Abstract

This paper is devoted to the normalized solutions of a planar $L^2$ -critical Schrödinger-Poisson system with an external potential $V\left(x\right)={\left|x\right|}^2$ and inhomogeneous attractive interactions $K\left(x\right)\in \left(0,1\right)$ . Applying the constraint variational method, we prove that the normalized solutions exist if and only if the interaction strength $a$ satisfies $a\in \left(0,a^{\text{*}}\right):={\Vert Q\Vert }_{L^2\left({ℝ}^2\right)}^2$ , where $Q$ is the unique positive solution of $\text{Δ}u-u+u^3=0$ in ${ℝ}^2$ . Particularly, the refined limiting behavior of positive minimizers is also analyzed as $a\nearrow a^{\text{*}}$ .

Keywords

Schrödinger-Poisson System, Logarithmic Convolution, Inhomogeneous Attractive Interaction, Normalized Solution

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1. Introduction

In this paper, we study the following inhomogeneous elliptic equation with a power potential and a logarithmic convolution potential

$-\text{Δ}u+\left({\left|x\right|}^2-\mu \right)u+\left(\text{ln}\left|x\right|\ast u^2\right)u=aK\left(x\right){\left|u\right|}^{2}u\text{in}{ℝ}^2,$ (1.1)

where $\mu \in ℝ$ is an uncertain Lagrange constant, $a>0$ denotes the strength of attractive interactions, and $K\left(x\right)>0$ gives the spatially inhomogeneous attractive interactions. Under the standing wave ansatz $\psi \left(x,t\right)={\text{e}}^{i\mu t}u\left(x\right)$ , where $i$ is the imaginary unit, it is well known that (1.1) can be obtained from the time-dependent Schrödinger-Poisson system

$\{\begin{array}{ll}i{\psi }_t-\text{Δ}\psi +{\left|x\right|}^2\psi +\lambda \omega \psi =aK\left(x\right){\left|\psi \right|}^2\psi \hfill & \text{in}{ℝ}^2\times ℝ,\hfill \\ \text{Δ}\omega ={\left|\psi \right|}^2\hfill & \text{in}{ℝ}^2\times ℝ,\hfill \end{array}$

where $\psi :{ℝ}^2\times ℝ\to ℂ$ is the time-dependent wave function and $\lambda \in ℝ$ is a parameter. The function $\omega$ represents an internal potential for a nonlocal self-interaction of the wave function $\psi$ , and the nonlinear term ${\left|\psi \right|}^2\psi$ is frequently used to model the interaction among particles [1]-[3]. In the past several decades, this system, as a cross-disciplinary model bridging quantum mechanics and classical electromagnetism, has garnered great attention owing to its physical relevance. It originates from quantum mechanics [4]-[9] and especially semiconductor physics [10] [11]. We would like to mention the results [12]-[15] for normalized solutions of inhomogeneous elliptic equations, and [15]-[18] with references therein for the Schrödinger-Poisson systems.

In order to investigate the normalized solutions of (1.1), we define the energy functional

$\begin{array}{c}{E}_a\left(u\right):=\frac{1}{2}{\displaystyle {\int }_{{ℝ}^2}\left[{\left|\nabla u\left(x\right)\right|}^2+{\left|x\right|}^2{u}^2\left(x\right)\right]\text{d}x}\\ +\frac{1}{4}{\displaystyle {\int }_{{ℝ}^2}{\displaystyle {\int }_{{ℝ}^2}\text{ln}\left|x-y\right|u^2\left(x\right)u^2\left(y\right)\text{d}x\text{d}y}}-\frac{a}{4}{\displaystyle {\int }_{{ℝ}^2}K}\left(x\right)u^4\left(x\right)\text{d}x.\end{array}$ (1.2)

Due to the power potential term and the logarithmic convolution term, $E_a$ is not well defined on $H^1\left({ℝ}^2\right)$ . Stimulated by [14], we consider the space $X$ satisfying

$X:=\left\{u\in H^1\left({ℝ}^2\right):{\Vert u\Vert }_{\text{*}}:={\left({\displaystyle {\int }_{{ℝ}^2}{\left|x\right|}^2{u}^2\left(x\right)\text{d}x}\right)}^{\frac{1}{2}}<\infty \right\}$

with the associated norm

${\Vert u\Vert }_X:={\left\{{\displaystyle {\int }_{{ℝ}^2}\left[{\left|\nabla u\right|}^2+\left(1+{\left|x\right|}^2\right)u^2\left(x\right)\right]\text{d}x}\right\}}^{\frac{1}{2}},u\in X.$

Recall from ([19], Lemma 3.1) that for any $p\in \left[2,\infty \right)$ ,

$X\text{is}\text{compactly}\text{embedded}\text{into}{L}^p\left({ℝ}^2\right).$ (1.3)

As performed in [20], we decompose the logarithmic convolution term as below:

$F_1\left(u\right):={\displaystyle {\int }_{{ℝ}^2}{\displaystyle {\int }_{{ℝ}^2}\text{ln}}}\left(1+\left|x-y\right|\right)u^2\left(x\right)u^2\left(y\right)\text{d}x\text{d}y,$

$F_2\left(u\right):={\displaystyle {\int }_{{ℝ}^2}{\displaystyle {\int }_{{ℝ}^2}\text{ln}}}\left(1+\frac{1}{\left|x-y\right|}\right)u^2\left(x\right)u^2\left(y\right)\text{d}x\text{d}y,$

and

$F_1\left(u\right)-F_2\left(u\right)={\displaystyle {\int }_{{ℝ}^2}{\displaystyle {\int }_{{ℝ}^2}\text{ln}}}\left|x-y\right|u^2\left(x\right)u^2\left(y\right)\text{d}x\text{d}y.$

