Limit Property of an L²-Normalized Solution for an L²-Subcritical Kirchhoff-Type Equation with a Variable Exponent

Abstract

This paper is concerned with the following $L^2$-subcritical Kirchhoff-type equation $-\left(a+b{\left({\int }_{{\mathbb{R}}^2}{|\nabla u|}^{2}dx\right)}^s\right)\Delta u+V\left(x\right)u=\mu u+\beta {\left|u\right|}^{2}u,x\in {\mathbb{R}}^2$, with ${\int }_{{\mathbb{R}}^2}{\left|u\right|}^{2}dx=1$. We give a detailed analysis of the limit property of the $L^2$-normalized solution when exponent s tends toward 0 from the right (i.e., $s\searrow 0$). Our research extends previous works, in which the authors have displayed the limit behavior of $L^2$-normalized solutions when $s=1$ as $a\searrow 0$ or $b\searrow 0$.

1. Introduction

In this paper, we study the following Kirchhoff-type equation with ${\int }_{{\mathbb{R}}^2}{\left|u\right|}^{2}dx=1$:

$$-\left(a+b{\left({\int }_{{\mathbb{R}}^2}{|\nabla u|}^{2}dx\right)}^s\right)\Delta u+V\left(x\right)u=\mu u+\beta {\left|u\right|}^{2}u,x\in {\mathbb{R}}^2,$$

(1)

where $a,b>0$ are constants, exponent $s>0$, parameter $\beta >0$, and $\mu$ denotes a Lagrange multiplier. The above ${\left({\int }_{{\mathbb{R}}^2}{|\nabla u|}^{2}dx\right)}^s\Delta u$ in (1) is called a variable nonlocal term, and ${\beta \left|u\right|}^{2}u$ is an $L^2$-subcritical term.

In recent decades, many different types of elliptic equations involved in (1) have been studied extensively through variational methods and functional analysis techniques. For instance, when $s=0$ and $\beta >0$ in (1), Equation (1) comes from the well-known Bose-Einstein condensates [1,2,3], which can be described by a Gross–Pitaevskii (G-P) energy functional [4,5]. Especially for $V\left(x\right)$ in ring-shaped, multi-well, sinusoidal, and periodic forms, the authors in [6,7,8,9] presented the existence, non-existence, and limit properties of $L^2$-normalized solutions when $\beta$ tends to a threshold value ${\beta }^{*}$. Furthermore, if $V\left(x\right)$ is a logarithmic or homogeneous function [10,11], the local uniqueness was also analyzed as $\beta$ approaches ${\beta }^{*}$.

For $s=1$, Equation (1) is regarded as a classic Kirchhoff-type equation. When $V\left(x\right)=0$, Ye [12,13] gave a quantitative classification of existence and nonexistence for the $L^2$-normalized solution. Meng and Zeng [14] have displayed a detailed limit behavior of the $L^2$-normalized solution when $V\left(x\right)$ is a periodic function. Somewhat similarly, there are many works [15,16,17,18] involved in the existence, non-existence, and limit properties of $L^2$-normalized solutions when (1) possesses an $L^2$-subcritical term. If $V\left(x\right)$ is a polynomial potential, Tang, Zhou, and their co-workers in [15,19] obtained some results on the refined limit behavior of $L^2$-normalized solutions as $b\searrow 0$ or as $a\searrow 0$.

Inspired by previous works, in the present paper, we shall study $L^2$-normalized solutions for the $L^2$-subcritical Kirchhoff-type Equation (1) with a variable nonlocal term. A well-known result shows that the $L^2$-normalized solution of (1) can be solved by dealing with the following constrained minimization problem

$$e(s,\beta ):=\underset{u\in \mathcal{K}}{inf}{E}_{s,\beta }\left(u\right),$$

(2)

where $E_{s,\beta }\left(u\right)$ fulfills

$$E_{s,\beta }\left(u\right)=a{\int }_{{\mathbb{R}}^2}{|\nabla u|}^{2}dx+{\displaystyle \frac{b}{s+1}}{\left({\int }_{{\mathbb{R}}^2}{|\nabla u|}^{2}dx\right)}^{s+1}+{\int }_{{\mathbb{R}}^2}V\left(x\right)u^{2}dx-{\displaystyle \frac{\beta }{2}}{\int }_{{\mathbb{R}}^2}{\left|u\right|}^{4}dx.$$

(3)

The above $\mathcal{K}$ in (2) satisfies

$$\mathcal{K}:=\left\{u\in \mathcal{H}:{\int }_{{\mathbb{R}}^2}{\left|u\right|}^2=1\right\}$$

(4)

and $\mathcal{H}$ such as

$$\mathcal{H}:=\left\{u\in H^1\left({\mathbb{R}}^2\right):{\int }_{{\mathbb{R}}^2}V\left(x\right){\left|u\right|}^{2}dx<\infty \right\}$$

with norm ${\parallel u\parallel }_{\mathcal{H}}:={\left({\int }_{{\mathbb{R}}^2}{|\nabla u|}^{2}dx+{\int }_{{\mathbb{R}}^2}\left(1+{V\left(x\right)\left|u\right|}^2\right)dx\right)}^{{\displaystyle \frac{1}{2}}}$.

Because many physicists are often interested in $L^2$-normalized solutions, we study minimization problem (1.2) with ${\int }_{{\mathbb{R}}^n}{\left|u\right|}^2=1$. In terms of physics, if ${\int }_{{\mathbb{R}}^n}{\left|u\right|}^2=c$, the constant c denotes the number of particles, mass, density, etc. The readers are encouraged to refer to the papers [12,13,20] for more details on physical aspects. Furthermore, we mention that, for ${\int }_{{\mathbb{R}}^2}{\left|u\right|}^2=c$, the techniques to deal with these problems are essentially the same. Without loss of generality, we only consider ${\int }_{{\mathbb{R}}^n}{\left|u\right|}^2=1$.

For convenience, the structure of our paper is arranged as follows. The main three theorems of this paper are introduced in Section 2. In Section 3, we shall give a detailed analysis of the limit properties of least energy $e(s,\beta )$ and minimizer $u_{s,\beta }$ as $s\searrow 0$ when $0<\beta <{\beta }^{*}$. In Section 4, for any $\beta >{\beta }^{*}$ and positive sequence $\left\{s_k\right\}$ with $s_k\searrow 0$ as $k\to \infty$, we are devoted not only to establishing a refined energy estimation of $e(s_k,\beta )$, but also analyzing the concrete limit behavior of constraint minimizer $u_{s_k,\beta }$.

