} - a)}]}^{\frac{1}{2}}$ in (2.10), we then obtain that

$\begin{array}{l} e (γ, α) \leq C \frac{τ^{2 - b}}{4 a^{*}} \int_{ℝ^{2}} {| x |}^{b} Q^{4} (x) d x + \frac{1}{2 a^{*} τ^{2}} \int_{ℝ^{2}} {| x |}^{2} Q^{2} (x) d x - \frac{γ θ (1 - θ)}{4 π} \ln τ \\ + \frac{γ θ (1 - θ)}{4 π {(a^{*})}^{2}} \int_{ℝ^{2}} \int_{ℝ^{2}} \ln | x - y | Q^{2} (x) Q^{2} (y) d x d y \\ \leq \frac{γ}{32 π} ln \frac{8 π (2 a^{*} - a)}{γ a^{*}} \\ + \frac{γ}{16 π {(a^{*})}^{2}} \int_{ℝ^{2}} \int_{ℝ^{2}} \ln | x - y | Q^{2} (x) Q^{2} (y) d x d y + C as a ↗ 2 a^{*} . \end{array}$ (3.50)

The same argument as (2.5) together with (3.5) yields that

$\begin{array}{l} \int_{ℝ^{2}} \int_{ℝ^{2}} \ln | x - y | v_{1}^{a} (x) v_{2}^{a} (x) v_{1}^{a} (y) v_{2}^{a} (y) d x d y \\ \to \frac{1}{{(2 a^{*})}^{2}} \int_{ℝ^{2}} \int_{ℝ^{2}} \ln | x - y | Q^{2} (x) Q^{2} (y) d x d y as a ↗ 2 a^{*} . \end{array}$ (3.51)

Following (1.12) and (1.15), we then obtain from (3.5), (3.7) and (3.51) that

$\begin{matrix} e (γ, a) = \frac{1}{2 ε_{a}^{2}} \int_{ℝ^{2}} ({| \nabla v_{1}^{a} |}^{2} + {| \nabla v_{2}^{a} |}^{2}) d x \\ - \frac{a^{*}}{4 ε_{a}^{2}} \int_{ℝ^{2}} {(\sqrt{k (ε_{a} x + ε_{a} y_{ε_{a}})} ({| v_{1}^{a} |}^{2} + {| v_{2}^{a} |}^{2}))}^{2} d x \\ + \frac{1}{2} \int_{ℝ^{2}} {| ε_{a} x + ε_{a} y_{ε_{a}} |}^{2} ({| v_{1}^{a} |}^{2} + {| v_{2}^{a} |}^{2}) d x \\ + \frac{γ}{4 π} \int_{ℝ^{2}} \int_{ℝ^{2}} \ln | x - y | v_{1}^{a} (x) v_{2}^{a} (x) v_{1}^{a} (y) v_{2}^{a} (y) d x d y \\ + \frac{γ}{4 π} \ln ε_{a} \int_{ℝ^{2}} v_{1}^{a} (x) v_{2}^{a} (x) d x \int_{ℝ^{2}} v_{1}^{a} (y) v_{2}^{a} (y) d y \\ + \frac{a^{*}}{4 ε_{a}^{2}} \int_{ℝ^{2}} {(\sqrt{k (ε_{a} x + ε_{a} y_{ε_{a}})} ({| v_{1}^{a} |}^{2} - {| v_{2}^{a} |}^{2}))}^{2} d x \\ + \frac{2 a^{*} - a}{2 ε_{a}^{2}} \int_{ℝ^{2}} k (ε_{a} x + ε_{a} y_{ε_{a}}) {| v_{1}^{a} |}^{2} {| v_{2}^{a} |}^{2} d x \\ \geq \frac{γ}{16 π {(a^{*})}^{2}} \int_{ℝ^{2}} \int_{ℝ^{2}} \ln | x - y | Q^{2} (x) Q^{2} (y) d x d y + o (1) \\ + \frac{γ}{16 π} (1 + o (1)) \ln ε_{a} + \frac{2 a^{*} - a}{4 a^{*} ε_{a}^{2}} (1 + o (1)) \\ \geq \frac{γ}{16 π {(a^{*})}^{2}} \int_{ℝ^{2}} \int_{ℝ^{2}} \ln | x - y | Q^{2} (x) Q^{2} (y) d x d y + o (1) \\ + \frac{γ}{32 π} [1 + \ln \frac{8 π (2 a^{*} - a)}{γ a^{*}}] (1 + o (1)), \end{matrix}$ (3.52)

where the identity in the last inequality is achieved at

$ε_{a} = {[\frac{8 π (2 a^{*} - a)}{γ a^{*}}]}^{\frac{1}{2}} (1 + o (1)) as a ↗ 2 a^{*} .$ (3.53)

Hence it follows from (3.50) and (3.52) that

$e (γ, a) \approx \frac{γ}{32 π} ln \frac{8 π (2 a^{*} - a)}{γ a^{*}} + C as a ↗ 2 a^{*},$

and $ε_{a}$ satisfies (3.53). We conclude from (3.38) and (3.53) that for $i = 1, 2$ and as $a ↗ 2 a^{*}$ ,

$\sqrt{\frac{8 π (2 a^{*} - a)}{γ a^{*}}} u_{i}^{a} (\sqrt{\frac{8 π (2 a^{*} - a)}{γ a^{*}}} x + x_{i}^{a}) \to \frac{Q (x)}{\sqrt{2 a^{*}}} in H \cap L^{\infty} (ℝ^{2}) .$

This completes the proof of Theorem 1.2.

Acknowledgements

The author is very grateful to the referee for many valuable suggestions which lead to the great improvements of the present paper.

Conflicts of Interest

The author declares no conflicts of interest.

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