In what follows, we use ${\Vert \cdot \Vert }_p$ to denote the standard Lebesgue norm on $L^p\left({ℝ}^2\right)$ . Since

$\text{ln}\left(1+\left|x-y\right|\right)\le \left|x-y\right|\le \left|x\right|+\left|y\right|,x,y\in {ℝ}^2,$

cwe derive from the Hölder inequality that

$F_1\left(u\right)\le {\displaystyle {\int }_{{ℝ}^2}{\displaystyle {\int }_{{ℝ}^2}\left(\left|x\right|+\left|y\right|\right)u^2\left(x\right)u^2\left(y\right)\text{d}x\text{d}y}}\le 2{\Vert u\Vert }_2^3{\Vert u\Vert }_{\ast }.$ (1.4)

From the fact that $0<\text{ln}\left(1+r\right)<r$ holds for all $r>0$ again, we can deduce from the Hardy-Littlewood-Sobolev inequality (cf. [21]):

${\displaystyle {\int }_{{ℝ}^2}{\displaystyle {\int }_{{ℝ}^2}\frac{\left|u\left(x\right)\right|\left|v\left(y\right)\right|}{\left|x-y\right|}\text{d}x\text{d}y}}\le C{\Vert u\Vert }_{\frac{4}{3}}{\Vert u\Vert }_{\frac{4}{3}},u,v\in L^{\frac{4}{3}}\left({ℝ}^2\right),$ (1.5)

that there exists a constant $C>0$ such that

$F_2\left(u\right)\le {\displaystyle {\int }_{{ℝ}^2}{\displaystyle {\int }_{{ℝ}^2}\frac{u^2\left(x\right)u^2\left(y\right)}{\left|x-y\right|}\text{d}x\text{d}y}}\le C{\Vert u\Vert }_{\frac{8}{3}}^4,u\in L^{\frac{8}{3}}\left({ℝ}^2\right).$ (1.6)

It follows from (1.4) and (1.6) that ${\displaystyle {\int }_{{ℝ}^2}{\displaystyle {\int }_{{ℝ}^2}\text{ln}}}\left|x-y\right|u^2\left(x\right)u^2\left(y\right)\text{d}x\text{d}y$ is well defined on $X$ .

Throughout the paper, we assume that the inhomogeneous attractive interactions $K\left(x\right)$ satisfy that

$0<K\left(x\right)\le 1\text{and}K\left(0\right)=\underset{x\in {ℝ}^2}{\text{sup}}K\left(x\right)=1.$ (1.7)

Hence $E_a$ is well defined on $X$ . In the following we focus on studying the minimizers of the constraint variational problem:

$e\left(a\right):=\underset{u\in S}{\text{inf}}{E}_a\left(u\right),$ (1.8)

where the manifold $S$ is defined by

$S:=\left\{u\in X:{\displaystyle {\int }_{{ℝ}^2}{u}^2\left(x\right)\text{d}x}=1\right\}.$

The main purpose of this paper is to prove the existence, nonexistence and the refined limiting behavior of minimizers for $e\left(a\right)$ . The proof is closely related to the unique (up to translations) positive solution $Q\left(x\right)=Q\left(\left|x\right|\right)$ of the following elliptic equation (cf. [22] [23]):

$-\text{Δ}u+u=u^3,u\in H^1\left({ℝ}^2\right).$ (1.9)

Note from [23] that the function $Q\left(x\right)$ satisfies exponential decay in the sense that

$Q\left(\left|x\right|\right),\left|\nabla Q\left(\left|x\right|\right)\right|=O\left({\left|x\right|}^{-\frac{1}{2}}{\text{e}}^{-\left|x\right|}\right)\text{as}\left|x\right|\to \infty .$ (1.10)

In addition, we also need the following Gagliardo-Nirenberg inequality (cf. [24]):

${\displaystyle {\int }_{{ℝ}^2}{\left|u\left(x\right)\right|}^4\text{d}x}\le \frac{2}{{\Vert Q\Vert }_2^2}{\displaystyle {\int }_{{ℝ}^2}{\left|\nabla u\left(x\right)\right|}^2\text{d}x}{\displaystyle {\int }_{{ℝ}^2}{\left|u\left(x\right)\right|}^2\text{d}x},u\in H^1\left({ℝ}^2\right),$ (1.11)

where the equality is achieved at $u\left(x\right)=Q\left(\left|x\right|\right)$ . We can derive from (1.9) and (1.11) that

${\displaystyle {\int }_{{ℝ}^2}{\left|\nabla Q\left(x\right)\right|}^2\text{d}x}={\displaystyle {\int }_{{ℝ}^2}{Q}^2}\left(x\right)\text{d}x=\frac{1}{2}{\displaystyle {\int }_{{ℝ}^2}{Q}^4}\left(x\right)\text{d}x,$ (1.12)

Applying the above facts, we can establish the existence and nonexistence of minimizers for $e\left(a\right)$ .

Theorem 1.1. Let $Q\left(x\right)=Q\left(\left|x\right|\right)$ be the unique positive solution of (1.9) and $a^{*}:={\Vert Q\Vert }_2^2$ .

1) If $a\in \left(0,a^{\text{*}}\right)$ , then there exists at least one minimizer of $e\left(a\right)$ ;

2) If $a\in \left(a^{\text{*}},\infty \right)$ , then there is no minimizer of $e\left(a\right)$ and $e\left(a\right)=-\infty$ ;

3) If $a=a^{\text{*}}$ and $1-K\left(x\right)=O\left({\left|x\right|}^2\right)$ as $x\to 0$ , then there is no minimizer of $e\left(a\right)$ and $e\left(a\right)=-\infty$ .