2. Main Results

To obtain the detailed limit behavior of the $L^2$-normalized solution, some results on the existence and non-existence of constraint minimizers for (2) are necessary. For this reason, the potential $V\left(x\right)$ is restricted to satisfying

$$V\left(x\right)\in L_{loc}^{\infty }\left({\mathbb{R}}^2\right)\mathrm{and}\underset{\left|x\right|\to \infty }{lim}V\left(x\right)=+\infty \left(V_1\right).$$

Next, we introduce a nonlinear scalar field equation [21],

$$-\Delta u+u-u^3=0,x\in {\mathbb{R}}^2,u\in H^1\left({\mathbb{R}}^2\right),$$

(5)

where (5) admits a unique (up to translation) positive radially symmetric solution $Q\in H^1\left({\mathbb{R}}^2\right)$. At the same time, one can obtain from (5) that

$${\parallel \nabla Q\parallel }_{L^2}^2={\parallel Q\parallel }_{L^2}^2={\displaystyle \frac{1}{2}}{\parallel Q\parallel }_{L^4}^4.$$

(6)

In view of [22] (Proposition 4.1), the $Q\left(x\right)$ satisfies

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

(7)

Furthermore, the following Gagliardo–Nirenberg (G-N) inequality [23] is necessary

$${\int }_{{\mathbb{R}}^2}{\left|u\right|}^{4}dx\le {\displaystyle \frac{2}{{\parallel Q\parallel }_{L^2}^2}}{\int }_{{\mathbb{R}}^2}{|\nabla u|}^{2}dx{\int }_{{\mathbb{R}}^2}{\left|u\right|}^{2}dx,u\in H^1\left({\mathbb{R}}^2\right),$$

(8)

where $Q\left(x\right)$ is given by (5).

Based on the assumption $\left(V_1\right)$, we recall from [20,24,25] that the following existence and non-existence results of the constraint minimizer for $e(s,\beta )$ hold.

Theorem 1. 

Assume that $\left(V_1\right)$ holds and $s=0$ in (1); then, there exists a critical value ${\beta }^{*}:=(a+b){\parallel Q\parallel }_{L^2}^2$ such that $e(0,\beta )$ has at least one minimizer if $0<\beta <{\beta }^{*}$, and $e(0,\beta )$ has no minimizer for $\beta \ge {\beta }^{*}$. Furthermore, for any $s>0$ and $\beta >0$, $e(s,\beta )$ has at least one minimizer.

Notice that $e(0,\beta )$ is a constrained minimization problem related to the classic G-P functional, and the conclusion of Theorem 1 comes from [20] (Theorem 1). Also for $s>0$, $e(s,\beta )$ is associated with a Kirchhoff-type equation, and the related results can be obtained from [24] (Theorem 1.2) or [25] (Theorem 1). We, however, state the results of Theorem 1 here for the reader’s convenience, and the detailed proof process of Theorem 1 can be found in [20,24,25].

The above Theorem 1 gives the fact that $e(0,\beta )$ has no minimizer for $\beta \ge {\beta }^{*}$, but for any $s>0$ and $\beta >0$, $e(s,\beta )$ has at least one minimizer. An interesting question is what happens on the constraint minimizer of $e(s,\beta )$ when s tends to 0 from the right. In truth, a priori analysis shows that the constant

$${\beta }^{*}=(a+b){\parallel Q\parallel }_{L^2}^2$$

(9)

is a critical threshold, which is the criterion used to judge the constraint minimizer of $e(s,\beta )$ blowing up or converging as $s\searrow 0$. According to Theorem 1, we set $u_{s,\beta }$ be a constraint minimizer of $e(s,\beta )$ for any $s>0$; then we always say that $|u_{s,\beta }|$ is also a nonnegative minimizer of $e(s,\beta )$ by the fact that $E\left(u_{s,\beta }\right)\ge E\left(\right|u_{s,\beta }\left|\right)$.

Next, we firstly establish some results on the limit properties of the least energy $e(s,\beta )$ and nonnegative minimizer $u_{s,\beta }$ as $s\searrow 0$ when $0<\beta <{\beta }^{*}$.

Theorem 2. 

Assume that $\left(V_1\right)$ holds and that for any $s>0$, $0<\beta <{\beta }^{*}$, let $u_{s,\beta }$ be a nonnegative minimizer of $e(s,\beta )$. Then, we have

$$\underset{s\searrow 0}{lim}e(s,\beta )=e(0,\beta ).$$

(10)

Furthermore, for any $s>0$, it admits a $u_0\in \mathcal{H}$ such that as $s\searrow 0$

$$u_{s,\beta }\to u_0\mathrm{strongly}\mathrm{in}\mathcal{H},$$

(11)

where $u_0$ is a nonnegative minimizer of $e(0,\beta )$.

Theorem 1 shows that for any $s>0$ and $\beta >{\beta }^{*}$, $e(s,{\beta }^{*})$ has at least one minimizer. However, for $s=0$ and $\beta >{\beta }^{*}$, $e(s,{\beta }^{*})$ has no minimizer. We next are concerned with the limit property of the minimizer as $s\searrow 0$ when $\beta >{\beta }^{*}$. For this goal, a more appropriate assumption about the potential $V\left(x\right)$ is given as follows:

$$V\left(x\right)\in C_{loc}^{\alpha }\left({\mathbb{R}}^2\right)\cap C^1\left({\mathbb{R}}^2\right),\alpha \in (0,1)\mathrm{and}\underset{{\mathbb{R}}^2}{min}V\left(x\right)=0\left(V_2\right).$$

Further, we denote the class of minima for $V\left(x\right)$ by

$$\mathcal{Z}=\{x_i:V\left(x_i\right)=0,i=1,2,3\cdots \}.$$

(12)

Theorem 3. 

Suppose that $\left(V_1\right)$ and $\left(V_2\right)$ hold. For any $\beta >{\beta }^{*}$ and positive sequence $\left\{s_k\right\}$ with $s_k\searrow 0$ as $k\to \infty$, set $u_{s_k,\beta }$ to be the nonnegative minimizer of $e(s_k,\beta )$, and then $u_{s_k,\beta }$ exists as a unique local maximum point $z_{s_k,\beta }\in {\mathbb{R}}^2$ such that $\left\{s_k\right\}$ has a subsequence (still denoted by $\left\{s_k\right\}$) satisfying, as $k\to \infty$,

$$\underset{k\to \infty }{lim}{z}_{s_k,\beta }=x_0\in \mathcal{Z}\mathrm{and}V\left(x_0\right)=0$$

(13)

and

$${ϵ}_{s_k,\beta }{u}_{s_k,\beta }({ϵ}_{s_k,\beta }x+z_{s_k,\beta })\to {\displaystyle \frac{Q\left(\right|x\left|\right)}{{\parallel Q\parallel }_{L^2}}}\mathit{strongly}\mathrm{in}{H}^1\left({\mathbb{R}}^2\right),$$

where Q is given by (5) and ${ϵ}_{s_k,\beta }:={\left({\int }_{{\mathbb{R}}^2}{|\nabla u_{s_k,\beta }|}^{2}dx\right)}^{-{\displaystyle \frac{1}{2}}}$. Moreover, the ${ϵ}_{s_k,\beta }$ and $e(s_k,\beta )$ satisfy $k\to \infty$

$${ϵ}_{s_k,\beta }^{-2s_k}\to {\displaystyle \frac{b\beta +a(\beta -{\beta }^{*})}{b{\beta }^{*}}}$$

(14)

and

$$e(s_k,\beta )\approx -{\displaystyle \frac{s_{k}b}{s_k+1}}{\left[{\displaystyle \frac{b\beta +a(\beta -{\beta }^{*})}{b{\beta }^{*}}}\right]}^{{\displaystyle \frac{s_k+1}{s_k}}},$$

(15)

where the function $f\left(s_k\right)\approx g\left(s_k\right)$ means $\underset{k\to \infty }{lim}{\displaystyle \frac{f\left(s_k\right)}{g\left(s_k\right)}}=1$.