Moreover, there holds $\underset{a\nearrow a^{\text{*}}}{\text{lim}}e\left(a\right)=-\infty$ when $1-K\left(x\right)=O\left({\left|x\right|}^2\right)$ as $x\to 0$ .

Suppose that $u_a$ is a minimizer of $e\left(a\right)$ for $a\in \left(0,a^{\text{*}}\right)$ , then according to the variational theory, $u_a$ satisfies the following Euler-Lagrange equation:

$-\text{Δ}{u}_a+{\left|x\right|}^2{u}_a+{\displaystyle {\int }_{{ℝ}^2}\text{ln}}\left|x-y\right|u_a^2\left(y\right)\text{d}y{u}_a={\mu }_a{u}_a+aK\left(x\right){\left|u_a\right|}^2{u}_a\text{in}{ℝ}^2,$ (1.13)

where ${\mu }_a\in ℝ$ denotes the Lagrange multiplier and satisfies

${\mu }_a=2e\left(a\right)+\frac{1}{2}{\displaystyle {\int }_{{ℝ}^2}{\displaystyle {\int }_{{ℝ}^2}\text{ln}}}\left|x-y\right|u_a^2\left(x\right)u_a^2\left(y\right)\text{d}x\text{d}y-\frac{a}{2}{\displaystyle {\int }_{{ℝ}^2}K}\left(x\right)u_a^4\text{d}x.$ (1.14)

That is, $u_a$ is a normalized solution of (1.1). Noticing $E_a\left(\left|u_a\right|\right)=E_a\left(u_a\right)$ , we get that $\left|u_a\right|$ is also a minimizer of $e\left(a\right)$ . Together with the strong maximum principle, we next mainly discuss the limiting behavior of positive minimizers.

Theorem 1.2. Assume that $u_a$ is a positive minimizer of $e\left(a\right)$ for $a\in \left(0,a^{*}\right)$ and $1-K\left(x\right)=O\left({\left|x\right|}^2\right)$ as $x\to 0$ . Then

$\underset{a\nearrow a^{\text{*}}}{\text{lim}}2\sqrt{\frac{a^{\text{*}}-a}{a^{\text{*}}}}{u}_a\left(2\sqrt{\frac{a^{\text{*}}-a}{a^{\text{*}}}}x+x_a\right)=Q\left(x\right)\text{in}X,$

where $x_a$ is the unique global maximum point of $u_a$ as $a\nearrow a^{\text{*}}$ .

The proof of Theorem 1.2 requires a series of analysis. We have to overcome the sign-changing property of the logarithmic convolution term. We shall derive the following crucial estimate: there exists a constant $C>0$ such that for any $x\in {ℝ}^2$ and for all $a\in \left(0,a^{\text{*}}\right)$ , there holds

${\displaystyle {\int }_{{ℝ}^2}\text{ln}}\left(1+\frac{1}{\left|x-y\right|}\right)v_a^2\left(y\right)\text{d}y\le C,$

where $v_a$ is a suitable scaled function of the minimizer $u_a$ .

We organize the next of this paper as follows. In Section 2, we prove Theorem 1.1 on the existence and nonexistence of minimizers for $e\left(a\right)$ . In Section 3, we prove Theorem 1.2 on the refined limiting behavior of positive minimizers for $e\left(a\right)$ as $a\nearrow a^{\text{*}}$ .

2. Existence and Nonexistence of Minimizers

In this section, we shall complete the proof of Theorem 1.1 by applying the Gagliardo-Nirenberg inequalities and the properties of $Q\left(x\right)$ .

Proof of Theorem 1.1. 1). For any $p\ge 2$ and $u\in H^1\left({ℝ}^2\right)$ , there results the Gagliardo-Nirenberg inequality (cf. [24]):

${\Vert u\Vert }_p\le {\left(\frac{p}{2{\Vert Q_p\Vert }_2^{p-2}}\right)}^{\frac{1}{p}}{\left({\displaystyle {\int }_{{ℝ}^2}{\left|\nabla u\right|}^2\text{d}x}\right)}^{\frac{p-2}{2p}}{\left({\displaystyle {\int }_{{ℝ}^2}{\left|u\right|}^2\text{d}x}\right)}^{\frac{1}{p}},$ (2.1)

where $Q_p$ is the positive ground state solution of the following elliptic equation

$-\frac{p-2}{2}\text{Δ}u+u=u^{p-1},u\in H^1\left({ℝ}^2\right).$

By (1.6) and (2.1), we derive that there exists a constant $C>0$ such that

$F_2\left(u\right)\le C{\left({\displaystyle {\int }_{{ℝ}^2}{\left|\nabla u\right|}^2\text{d}x}\right)}^{\frac{1}{2}},u\in S.$ (2.2)

Under the assumption (1.7), we deduce from (1.11) that

${\displaystyle {\int }_{{ℝ}^2}K}\left(x\right)u^4\text{d}x\le {\displaystyle {\int }_{{ℝ}^2}{u}^4\text{d}x}\le \frac{2}{a^{\text{*}}}{\displaystyle {\int }_{{ℝ}^2}{\left|\nabla u\right|}^2\text{d}x},u\in S.$ (2.3)

Notice that $F_1\left(u\right)\ge 0$ , we infer from (2.2) and (2.3) that for $u\in S$ ,

$E_a\left(u\right)\ge \left(\frac{1}{2}-\frac{a}{2a^{\text{*}}}\right){\displaystyle {\int }_{{ℝ}^2}{\left|\nabla u\right|}^2\text{d}x}+\frac{1}{2}{\displaystyle {\int }_{{ℝ}^2}{\left|x\right|}^2{u}^2\text{d}x}-C{\left({\displaystyle {\int }_{{ℝ}^2}{\left|\nabla u\right|}^2\text{d}x}\right)}^{\frac{1}{2}},$ (2.4)

which implies that $E_a\left(u\right)$ is bounded from below on $S$ when $a\in \left(0,a^{\text{*}}\right)$ .