3. Limit Properties of $\mathbf{0}<\mathit{\beta }<{\mathit{\beta }}^{*}$

As stated in Theorem 1, $e(s,\beta )$ has at least one minimizer for any $0<\beta <{\beta }^{*}$ and $s\ge 0$. In the following section, we shall give the proof of Theorem 2, which is related to the limit properties of least energy $e(s,\beta )$ and nonnegative minimizer $u_{s,\beta }$ as $s\searrow 0$. Before this, we introduce a well-known compact embedding theorem [26] (Theorem 2.1) such that if $V\left(x\right)$ satisfies $\left(V_1\right)$, then

$$\mathcal{H}\hookrightarrow L^q\left({\mathbb{R}}^2\right)(2\le q<\infty )\mathrm{is}\mathrm{compact}.$$

(16)

Proof of Theorem 2. 

For $0<\beta <{\beta }^{*}$ and $s>0$, we define the constrained minimization problem without nonlinear terms such as

$$d\left(s\right):=\underset{u\in \mathcal{K}}{inf}{E}_s\left(u\right)$$

(17)

and

$$E_s\left(u\right)=a{\int }_{{\mathbb{R}}^2}{|\nabla u|}^{2}dx+{\displaystyle \frac{b}{s+1}}{\left({\int }_{{\mathbb{R}}^2}{|\nabla u|}^{2}dx\right)}^{s+1}+{\int }_{{\mathbb{R}}^2}V\left(x\right)u^{2}dx.$$

(18)

We first claim that (17) has at least one minimizer. In truth, it is easy to know from (18) that for any $s>0$, $E_s\left(u\right)$ is bounded from below on $\mathcal{K}$. Let $\left\{u_n\right\}$ be a minimizing sequence of $d\left(s\right)$; one can obtain from (17) and (18) that $\left\{u_n\right\}$ is bounded in $\mathcal{H}$. Furthermore, the (16) yields the fact that there exists a $u_s\in \mathcal{K}$ and $\left\{u_n\right\}$ has a subsequence (still denoted by $\left\{u_n\right\}$) such that

$$u_n\rightharpoonup u_s\mathrm{weakly}\mathrm{in}\mathcal{K},u_n\to u_s\mathrm{strongly}\mathrm{in}{L}^q(2\le q<\infty ),$$

which together with

$$\underset{n\to \infty }{lim\; inf}{\int }_{{\mathbb{R}}^2}|\nabla u_n{|}^{2}dx\ge {\int }_{{\mathbb{R}}^2}{|\nabla u_s|}^{2}dx$$

gives

$$d\left(s\right)=\underset{n\to \infty }{lim\; inf}{E}_s\left(u_n\right)\ge E_s\left(u_s\right)\ge d\left(s\right).$$

(19)

Therefore, we conclude that $u_s$ is a minimizer of $d\left(s\right)$ for any $s>0$.

For any $s>0$, set $u_s$ be a minimizer of $d\left(s\right)$, then one can obtain from (17) that

$$d\left(s\right)=E_s\left(u_s\right).$$

(20)

In fact, from a fixed function $\eta \left(x\right)\in C_0^{\infty }\left({\mathbb{R}}^2\right)\cap \mathcal{K}$, we obtain from (18) and (20) that there exists a constant $M>0$ satisfying

$$0\le \underset{s\searrow 0}{lim}d\left(s\right)=\underset{s\searrow 0}{lim}{E}_s\left(u_s\right)\le \underset{s\searrow 0}{lim}{E}_s\left(\eta \right)\le M.$$

(21)

For $0<\beta <{\beta }^{*}$, applying the definitions of $d\left(s\right)$ and $e(s,\beta )$, we then deduce from (20) and (21) that as $s\searrow 0$,

$$e(s,\beta )\le E_{s,\beta }\left(u_s\right)\le E_s\left(u_s\right)=d\left(s\right)\le E_s\left(\eta \right)\le M.$$

(22)

In view of Theorem 1, set $u_{s,\beta }$ be a nonnegative minimizer of $e(s,\beta )$; then, we have

$$\begin{array}{cc}\hfill e(s,\beta )=E_{s,\beta }\left(u_{s,\beta }\right)=& a{\int }_{{\mathbb{R}}^2}{|\nabla u_{s,\beta }|}^{2}dx+{\displaystyle \frac{b}{s+1}}{\left({\int }_{{\mathbb{R}}^2}{|\nabla u_{s,\beta }|}^{2}dx\right)}^{s+1}\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^2}V\left(x\right)u_{s,\beta }^{2}dx-{\displaystyle \frac{\beta }{2}}{\int }_{{\mathbb{R}}^2}{\left|u_{s,\beta }\right|}^{4}dx.\hfill \end{array}$$

(23)

Next, we claim that there exists a constant $D>0$ such that

$$\underset{s\searrow 0}{lim}{\int }_{{\mathbb{R}}^2}{|\nabla u_{s,\beta }|}^{2}dx\le D.$$

(24)

If (24) is false, that is, ${\int }_{{\mathbb{R}}^2}{|\nabla u_{s,\beta }|}^{2}dx\to \infty$ as $s\searrow 0$. Since $0<\beta <{\beta }^{*}$, applying the G-N inequality (8), we then deduce from (9) and (23) that there exists an arbitrarily large constant $\mathcal{P}>0$ satisfying

$$\begin{array}{cc}\hfill e(s,\beta )=E_{s,\beta }\left(u_{s,\beta }\right)=& a{\int }_{{\mathbb{R}}^2}{|\nabla u_{s,\beta }|}^{2}dx+{\displaystyle \frac{b}{s+1}}{\left({\int }_{{\mathbb{R}}^2}{|\nabla u_{s,\beta }|}^{2}dx\right)}^{s+1}\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^2}V\left(x\right)u_{s,\beta }^{2}dx-{\displaystyle \frac{\beta (a+b)}{{\beta }^{*}}}{\int }_{{\mathbb{R}}^2}{|\nabla u_{s,\beta }|}^{2}dx\hfill \\ \hfill \ge & (a+b)\left({\displaystyle \frac{{\beta }^{*}-\beta }{{\beta }^{*}}}\right){\int }_{{\mathbb{R}}^2}{|\nabla u_{s,\beta }|}^{2}dx\ge \mathcal{P},\hfill \end{array}$$

(25)

which is a contradiction with (22). Therefore, (24) holds, and then $\left\{u_{s,\beta }\right\}$ is bounded in $\mathcal{H}$.

Using again (16), it implies that there exists a $u_0\in \mathcal{K}$, and $\left\{u_{s,\beta }\right\}$ has a subsequence (still denoted by $\left\{u_{s,\beta }\right\}$) such that as $s\searrow 0$,

$$u_{s,\beta }\rightharpoonup u_0\mathrm{weakly}\mathrm{in}\mathcal{K},u_{s,\beta }\to u_0\mathrm{strongly}\mathrm{in}{L}^q(2\le q<\infty ).$$

(26)

Combining (23) and (26), one can obtain

$$\underset{s\searrow 0}{lim}e(s,\beta )=\underset{s\searrow 0}{lim}{E}_{s,\beta }\left(u_{s,\beta }\right)\ge \underset{s\searrow 0}{lim\; inf}{E}_{s,\beta }\left(u_{s,\beta }\right)\ge E_{0,\beta }\left(u_0\right)\ge e(0,\beta ).$$

(27)

Additionally, by the definition of $e(0,\beta )$ and (22), for any $ϵ>0$, there exists a $u_{ϵ}\in \mathcal{K}$ satisfying

$$E_{0,\beta }\left(u_{ϵ}\right)\le e(0,\beta )+ϵ\le M+ϵ.$$

(28)

Similar to the proof of (24), we deduce from (28) that $u_{ϵ}$ is bounded in $\mathcal{H}$. By this fact, we have

$$\underset{s\searrow 0}{lim}\left[{\displaystyle \frac{b}{s+1}}{\left({\int }_{{\mathbb{R}}^2}{|\nabla u_{ϵ}|}^{2}dx\right)}^{s+1}-b{\int }_{{\mathbb{R}}^2}{|\nabla u_{ϵ}|}^{2}dx\right]=0,$$

(29)

which, together with (28) and (29), gives that as $s\searrow 0$,

$$\begin{array}{cc}\hfill e(s,\beta )\le E_{s,\beta }\left(u_{ϵ}\right)=& E_{0,\beta }\left(u_{ϵ}\right)+{\displaystyle \frac{b}{s+1}}{\left({\int }_{{\mathbb{R}}^2}{|\nabla u_{ϵ}|}^{2}dx\right)}^{s+1}\hfill \\ \hfill & -b{\int }_{{\mathbb{R}}^2}{|\nabla u_{ϵ}|}^{2}dx\le e(0,\beta )+ϵ.\hfill \end{array}$$

(30)

Combining with (27)–(30), we have

$$e(0,\beta )\le E_{0,\beta }\left(u_0\right)\le \underset{s\searrow 0}{lim}e(s,\beta )\le e(0,\beta )+ϵ.$$

(31)

Letting $ϵ\to 0$, we then conclude from (31) that $u_0$ is a minimizer of $e(0,\beta )$ and $u_{s,\beta }\to u_0$ strongly in $\mathcal{H}$ as $s\searrow 0$. We thereby complete the proof of Theorem 2. □

4. Limit Behavior Analysis of $\mathit{\beta }>{\mathit{\beta }}^{*}$

In this section, we are concerned with the limit property of minimizer as $s\searrow 0$ when $\beta >{\beta }^{*}$. Before this, we define a constrained minimization problem without potential such as

$$\overline{e}(s,\beta ):=\underset{u\in \mathcal{S}}{inf}{\overline{E}}_{s,\beta }\left(u\right),$$

(32)

where ${\overline{E}}_{s,\beta }\left(u\right)$ and S satisfy

$${\overline{E}}_{s,\beta }\left(u\right)=a{\int }_{{\mathbb{R}}^2}{|\nabla u|}^{2}dx+{\displaystyle \frac{b}{s+1}}{\left({\int }_{{\mathbb{R}}^2}{|\nabla u|}^{2}dx\right)}^{s+1}-{\displaystyle \frac{\beta }{2}}{\int }_{{\mathbb{R}}^2}{\left|u\right|}^{4}dx,$$

$$\mathcal{S}:=\left\{u\in H^1\left({\mathbb{R}}^2\right):{\int }_{{\mathbb{R}}^2}{\left|u\right|}^{2}dx=1\right\}.$$

In order to attain our goal, the following existence result of constrained minimizer for (32) is established for any $s>0$ when $\beta >{\beta }^{*}$. Note from [24] (Theorem 1.1) that for $s=0$, a similar result can also be found.

Lemma 1. 

For any $s>0$ and $\beta >{\beta }^{*}$, $\overline{e}(s,\beta )$ has at least one nonnegative minimizer.

Proof. 

We firstly claim that for any $s>0$ and $\beta >{\beta }^{*}$,

$$\overline{e}(s,\beta )<0.$$

(33)

In truth, we choose a test function such as

$$u_{\theta }\left(x\right)={\displaystyle \frac{\theta Q\left(\theta x\right)}{{\parallel Q\parallel }_{L^2}}}(\theta >0)\mathrm{and}Q\left(x\right)\mathrm{is}\mathrm{given}\mathrm{by}\left(8\right).$$

(34)

It is derived from (6), (34) that $u_{\theta }\in \mathcal{S}$ and

$${\int }_{{\mathbb{R}}^2}|\nabla u_{\theta }{|}^{2}dx={\theta }^2,{\left({\int }_{{\mathbb{R}}^2}{|\nabla u_{\theta }|}^{2}dx\right)}^{s+1}={\theta }^{2(s+1)},{\displaystyle \frac{\beta }{2}}{\int }_{{\mathbb{R}}^2}{\left|u_{\theta }\right|}^{4}dx={\displaystyle \frac{\beta }{{\parallel Q\parallel }_{L^2}^2}}{\theta }^2.$$

(35)

Hence, it follows from (9) and (35) that

$$\begin{array}{cc}\hfill {\overline{E}}_{s,\beta }\left(u_{\theta }\right)& =a{\theta }^2+{\displaystyle \frac{b}{s+1}}{\theta }^{2(s+1)}-{\displaystyle \frac{\beta }{{\parallel Q\parallel }_{L^2}^2}}{\theta }^2\hfill \\ \hfill & =a{\theta }^2+{\displaystyle \frac{b}{s+1}}{\theta }^{2(s+1)}-{\displaystyle \frac{(a+b)\beta }{{\beta }^{*}}}{\theta }^2.\hfill \end{array}$$

(36)

For $s>0$ and $\beta >{\beta }^{*}$, choosing $\theta ={\left({\displaystyle \frac{\beta }{{\beta }^{*}}}\right)}^{{\displaystyle \frac{1}{2s}}}$ and putting it into (36), we have

$$\overline{e}(s,\beta )\le {\overline{E}}_{s,\beta }\left(u_{\theta }\right)=-a{\left({\displaystyle \frac{\beta }{{\beta }^{*}}}\right)}^{{\displaystyle \frac{1}{s}}}\left({\displaystyle \frac{\beta -{\beta }^{*}}{\beta }}\right)-{\displaystyle \frac{sb}{s+1}}{\left({\displaystyle \frac{\beta }{{\beta }^{*}}}\right)}^{{\displaystyle \frac{s+1}{s}}}<0.$$

(37)

Therefore, (33) holds.