Letting $\left\{u_n\right\}\subset S$ be a minimizing sequence of $e\left(a\right)$ for $a\in \left(0,a^{\text{*}}\right)$ , we can know from (2.4) that ${\displaystyle {\int }_{{ℝ}^2}{\left|\nabla u_n\right|}^2\text{d}x}$ and ${\displaystyle {\int }_{{ℝ}^2}{\left|x\right|}^2{u}_n^2\text{d}x}$ are bounded uniformly with respect to $n$ . Since ${\displaystyle {\int }_{{ℝ}^2}{u}_n^2\text{d}x}=1$ , we then obtain that $\left\{u_n\right\}$ is bounded uniformly in $X$ . By (1.3), there exists a function $u\in X$ such that

$u_n\rightharpoonup u\text{in}X\text{and}{u}_n\to u\text{in}{L}^p\left({ℝ}^2\right)\text{for}p\in \left[2,\infty \right),$

which implies that

${\displaystyle {\int }_{{ℝ}^2}{u}^2\text{d}x}=1\text{and}\underset{n\to \infty }{\text{lim}}{\displaystyle {\int }_{{ℝ}^2}K}\left(x\right)u_n^4\text{d}x={\displaystyle {\int }_{{ℝ}^2}K}\left(x\right)u^4\text{d}x.$

Then we obtain that $u\in S$ . Furthermore, according to ([20], Lemma 2.2), we have

${\displaystyle {\int }_{{ℝ}^2}{\displaystyle {\int }_{{ℝ}^2}\ln }}\left|x-y\right|u^2\left(x\right)u^2\left(y\right)\text{d}x\text{d}y\le \underset{n\to \infty }{\lim \inf }{\displaystyle {\int }_{{ℝ}^2}{\displaystyle {\int }_{{ℝ}^2}\ln }}\left|x-y\right|u_n^2\left(x\right)u_n^2\left(y\right)\text{d}x\text{d}y.$

Together with the weak lower semicontinuity of norm, we then deduce from above that

$e\left(a\right)\le E_a\left(u\right)\le \underset{n\to \infty }{\lim \inf }{E}_a\left(u_n\right)=e\left(a\right),$

which yields that $E_a\left(u\right)=e\left(a\right)$ . This indicates that $u$ is a minimizer of $e\left(a\right)$ for $a\in \left(0,a^{\text{*}}\right)$ .

2). Consider the function

$u_{\tau }\left(x\right)=\frac{\tau }{\sqrt{a^{\text{*}}}}Q\left(\tau x\right),\tau >0.$

Then $u_{\tau }\in S$ for all $\tau >0$ . We deduce from (1.12) that

$\begin{array}{l}e\left(a\right)\le E_a\left(u_{\tau }\right)=\frac{{\tau }^2}{4a^{\text{*}}}{\displaystyle {\int }_{{ℝ}^2}\left[1-\frac{a}{a^{\text{*}}}K\left(\frac{x}{\tau }\right)\right]Q^4\left(x\right)\text{d}x}+\frac{1}{2{\tau }^2{a}^{\text{*}}}{\displaystyle {\int }_{{ℝ}^2}{\left|x\right|}^2{Q}^2\left(x\right)\text{d}x}\\ +\frac{1}{4{\left(a^{\text{*}}\right)}^2}{\displaystyle {\int }_{{ℝ}^2}{\displaystyle {\int }_{{ℝ}^2}\text{ln}}}\left|x-y\right|Q^2\left(x\right)Q^2\left(y\right)\text{d}x\text{d}y-\frac{1}{4}\text{ln}\tau .\end{array}$ (2.5)

Through (1.4), (1.10), (2.2) and the assumption (1.7), we obtain from (2.5) that

$e\left(a\right)\le \underset{\tau \to \infty }{\text{lim}}{E}_a\left(u_{\tau }\right)=-\infty \text{for}a>a^{\text{*}},$

which implies that there is no minimizer of $e\left(a\right)$ and $e\left(a\right)=-\infty$ when $a>a^{\text{*}}$ .

3). For the case $a=a^{\text{*}}$ , we infer from (2.5) that

$\begin{array}{l}e\left(a\right)\le E_a\left(u_{\tau }\right)=\frac{{\tau }^2}{4a^{\text{*}}}{\displaystyle {\int }_{{ℝ}^2}\left[1-K\left(\frac{x}{\tau }\right)\right]Q^4\left(x\right)\text{d}x}+\frac{1}{2{\tau }^2{a}^{\text{*}}}{\displaystyle {\int }_{{ℝ}^2}{\left|x\right|}^2{Q}^2\left(x\right)\text{d}x}\\ +\frac{1}{4{\left(a^{\text{*}}\right)}^2}{\displaystyle {\int }_{{ℝ}^2}{\displaystyle {\int }_{{ℝ}^2}\text{ln}}}\left|x-y\right|Q^2\left(x\right)Q^2\left(y\right)\text{d}x\text{d}y-\frac{1}{4}\ln \tau .\end{array}$ (2.6)

In virtue of (1.10), we can take $\delta >0$ satisfying ${\displaystyle {\int }_{\left|x\right|>\delta }{Q}^4\text{d}x}<{\tau }^{-2}$ . Hence we get from (1.7) that

$\frac{{\tau }^2}{4a^{\text{*}}}{\displaystyle {\int }_{\left|x\right|>\delta }\left[1-K\left(\frac{x}{\tau }\right)\right]Q^4\left(x\right)\text{d}x}\le \frac{1}{2a^{\text{*}}}.$

By the assumption that $1-K\left(x\right)=O\left({\left|x\right|}^2\right)$ as $x\to 0$ , we have

$\frac{{\tau }^2}{4a^{\text{*}}}{\displaystyle {\int }_{\left|x\right|\le \delta }\left[1-K\left(\frac{x}{\tau }\right)\right]Q^4\left(x\right)\text{d}x}\le C\text{as}\tau \to \infty .$

Therefore, we obtain from (1.4), (1.10), (2.2) and (2.6) that

$e\left(a^{\text{*}}\right)\le \underset{\tau \to \infty }{\text{lim}}{E}_{a^{\text{*}}}\left(u_{\tau }\right)=-\infty ,$

which means that there is no minimizer of $e\left(a^{\text{*}}\right)$ and $e\left(a^{\text{*}}\right)=-\infty$ .

In addition, for $a\in \left(0,a^{\text{*}}\right)$ , choosing $\tau ={\left(a^{\text{*}}-a\right)}^{-\frac{1}{2}}$ in (2.5), we get

$\begin{array}{c}e\left(a\right)\le \frac{1}{2a^{\text{*}}}+\frac{{\tau }^{2}a}{4{\left(a^{\text{*}}\right)}^2}{\displaystyle {\int }_{{ℝ}^2}\left[1-K\left(\frac{x}{\tau }\right)\right]Q^4\left(x\right)\text{d}x}+\frac{1}{2{\tau }^2{a}^{\text{*}}}{\displaystyle {\int }_{{ℝ}^2}{\left|x\right|}^2{Q}^2\left(x\right)\text{d}x}\\ +\frac{1}{4{\left(a^{\text{*}}\right)}^2}{\displaystyle {\int }_{{ℝ}^2}{\displaystyle {\int }_{{ℝ}^2}\text{ln}}}\left|x-y\right|Q^2\left(x\right)Q^2\left(y\right)\text{d}x\text{d}y-\frac{1}{4}\text{ln}\tau .\end{array}$ (2.7)

One can also obtain from the assumption $1-K\left(x\right)=O\left({\left|x\right|}^2\right)$ as $x\to 0$ that

$\frac{{\tau }^{2}a}{4{\left(a^{\text{*}}\right)}^2}{\displaystyle {\int }_{{ℝ}^2}\left[1-K\left(\frac{x}{\tau }\right)\right]Q^4\left(x\right)\text{d}x}\le C\text{as}a\nearrow a^{\text{*}}.$

Thus we obtain from (1.4), (1.10), (2.2) and (2.7) that $\underset{a\nearrow a^{\text{*}}}{\text{lim}}e\left(a\right)=-\infty .$ This completes the proof of Theorem 1.1.

3. Limiting Behavior of Minimizers

In this section, we shall prove Theorem 1.2 on the limiting behavior of positive minimizers for $e\left(a\right)$ as $a\nearrow a^{\text{*}}$ . We first establish some estimates for the positive minimizers of $e\left(a\right)$ as $a\nearrow a^{\text{*}}$ .

Lemma 3.1. Assume that $u_a$ is a positive minimizer of $e\left(a\right)$ for $a\in \left(0,a^{*}\right)$ and $1-K\left(x\right)=O\left({\left|x\right|}^2\right)$ as $x\to 0$ . Let

${\epsilon }_a:={\left({\displaystyle {\int }_{{ℝ}^2}{\left|\nabla u_a\right|}^2\text{d}x}\right)}^{-\frac{1}{2}},$ (3.1)

$v_a\left(x\right):={\epsilon }_a{u}_a\left({\epsilon }_{a}x+x_a\right)\text{in}{ℝ}^2,$ (3.2)

where $x_a$ is a global maximum point of $u_a$ . Then [(1)]

1) ${\epsilon }_a>0$ satisfies

${\epsilon }_a\to 0\text{and}{\mu }_a{\epsilon }_a^2\to -1\text{as}a\nearrow a^{\text{*}};$ (3.3)

2) There exists a constant $\eta >0$ , independent of $a\in \left(0,a^{\text{*}}\right)$ , such that

${\displaystyle {\int }_{B_2\left(0\right)}{v}_a^2\left(x\right)\text{d}x}\ge \eta \text{as}a\nearrow a^{\text{*}};$ (3.4)

3) $v_a$ satisfies

$v_a\left(x\right)\to \frac{1}{\sqrt{a^{\text{*}}}}Q\left(\left|x\right|\right)\text{in}{H}^1\left({ℝ}^2\right)\text{as}a\nearrow a^{\text{*}};$ (3.5)

4) There exist a large constant $R>0$ and a constant $C>0$ , independent of $a$ , such that

$\left|v_a\left(x\right)\right|\le C{\text{e}}^{-\frac{2\left|x\right|}{3}}\text{for}\left|x\right|\ge R\text{as}a\nearrow a^{\text{*}}.$ (3.6)

Proof. 1). Through (2.2) and (2.3), we have

$e\left(a\right)=E_a\left(u_a\right)\ge \frac{a^{\text{*}}-a}{2a^{\text{*}}}{\epsilon }_a^{-2}-C{\epsilon }_a^{-1}\ge -C{\epsilon }_a^{-1}.$

Together with the fact $\underset{a\nearrow a^{\text{*}}}{\text{lim}}e\left(a\right)=-\infty$ in Theorem 1.1, we obtain that ${\epsilon }_a\to 0$ as $a\nearrow a^{\text{*}}$ .