By applying (8), one obtains

$${\overline{E}}_{s,\beta }\left(u\right)\ge a{\int }_{{\mathbb{R}}^2}{|\nabla u|}^{2}dx+{\displaystyle \frac{b}{s+1}}{\left({\int }_{{\mathbb{R}}^2}{|\nabla u|}^{2}dx\right)}^{s+1}-{\displaystyle \frac{(a+b)\beta }{{\beta }^{*}}}{\int }_{{\mathbb{R}}^2}{|\nabla u|}^{2}dx$$

(38)

which yields that for any fixed $s>0$, ${\overline{E}}_{s,\beta }\left(u\right)$ is bounded from below on $\mathcal{S}$. According to (33) and (38), and choosing $\left\{{\overline{u}}_n\right\}$ as a minimizing sequence of $\overline{e}(s,\beta )$, it then follows from (33) and (38) that $\left\{{\overline{u}}_n\right\}$ is bounded in $H^1\left({\mathbb{R}}^2\right)$. Recall from [27] (Appendix A.III) and [24] (Lemma 2.3), we see that there exists a nonnegative and non-increasing sequence $\left\{{\overline{u}}_n^{*}\right\}\subseteq \mathcal{S}\cap H_r^1\left({\mathbb{R}}^2\right)$ such that $\left\{{\overline{u}}_n^{*}\right\}$ is also a minimizing sequence of $\overline{e}(s,\beta )$. At the same time, $\left\{{\overline{u}}_n^{*}\right\}$ is bounded in $H_r^1\left({\mathbb{R}}^2\right)$. Applying [28] (Proposition 1.7.1), there exists a subsequence of $\left\{{\overline{u}}_n^{*}\right\}$ (still denoted by $\left\{{\overline{u}}_n^{*}\right\}$) and a function ${\overline{u}}_0\in H_r^1\left({\mathbb{R}}^2\right)$ satisfying

$${\overline{u}}_n^{*}\rightharpoonup {\overline{u}}_0\mathrm{weakly}\mathrm{in}{H}_r^1\left({\mathbb{R}}^2\right),{\overline{u}}_n^{*}\to {\overline{u}}_0\mathrm{strongly}\mathrm{in}{L}^q\left({\mathbb{R}}^2\right)(2<q<\infty ).$$

(39)

In the following, we prove that ${\int }_{{\mathbb{R}}^2}{\left|{\overline{u}}_0\right|}^{2}dx=1$. Firstly, one can derive from (39) that ${\int }_{{\mathbb{R}}^2}{\left|{\overline{u}}_0\right|}^{2}dx\ne 0$. If not, we have, as $n\to \infty$,

$${\int }_{{\mathbb{R}}^2}{\left|{\overline{u}}_n^{*}\right|}^{4}dx\to 0,$$

which, together with (32) and (33), yields that

$$0>\overline{e}(s,\beta )=\underset{n\to \infty }{lim}\left[a{\int }_{{\mathbb{R}}^2}{|\nabla {\overline{u}}_n^{*}|}^{2}dx+{\displaystyle \frac{b}{s+1}}{\left({\int }_{{\mathbb{R}}^2}{|\nabla {\overline{u}}_n^{*}|}^{2}dx\right)}^{s+1}\right]\ge 0.$$

(40)

However, this is a contradiction, and hence, ${\int }_{{\mathbb{R}}^2}{\left|{\overline{u}}_0\right|}^{2}dx\ne 0$. Using the fact, we obtain from (33) and (39) that

$$0>\overline{e}(s,\beta )=\underset{n\to \infty }{lim}{\overline{E}}_{s,\beta }\left({\overline{u}}_n^{*}\right)\ge {\overline{E}}_{s,\beta }\left({\overline{u}}_0\right).$$

(41)

Denote $l:={\int }_{{\mathbb{R}}^2}{\left|{\overline{u}}_0\right|}^{2}dx$; then the above results show that $l\in (0,1]$. Setting ${\overline{u}}_l\left(x\right):={\overline{u}}_0(l^{{\displaystyle \frac{1}{2}}}x)\in \mathcal{S}$, we obtain from (32) and (41) that

$$\begin{array}{cc}\hfill \overline{e}(s,\beta )& \le {\overline{E}}_{s,\beta }\left({\overline{u}}_l\right)\hfill \\ \hfill & =a{\int }_{{\mathbb{R}}^2}|\nabla {\overline{u}}_0{|}^{2}dx+{\displaystyle \frac{b}{s+1}}{\left({\int }_{{\mathbb{R}}^2}{|\nabla {\overline{u}}_0|}^{2}dx\right)}^{s+1}-{\displaystyle \frac{\beta }{2l}}{\int }_{{\mathbb{R}}^2}{\left|{\overline{u}}_0\right|}^{4}dx\hfill \\ \hfill & ={\displaystyle \frac{1}{l}}\left[al{\int }_{{\mathbb{R}}^2}|\nabla {\overline{u}}_0{|}^{2}dx+{\displaystyle \frac{bl}{s+1}}{\left({\int }_{{\mathbb{R}}^2}{|\nabla {\overline{u}}_0|}^{2}dx\right)}^{s+1}-{\displaystyle \frac{\beta }{2}}{\int }_{{\mathbb{R}}^2}{\left|{\overline{u}}_0\right|}^{4}dx\right]\hfill \\ \hfill & \le {\displaystyle \frac{1}{l}}{\overline{E}}_{s,\beta }\left({\overline{u}}_0\right)\le {\displaystyle \frac{1}{l}}\overline{e}(s,\beta ),\hfill \end{array}$$

(42)

this yields from (33) that $l\ge 1$, that is, ${\int }_{{\mathbb{R}}^2}{\left|{\overline{u}}_0\right|}^{2}dx\ge 1$. Since $l={\int }_{{\mathbb{R}}^2}{\left|{\overline{u}}_0\right|}^{2}dx\in (0,1]$, one has ${\int }_{{\mathbb{R}}^2}{\left|{\overline{u}}_0\right|}^{2}dx=1$, which, together with (41) and (42), implies that ${\overline{u}}_0$ is a minimizer of $\overline{e}(s,\beta )$. By the definition of $\overline{e}(s,\beta )$, we see that $|{\overline{u}}_0|$ is a minimizer, too. Therefore, we always say that $\overline{e}(s,\beta )$ has at least one nonnegative minimizer. □

Lemma 2. 

For any fixed $\beta >{\beta }^{*}$, set ${\overline{u}}_{s,\beta }$ as a nonnegative minimizer of $\overline{e}(s,\beta )$; then, $\overline{e}(s,\beta )$ satisfies, as $s\searrow 0$,

$$\overline{e}(s,\beta )\approx -{\displaystyle \frac{sb}{s+1}}{\left[{\displaystyle \frac{b\beta +a(\beta -{\beta }^{*})}{b{\beta }^{*}}}\right]}^{{\displaystyle \frac{s+1}{s}}},$$

(43)

where $f\left(s\right)\approx g\left(s\right)$ means $\underset{s\to 0}{lim}{\displaystyle \frac{f\left(s\right)}{g\left(s\right)}}=1$. As $s\searrow 0$, the minimizer ${\overline{u}}_{s,\beta }$ behaves like

$${\overline{u}}_{s,\beta }={\displaystyle \frac{\theta Q\left(\theta x\right)}{{\parallel Q\parallel }_{L^2}}}\mathrm{and}\theta ={\left[{\displaystyle \frac{b\beta +a(\beta -{\beta }^{*})}{b{\beta }^{*}}}\right]}^{{\displaystyle \frac{1}{2s}}},$$

(44)

where Q satisfies Equation (5).