By (3.1), we have

$\begin{array}{c}{\epsilon }_a^{2}e\left(a\right)=\frac{1}{2}+\frac{{\epsilon }_a^2}{2}{\displaystyle {\int }_{{ℝ}^2}{\left|x\right|}^2{u}_a^2\left(x\right)\text{d}x}+\frac{{\epsilon }_a^2}{4}{\displaystyle {\int }_{{ℝ}^2}{\displaystyle {\int }_{{ℝ}^2}\text{ln}}}\left(1+\left|x-y\right|\right)u_a^2\left(x\right)u_a^2\left(y\right)\text{d}x\text{d}y\\ -\frac{{\epsilon }_a^2}{4}{\displaystyle {\int }_{{ℝ}^2}{\displaystyle {\int }_{{ℝ}^2}\text{ln}}}\left(1+\frac{1}{\left|x-y\right|}\right)u_a^2\left(x\right)u_a^2\left(y\right)\text{d}x\text{d}y-\frac{a{\epsilon }_a^2}{4}{\displaystyle {\int }_{{ℝ}^2}K}\left(x\right)u_a^4\left(x\right)\text{d}x.\end{array}$ (3.7)

Since ${\epsilon }_a\to 0$ as $a\nearrow a^{\text{*}}$ , we derive from (2.2) that

$0\le {\epsilon }_a^2{\displaystyle {\int }_{{ℝ}^2}{\displaystyle {\int }_{{ℝ}^2}\ln }}\left(1+\frac{1}{\left|x-y\right|}\right)u_a^2\left(x\right)u_a^2\left(y\right)\text{d}x\text{d}y\le C{\epsilon }_a\to 0\text{as}a\nearrow a^{\text{*}}.$ (3.8)

Note from (2.3) that

$\frac{1}{2}\left(1-\frac{a{\epsilon }_a^2}{2}{\displaystyle {\int }_{{ℝ}^2}K}\left(x\right)u_a^4\left(x\right)\text{d}x\right)\ge 0.$ (3.9)

Hence we deduce from (3.7)-(3.9) that $\underset{a\nearrow a^{*}}{\lim \inf }{\epsilon }_a^{2}e\left(a\right)\ge 0$ . Using the fact that $\underset{a\nearrow a^{\text{*}}}{\text{lim}}e\left(a\right)=-\infty$ again, we have $\underset{a\nearrow a^{*}}{\lim \sup }{\epsilon }_a^{2}e\left(a\right)\le 0$ . Therefore, we conclude that

$\underset{a\nearrow a^{\text{*}}}{\text{lim}}{\epsilon }_a^{2}e\left(a\right)=0.$ (3.10)

Furthermore, one can obtain from (3.7)-(3.10) that

$\underset{a\nearrow a^{\text{*}}}{\text{lim}}{\epsilon }_a^2{\displaystyle {\int }_{{ℝ}^2}K}\left(x\right)u_a^4\text{d}x=\frac{2}{a^{\text{*}}},\underset{a\nearrow a^{\text{*}}}{\text{lim}}{\epsilon }_a^2{\displaystyle {\int }_{{ℝ}^2}{\left|x\right|}^2{u}_a^2\text{d}x}=0,$ (3.11)

$\underset{a\nearrow a^{\text{*}}}{\text{lim}}{\epsilon }_a^2{\displaystyle {\int }_{{ℝ}^2}{\displaystyle {\int }_{{ℝ}^2}\ln }}\left(1+\left|x-y\right|\right)u_a^2\left(x\right)u_a^2\left(y\right)\text{d}x\text{d}y=0.$ (3.12)

Note from (1.14) that

$\begin{array}{c}{\mu }_a{\epsilon }_a^2=2{\epsilon }_a^{2}e\left(a\right)+\frac{{\epsilon }_a^2}{2}{\displaystyle {\int }_{{ℝ}^2}{\displaystyle {\int }_{{ℝ}^2}\ln }}\left|x-y\right|u_a^2\left(x\right)u_a^2\left(y\right)\text{d}x\text{d}y\\ -\frac{a{\epsilon }_a^2}{2}{\displaystyle {\int }_{{ℝ}^2}K}\left(x\right)u_a^4\left(x\right)\text{d}x.\end{array}$ (3.13)

Together with (3.8) and (3.10)-(3.12), we conclude from (3.13) that ${\mu }_a{\epsilon }_a^2\to -1$ as $a\nearrow a^{\text{*}}$ .

2). Due to (1.13) and (3.2), we see that $v_a$ satisfies

$\begin{array}{l}-\Delta v_a+{\epsilon }_a^2{\left|{\epsilon }_{a}x+x_a\right|}^2{v}_a+{\epsilon }_a^2\left({\displaystyle {\int }_{{ℝ}^2}\ln }\left|x-y\right|v_a^2\left(y\right)\text{d}y\right)v_a+{\epsilon }_a^2\ln {\epsilon }_a{v}_a\\ ={\epsilon }_a^2{\mu }_a{v}_a+aK\left({\epsilon }_{a}x+x_a\right)v_a^3\text{in}{ℝ}^2.\end{array}$ (3.14)

We use $\Vert \cdot \Vert$ to denote the standard norm on $H^1\left({ℝ}^2\right)$ . Note that

${\Vert v_a\Vert }^2={\epsilon }_a^2{\displaystyle {\int }_{{ℝ}^2}{\left|\nabla u_a\right|}^2\text{d}x}+{\displaystyle {\int }_{{ℝ}^2}{u}_a^2\text{d}x}=2.$ (3.15)

There exists a constant $C>0$ such that for any $x\in {ℝ}^2$ and $a\in \left(0,a^{\text{*}}\right)$ ,