Proof. 

Choosing the same test function as (34), similar to the estimations (35) and (36), we have

$${\overline{E}}_{s,\beta }\left(u_{\theta }\right)=a{\theta }^2+{\displaystyle \frac{b}{s+1}}{\theta }^{2(s+1)}-(a+b){\displaystyle \frac{\beta }{{\beta }^{*}}}{\theta }^2.$$

(45)

For any fixed $\beta >{\beta }^{*}$, we choose ${\theta }^2={\left[{\displaystyle \frac{b\beta +a(\beta -{\beta }^{*})}{b{\beta }^{*}}}\right]}^{{\displaystyle \frac{1}{s}}}\to \infty$ as $s\searrow 0$. Taking $\theta$ into (45), it then yields that as $s\searrow 0$

$$\overline{e}(s,\beta )\le {\overline{E}}_{s,\beta }\left(u_{\theta }\right)\le -{\displaystyle \frac{sb}{s+1}}{\left[{\displaystyle \frac{b\beta +a(\beta -{\beta }^{*})}{b{\beta }^{*}}}\right]}^{{\displaystyle \frac{s+1}{s}}}.$$

(46)

We next give the lower energy estimation of $\overline{e}(s,\beta )$ when $\beta >{\beta }^{*}$ as $s\searrow 0$. In view of Lemma 1, we assume that ${\overline{u}}_{s,\beta }\in \mathcal{S}$ is a nonnegative minimizer of $\overline{e}(s,\beta )$. Applying the G-N inequality (8) and (9), it is deduced that

$$\begin{array}{cc}\hfill \overline{e}(s,\beta )& =a{\int }_{{\mathbb{R}}^2}|\nabla {\overline{u}}_{s,\beta }{|}^{2}dx+{\displaystyle \frac{b}{s+1}}{\left({\int }_{{\mathbb{R}}^2}{|\nabla {\overline{u}}_{s,\beta }|}^{2}dx\right)}^{s+1}-{\displaystyle \frac{\beta }{2}}{\int }_{{\mathbb{R}}^2}{\left|{\overline{u}}_{s,\beta }\right|}^{4}dx\hfill \\ \hfill & \ge \left[a-(a+b){\displaystyle \frac{\beta }{{\beta }^{*}}}\right]{\int }_{{\mathbb{R}}^2}{|\nabla {\overline{u}}_{s,\beta }|}^{2}dx+{\displaystyle \frac{b}{s+1}}{\left({\int }_{{\mathbb{R}}^2}{|\nabla {\overline{u}}_{s,\beta }|}^{2}dx\right)}^{s+1}.\hfill \end{array}$$

(47)

Defining a function $f\left(t\right):=\left[a-(a+b){\displaystyle \frac{\beta }{{\beta }^{*}}}\right]t+{\displaystyle \frac{b}{s+1}}{t}^{s+1}$ and $t={\int }_{{\mathbb{R}}^2}{|\nabla {\overline{u}}_{s,\beta }|}^{2}dx\in (0,+\infty )$, a simple calculation shows that $f\left(t\right)$ attains its unique minimum at

$$t=t_s={\left[{\displaystyle \frac{b\beta +a(\beta -{\beta }^{*})}{b{\beta }^{*}}}\right]}^{{\displaystyle \frac{1}{s}}}.$$

(48)

It then follows from (47) that as $s\searrow 0$

$$\overline{e}(s,\beta )=f\left(t\right)\ge f\left(t_s\right)\ge -{\displaystyle \frac{sb}{s+1}}{\left[{\displaystyle \frac{b\beta +a(\beta -{\beta }^{*})}{b{\beta }^{*}}}\right]}^{{\displaystyle \frac{s+1}{s}}}.$$

(49)

Moreover, the procedures of (46) and (49) yield that $\overline{e}(s,\beta )$ attains its minimum at ${\overline{u}}_{s,\beta }={\displaystyle \frac{\theta Q\left(\theta x\right)}{{\parallel Q\parallel }_{L^2}}}$ as $s\searrow 0$, where $\theta ={\left[{\displaystyle \frac{b\beta +a(\beta -{\beta }^{*})}{b{\beta }^{*}}}\right]}^{{\displaystyle \frac{1}{2s}}}$. □

To obtain more detailed information about the minimizer and $e(s,\beta )$ as $s\searrow 0$, we always assume that $u_{s,\beta }$ is a nonnegative minimizer of $e(s,\beta )$. It then follows that $u_{s,\beta }$ satisfies the elliptic equation

$$-\left(a+b{\left({\int }_{{\mathbb{R}}^2}{|\nabla u_{s,\beta }|}^{2}dx\right)}^s\right)\Delta u_{s,\beta }+V\left(x\right)u_{s,\beta }={\mu }_{s,\beta }{u}_{s,\beta }+\beta {\left|u_{s,\beta }\right|}^2{u}_{s,\beta },x\in {\mathbb{R}}^2,$$

(50)

where ${\mu }_{s,\beta }\in \mathbb{R}$ denotes a Lagrange multiplier. Furthermore, we set

$${ϵ}_{s,\beta }:={\left({\int }_{{\mathbb{R}}^2}{|\nabla u_{s,\beta }|}^{2}dx\right)}^{-{\displaystyle \frac{1}{2}}}>0.$$

(51)

Lemma 3. 

For $\beta >{\beta }^{*}$, the ${ϵ}_{s,\beta }$, $e(s,\beta )$ and potential energy satisfy $s\searrow 0$

$${ϵ}_{s,\beta }\to 0,e(s,\beta )-\overline{e}(s,\beta )\to 0$$

(52)

and

$${\int }_{{\mathbb{R}}^2}V\left(x\right)u_{s,\beta }^{2}dx\to 0.$$

(53)

Moreover, as $s\searrow 0$,

$${ϵ}_{s,\beta }^{-2s}\to {\displaystyle \frac{b\beta +a(\beta -{\beta }^{*})}{b{\beta }^{*}}}.$$

(54)

Proof. 

If ${ϵ}_{s,\beta }\nrightarrow 0$, then $\left\{u_{s,\beta }\right\}$ is bounded in $\mathcal{H}$. Applying (16), there admits a $\widehat{u}\in \mathcal{K}$ and $\left\{u_{s,\beta }\right\}$ exists as a subsequence (still denoted by $\left\{u_{s,\beta }\right\}$) satisfying, as $s\searrow 0$,

$$u_{s,\beta }\rightharpoonup \widehat{u}\mathrm{weakly}\mathrm{in}\mathcal{K},u_{s,\beta }\to \widehat{u}\mathrm{strongly}\mathrm{in}{L}^q(2\le q<\infty ).$$

(55)

Similar to (27) and (31), it is deduced that

$$e(0,\beta )\le E_{0,\beta }\left(\widehat{u}\right)\le \underset{s\searrow 0}{lim}e(s,\beta )\le e(0,\beta ),$$

(56)

which yields that $\widehat{u}$ is a minimizer of $e(0,\beta )$. But this is impossible because Theorem 1 shows that $e(0,\beta )$ has no minimizer for $\beta >{\beta }^{*}$. Hence, we have ${ϵ}_{s,\beta }\to 0$ as $s\searrow 0$.