$\begin{array}{l}{\displaystyle {\int }_{{ℝ}^2}\text{ln}}\left(1+\frac{1}{\left|x-y\right|}\right)v_a^2\left(y\right)\text{d}y\\ \le {\displaystyle {\int }_{\left|x-y\right|<1}\frac{v_a^2\left(y\right)}{\left|x-y\right|}\text{d}y}+{\displaystyle {\int }_{\left|x-y\right|\ge 1}\frac{v_a^2\left(y\right)}{\left|x-y\right|}\text{d}y}\\ \le {\left({\displaystyle {\int }_{\left|x-y\right|<1}\frac{1}{{\left|x-y\right|}^{\frac{3}{2}}}\text{d}y}\right)}^{\frac{2}{3}}{\left({\displaystyle {\int }_{\left|x-y\right|<1}{v}_a^6\left(y\right)\text{d}y}\right)}^{\frac{1}{3}}+{\displaystyle {\int }_{\left|x-y\right|\ge 1}{v}_a^2\left(y\right)\text{d}y}\\ \le C{\Vert v_a\Vert }^2\le C,\end{array}$ (3.16)

Thus we infer from (1.7), (3.3), (3.14) and (3.16) that

$-\text{Δ}{v}_a+\frac{5}{9}{v}_a-a^{\text{*}}{v}_a^3\le 0\text{in}{ℝ}^2\text{as}a\nearrow a^{\text{*}}.$ (3.17)

Since $x_a$ is a maximum point of $u_a$ , the origin is a maximum point of $v_a$ for $a\in \left(0,a^{\text{*}}\right)$ , which illustrates that $-\text{Δ}{v}_a\left(0\right)\ge 0$ for $a\in \left(0,a^{\text{*}}\right)$ . Thus we get from (3.17) that there exists some constant $\beta >0$ , independent of $a$ , such that $v_a\left(0\right)\ge \beta >0$ as $a\nearrow a^{\text{*}}$ . Applying the De Giorgi-Nash-Moser theory [25], we derive from (3.17) that there exists a constant $C>0$ such that

${\left({\displaystyle {\int }_{B_2\left(0\right)}{v}_a^2\text{d}x}\right)}^{\frac{1}{2}}\ge C\underset{x\in B_1\left(0\right)}{\max }{v}_a\ge C\beta :=\sqrt{\eta }>0\text{as}a\nearrow a^{\text{*}}.$

3). In view of (3.15), up to a subsequence if necessary, there exists a function $v_0\in H^1\left({ℝ}^2\right)$ such that $v_a\rightharpoonup v_0$ in $H^1\left({ℝ}^2\right)$ , $v_a\to v_0$ in $L_{\text{loc}}^p\left({ℝ}^2\right)$ for $p\in \left[2,\infty \right)$ , and $v_a\to v_0$ almost everywhere in ${ℝ}^2$ as $a\nearrow a^{\text{*}}$ . Furthermore, we get $v_0\cancel{\equiv }0$ from (3.4). Let $o\left(1\right)$ denote the infinitesimal quantities as $a\nearrow a^{\text{*}}$ . Based on the Brézis-Lieb lemma (cf. [26]), we obtain that as $a\nearrow a^{\text{*}}$ ,

$1={\Vert v_a\Vert }_2^2={\Vert v_0\Vert }_2^2+{\Vert v_a-v_0\Vert }_2^2+o\left(1\right),$

${\Vert K^{\frac{1}{4}}\left({\epsilon }_{a}x+x_a\right)v_a\Vert }_4^4={\Vert K^{\frac{1}{4}}\left({\epsilon }_{a}x+x_a\right)v_0\Vert }_4^4+{\Vert K^{\frac{1}{4}}\left({\epsilon }_{a}x+x_a\right)\left(v_a-v_0\right)\Vert }_4^4+o\left(1\right),$

and

$1={\Vert \nabla v_a\Vert }_2^2={\Vert \nabla v_0\Vert }_2^2+{\Vert \nabla v_a-\nabla v_0\Vert }_2^2+o\left(1\right).$

Together with (1.7), (1.11) and (3.11), it yields that

$\begin{array}{c}0=\underset{a\nearrow a^{\text{*}}}{\text{lim}}\left({\displaystyle {\int }_{{ℝ}^2}{\left|\nabla v_a\right|}^2\text{d}x}-\frac{a}{2}{\displaystyle {\int }_{{ℝ}^2}K}\left({\epsilon }_{a}x+x_a\right)v_a^4\text{d}x\right)\\ ={\displaystyle {\int }_{{ℝ}^2}{\left|\nabla v_0\right|}^2\text{d}x}-\underset{a\nearrow a^{\text{*}}}{\text{lim}}\frac{a}{2}{\displaystyle {\int }_{{ℝ}^2}K}\left({\epsilon }_{a}x+x_a\right)v_0^4\text{d}x\\ +\underset{a\nearrow a^{\text{*}}}{\text{lim}}\left({\displaystyle {\int }_{{ℝ}^2}{\left|\nabla v_a-\nabla v_0\right|}^2\text{d}x}-\frac{a}{2}{\displaystyle {\int }_{{ℝ}^2}K}\left({\epsilon }_{a}x+x_a\right){\left|v_a-v_0\right|}^4\text{d}x\right)\\ \ge \frac{a^{\text{*}}}{2}\left({\Vert v_0\Vert }_2^{-2}-1\right){\displaystyle {\int }_{{ℝ}^2}{v}_0^4\text{d}x}+\underset{a\nearrow a^{\text{*}}}{\text{lim}}\left({\displaystyle {\int }_{{ℝ}^2}{\left|\nabla v_a-\nabla v_0\right|}^2\text{d}x}-\frac{a}{2}{\displaystyle {\int }_{{ℝ}^2}{\left|v_a-v_0\right|}^4\text{d}x}\right)\\ \ge \underset{a\nearrow a^{\text{*}}}{\text{lim}}\left(1-{\displaystyle {\int }_{{ℝ}^2}{\left|v_a-v_0\right|}^2\text{d}x}\right){\displaystyle {\int }_{{ℝ}^2}{\left|\nabla v_a-\nabla v_0\right|}^2\text{d}x}\ge 0.\end{array}$ (3.18)