On the one hand, by the definitions of $e(s,\beta )$ and $\overline{e}(s,\beta )$, one directly derives that

$$e(s,\beta )-\overline{e}(s,\beta )\ge 0.$$

(57)

On the other hand, we turn to estimate an upper bound of $e(s,\beta )-\overline{e}(s,\beta )$ as $s\searrow 0$. Toward this goal, we choose a cut-off function $\phi \left(x\right)\in C_0^{\infty }\left({\mathbb{R}}^2\right)$ such as $0\le \phi \left(x\right)\le 1$, and $\phi \left(x\right)=1$ for $\left|x\right|\le 1$, $\phi \left(x\right)=0$ for $\left|x\right|\ge 2$, $|\nabla \phi (x\left)\right|\le 2$ for $x\in {\mathbb{R}}^2$. Define

$${\widehat{u}}_{s,\beta }:=A_{s,\beta }\phi (x-x_0){\overline{u}}_{s,\beta }(x-x_0),$$

(58)

where $x_0$ satisfies $V\left(x_0\right)=0$ and ${\overline{u}}_{s,\beta }$ is given by (44). The above $A_{s,\beta }$ is chosen so that ${\int }_{{\mathbb{R}}^2}{\left|{\widehat{u}}_{s,\beta }\right|}^{2}dx=1$. Applying (7), one can calculate from (58) that

$$1\le A_{s,\beta }^2\le 1+C{e}^{-2\theta }\mathrm{as}s\searrow 0.$$

(59)

Based on this fact, we then have

$${\int }_{{\mathbb{R}}^2}|\nabla {\widehat{u}}_{s,\beta }{|}^{2}dx\le {\int }_{{\mathbb{R}}^2}{|\nabla {\overline{u}}_{s,\beta }|}^{2}dx+C{e}^{-\theta }\mathrm{as}s\searrow 0,$$

(60)

$${\int }_{{\mathbb{R}}^2}|{\widehat{u}}_{s,\beta }{|}^{4}dx\ge {\int }_{{\mathbb{R}}^2}{\left|{\overline{u}}_{s,\beta }\right|}^{4}dx-C{e}^{-2\theta }\mathrm{as}s\searrow 0$$

(61)

and

$${\int }_{{\mathbb{R}}^2}V\left(x\right){\left|{\widehat{u}}_{s,\beta }\right|}^{2}dx=V\left(x_0\right)+o\left(1\right)=o\left(1\right),$$

(62)

where $o\left(1\right)\to 0$ as $s\searrow 0$. Equations (60)–(62) together with (57) yield that

$$\begin{array}{cc}\hfill 0\le e(s,\beta )-\overline{e}(s,\beta )\le & E_{s,\beta }\left({\widehat{u}}_{s,\beta }\right)-{\overline{E}}_{s,\beta }\left({\overline{u}}_{s,\beta }\right)+{\int }_{{\mathbb{R}}^2}V\left(x\right){\left|{\widehat{u}}_{s,\beta }\right|}^{2}dx\hfill \\ \hfill \le & C{e}^{-{\displaystyle \frac{1}{2}}\theta }+o\left(1\right)\to 0\mathrm{as}s\searrow 0\hfill \end{array}$$

which then completes the proof of (52). Furthermore, we obtain from the definitions of $e(s,\beta )$ and $\overline{e}(s,\beta )$ that

$${\int }_{{\mathbb{R}}^2}V\left(x\right)u_{s,\beta }^{2}dx=E_{s,\beta }\left(u_{s,\beta }\right)-{\overline{E}}_{s,\beta }\left(u_{s,\beta }\right)\le e(s,\beta )-\overline{e}(s,\beta )\to 0\mathrm{as}s\searrow 0,$$

and hence (53) holds. Furthermore, the estimation of (54) is deduced directly from Lemma 2 and (52) and (53). □

Lemma 4. 

For $\beta >{\beta }^{*}$ and any positive sequence $\left\{s_k\right\}$ with $s_k\searrow 0$ as $k\to \infty$, the nonnegative minimizer $u_{s_k,\beta }$ has at least one local maximum point $z_{s_k,\beta }$. Define an $L^2$-normalized function

$$w_{s_k,\beta }\left(x\right):={ϵ}_{s_k,\beta }{u}_{s_k,\beta }({ϵ}_{s_k,\beta }x+z_{s_k,\beta }),{ϵ}_{s_k,\beta }$$

(63)

as given by (51); then, there exists a finite ball $B_2\left(0\right)$ and a constant $\eta >0$ such that

$${\int }_{B_2\left(0\right)}{w}_{s_k,\beta }^2\left(x\right)dx\ge \eta >0.$$

(64)

Furthermore, $z_{s_k,\beta }$ admits a convergent subsequence (still denoted by $z_{s_k,\beta }$) such that

$$z_{s_k,\beta }\to z_0\mathrm{as}k\to \infty ,$$

where $z_0$ satisfies $V\left(z_0\right)=0$, that is, where $z_0$ is a global minimum point of $V\left(x\right)$.

Proof. 

Since $u_{s_k,\beta }$ is a nonnegative minimizer of $e(s_k,\beta )$, it satisfies (50). Using the results of Lemma 2 and (52), we have, as $k\to \infty$,

$$e(s_k,\beta )\approx -{\displaystyle \frac{s_{k}b}{s_k+1}}{\left[{\displaystyle \frac{b\beta +a(\beta -{\beta }^{*})}{b{\beta }^{*}}}\right]}^{{\displaystyle \frac{s_k+1}{s_k}}}$$

which then yields from (54) that, as $k\to \infty$,

$${ϵ}_{s_k,\beta }^{2(s_k+1)}e(s_k,\beta )\approx {\displaystyle \frac{s_{k}b}{s_k+1}}\to 0.$$

(65)

Define an $L^2$-normalized function

$${\overline{w}}_{s_k,\beta }\left(x\right):={ϵ}_{s_k,\beta }{u}_{s_k,\beta }\left({ϵ}_{s_k,\beta }x\right),{ϵ}_{s_k,\beta }\mathrm{is}\mathrm{given}\mathrm{by}\left(51\right).$$

(66)

One then obtains from (53) and (65) that, as $k\to \infty$,

$${ϵ}_{s_k,\beta }^{2(s_k+1)}e(s_k,\beta )=a{ϵ}_{s_k,\beta }^{2s_k}+{\displaystyle \frac{b}{s_k+1}}-{\displaystyle \frac{\beta }{2}}{ϵ}_{s_k,\beta }^{2s_k}{\int }_{{\mathbb{R}}^2}{\left|{\overline{w}}_{s_k,\beta }\right|}^{4}dx\to 0$$

which derives from (54) that

$${\int }_{{\mathbb{R}}^2}{\left|{\overline{w}}_{s_k,\beta }\right|}^{4}dx\to {\displaystyle \frac{2(a+b)}{{\beta }^{*}}}\mathrm{as}k\to \infty .$$

(67)