Therefore, we conclude from (3.18) that

${\Vert v_0\Vert }_2=1\text{and}{\displaystyle {\int }_{{ℝ}^2}{\left|\nabla v_a-\nabla v_0\right|}^2\text{d}x}\to 0\text{as}a\nearrow a^{\text{*}},$

which further imply that

$v_a\to v_0\text{in}{H}^1\left({ℝ}^2\right)\text{as}a\nearrow a^{\text{*}}.$

Additionally, the first equality of (3.18) yields that

${\displaystyle {\int }_{{ℝ}^2}{\left|\nabla v_0\right|}^2\text{d}x}=\frac{a^{\text{*}}}{2}\underset{a\nearrow a^{\text{*}}}{\text{lim}}K\left(x_a\right){\displaystyle {\int }_{{ℝ}^2}{v}_0^4\text{d}x}.$

Thus we derive from the Lagrange multiplier rule that $v_0$ satisfies

$-△v_0+v_0-a^{\text{*}}\underset{a\nearrow a^{\text{*}}}{\text{lim}}K\left(x_a\right)v_0^3=0\text{in}{ℝ}^2.$

The strong maximum principle implies that $v_0>0$ in ${ℝ}^2$ . Through a simple scaling, the uniqueness (up to translations) of the positive solution of (1.9) ensure that there exists a point $y_0\in {ℝ}^2$ such that

$v_0\left(x\right)={\left(a^{\text{*}}\underset{a\nearrow a^{\text{*}}}{\text{lim}}K\left(x_a\right)\right)}^{-\frac{1}{2}}Q\left(\left|x-y_0\right|\right).$

It follows from ${\Vert v_0\Vert }_2=1$ that $\underset{a\nearrow a^{\text{*}}}{\text{lim}}K\left(x_a\right)=1$ . Since the origin is a global maximum point of $v_a$ , it is also a global maximum point of $v_0$ . This indicates that $y_0=0$ . Hence we get

$v_a\left(x\right)\to \frac{1}{\sqrt{a^{\text{*}}}}Q\left(\left|x\right|\right)\text{in}{H}^1\left({ℝ}^2\right)\text{as}a\nearrow a^{\text{*}}.$

This convergence is independent of the choice of subsequences and holds true for the whole sequence as well.

4). Using the De Giorgi-Nash-Moser theory (cf. [25]), we derive from (3.15) and (3.17) that

$\underset{x\in B_1\left(\xi \right)}{\max }{v}_a\le C{\left({\displaystyle {\int }_{B_2\left(\xi \right)}{v}_a^2\text{d}x}\right)}^{\frac{1}{2}}\text{for}\text{any}\xi \in {ℝ}^2,$

where $C>0$ is a constant independent of $a$ and $\xi$ . Together with (3.5) and (3.15), we get that $\left\{v_a\right\}$ is bounded uniformly in $L^{\infty }\left({ℝ}^2\right)$ and

$v_a\left(x\right)\to 0\text{as}\left|x\right|\to \infty \text{uniformly}\text{in}a\nearrow a^{\text{*}},$ (3.19)

Combining (3.19) with (3.17) then yields that there exists a large constant $R>0$ such that

$-\text{Δ}{v}_a+\frac{4}{9}{v}_a\le 0\text{for}\left|x\right|\ge R\text{uniformly}\text{in}a\nearrow a^{\text{*}}.$ (3.20)

By applying the comparison principle to (3.20), we can conclude that there exists a positive constant $C>0$ such that

$\left|v_a\left(x\right)\right|\le C{\text{e}}^{-\frac{2\left|x\right|}{3}}\text{for}\left|x\right|\ge R\text{uniformly}\text{in}a\nearrow a^{\text{*}}.$ (3.21)

Thus we complete the proof of Lemma 3.1.

Now we prove the refined limiting behavior of positive minimizers of $e\left(a\right)$ in $X$ as $a\nearrow a^{\text{*}}$ .

Proof of Theorem 1.2. Using the exponential decays (1.10) and (3.6), we get that for any $ϵ>0$ , there exists a constant $R>0$ such that

$2{\displaystyle {\int }_{B_R^c}{\left|x\right|}^2{v}_a^2\left(x\right)\text{d}x}+2{\displaystyle {\int }_{B_R^c}{\left|x\right|}^2\frac{Q^2\left(x\right)}{a^{\text{*}}}\text{d}x}<\frac{ϵ}{2}.$

Combining this with (3.5) we obtain that

$\begin{array}{l}{\displaystyle {\int }_{{ℝ}^2}{\left|x\right|}^2{\left(v_a\left(x\right)-\frac{Q\left(x\right)}{\sqrt{a^{\text{*}}}}\right)}^2\text{d}x}\\ ={\displaystyle {\int }_{B_R}{\left|x\right|}^2{\left(v_a\left(x\right)-\frac{Q\left(x\right)}{\sqrt{a^{\text{*}}}}\right)}^2\text{d}x}+{\displaystyle {\int }_{B_R^c}{\left|x\right|}^2{\left(v_a\left(x\right)-\frac{Q\left(x\right)}{\sqrt{a^{\text{*}}}}\right)}^2\text{d}x}\\ \le {\left|R\right|}^2{\displaystyle {\int }_{B_R}{\left(v_a\left(x\right)-\frac{Q\left(x\right)}{\sqrt{a^{\text{*}}}}\right)}^2\text{d}x}+2{\displaystyle {\int }_{B_R^c}{\left|x\right|}^2{v}_a^2\left(x\right)\text{d}x}+2{\displaystyle {\int }_{B_R^c}{\left|x\right|}^2\frac{Q^2\left(x\right)}{a^{\text{*}}}\text{d}x}\\ <ϵ\text{as}a\nearrow a^{\text{*}}.\end{array}$