Because $u_{s_k,\beta }$ satisfies (50), by the definition of $e(s_k,\beta )$, it follows that ${\mu }_{s_k,\beta }$ fulfills

$${\mu }_{s_k,\beta }=e(s_k,\beta )+{\displaystyle \frac{s_{k}b}{s_k+1}}{\left({\int }_{{\mathbb{R}}^2}{|\nabla u_{s_k,\beta }|}^{2}dx\right)}^{s_k+1}-{\displaystyle \frac{\beta }{2}}{\int }_{{\mathbb{R}}^2}{\left|u_{s_k,\beta }\right|}^{4}dx,$$

which, together with (54), (65) and (67), yields that as $k\to \infty$,

$$\begin{array}{cc}\hfill {\mu }_{s_k,\beta }{ϵ}_{s_k,\beta }^{2(s_k+1)}=& {ϵ}_{s_k,\beta }^{2(s_k+1)}e(s_k,\beta )+{\displaystyle \frac{s_{k}b}{s_k+1}}\hfill \\ \hfill & -{\displaystyle \frac{\beta }{2}}{ϵ}_{s_k,\beta }^{2s_k}{\int }_{{\mathbb{R}}^2}{\left|{\overline{w}}_{s_k,\beta }\right|}^{4}dx\to -{\displaystyle \frac{b\beta (a+b)}{b\beta +a(\beta -{\beta }^{*})}}.\hfill \end{array}$$

(68)

Based on $\left(V_1\right)$, $\left(V_2\right)$ and (68), one can use the method of [29] (Lemma 3.5) to acquire that $u_{s_k,\beta }$ has at least one local maximum point by applying the standard regularity theory and comparison principle to elliptic Equation (50). Set $z_{s_k,\beta }$ as the local maximum point and define the $L^2$-normalized function $w_{s_k,\beta }$ such as (63); then, it follows from (50), (54) and (67) that

$${\int }_{{\mathbb{R}}^2}{\left|w_{s_k,\beta }\right|}^{4}dx\to {\displaystyle \frac{2(a+b)}{{\beta }^{*}}}>0\mathrm{as}k\to \infty$$

(69)

and $w_{s_k,\beta }$ satisfies

$$\begin{array}{cc}\hfill & -\Delta w_{s_k,\beta }+A_{s_k}{ϵ}_{s_k,\beta }^{2(s_k+1)}V({ϵ}_{s_k,\beta }x+z_{s_k,\beta })w_{s_k,\beta }\hfill \\ \hfill & =A_{s_k}{\mu }_{s_k,\beta }{ϵ}_{s_k,\beta }^{2(s_k+1)}{w}_{s_k,\beta }+A_{s_k}\beta {ϵ}_{s_k,\beta }^{2s_k}{w}_{s_k,\beta }^3,x\in {\mathbb{R}}^2\hfill \end{array}$$

(70)

where $\underset{k\to \infty }{lim}{A}_{s_k}={\displaystyle \frac{b\beta +a(\beta -{\beta }^{*})}{b\beta (a+b)}}>0$. Thus, for $s_k$ small enough, we have

$$-\Delta w_{s_k,\beta }-c\left(x\right)w_{s_k,\beta }\le 0\mathrm{in}{\mathbb{R}}^2,$$

where $c\left(x\right)=A_{s_k}\beta {ϵ}_{s_k,\beta }^{2s_k}{w}_{s_k,\beta }^2$. Applying the De Giorgi–Nash–Moser theory [30] (Theorem 4.1), one obtains that

$$\underset{B_1\left(0\right)}{max}{w}_{s_k,\beta }\le C{\left({\int }_{B_2\left(0\right)}{\left|w_{s_k,\beta }\right|}^{2}dx\right)}^{{\displaystyle \frac{1}{2}}},$$

(71)

where $C>0$ is a constant dependent on the upper bound of $\parallel w_{s_k,\beta }{\parallel }_{L^4\left(B_2\left(0\right)\right)}$.

In fact, we see that 0 is a local maximum point of $w_{s_k,\beta }$ due to $z_{s_k,\beta }$, which is a local maximum point of $u_{s_k,\beta }$. We next argue that there exists a constant $\eta >0$ satisfying

$$w_{s_k,\beta }\left(0\right)\ge \eta >0\mathrm{as}k\to \infty .$$

(72)

If (72) is not true, then for any $R>0$, one obtain

$$\underset{y\in {\mathbb{R}}^2}{sup}{\int }_{B_R\left(y\right)}{\left|w_{s_k,\beta }\right|}^{2}dx\to 0\mathrm{as}k\to \infty$$

which, together with the vanishing lemma [31] (Lemma 1.1), yields that ${\int }_{{\mathbb{R}}^2}{\left|w_{s_k,\beta }\right|}^{4}dx\to 0$ as $k\to \infty$. However, this is a contradiction with (69). Hence, (72) holds, and then (64) follows from (71) and (72).

Next, we need to prove that $z_{s_k,\beta }$ is bounded uniformly as $k\to \infty$. If this is false, then there exists a subsequence $\left\{s_k\right\}$ (still denoted by $\left\{s_k\right\}$) with $s_k\searrow 0$ as $k\to \infty$ such that $z_{s_k,\beta }\to \infty$; one then obtains from (63) and (64) that

$$\begin{array}{cc}\hfill {\int }_{{\mathbb{R}}^2}V\left(x\right)u_{s_k,\beta }^{2}dx& ={\int }_{{\mathbb{R}}^2}V({ϵ}_{s_k,\beta }x+z_{s_k,\beta }){\left|w_{s_k,\beta }\right|}^{2}dx\hfill \\ \hfill & \ge {\int }_{B_2\left(0\right)}V({ϵ}_{s_k,\beta }x+z_{s_k,\beta }){\left|w_{s_k,\beta }\right|}^{2}dx\ge \mathcal{D}\eta \hfill \end{array}$$

(73)

where $\mathcal{D}$ is an arbitrarily large constant. However, this contradicts (53). Hence, $\left\{z_{s_k,\beta }\right\}$ is a bounded sequence in ${\mathbb{R}}^2$.

Passing to a subsequence if necessary, there is a $z_0\in {\mathbb{R}}^2$ such that $z_{s_k,\beta }\to z_0$. In truth, we can prove that $V\left(z_0\right)=0$. If not, one denotes $K:=V\left(z_0\right)>0$; then, (64), together with Fatou’s Lemma, gives

$$\begin{array}{cc}\hfill & \underset{k\to \infty }{lim\; inf}{\int }_{{\mathbb{R}}^2}V({ϵ}_{s_k,\beta }x+z_{s_k,\beta }){\left|w_{s_k,\beta }\right|}^{2}dx\hfill \\ \hfill & \ge {\int }_{B_2\left(0\right)}\underset{k\to \infty }{lim\; inf}V({ϵ}_{s_k,\beta }x+z_{s_k,\beta }){\left|w_{s_k,\beta }\right|}^{2}dx\ge V\left(z_0\right)\eta =K\eta >0\hfill \end{array}$$

(74)

which is a contradiction with (53). We thereby complete the proof of Lemma 4. □

In the final part, we shall give the proof of Theorem 3.

Proof of Theorem 3. 

Let $u_{s_k,\beta }$ be a nonnegative minimizer of $e(s_k,\beta )$ and $z_{s_k,\beta }$ be is its local maximum point. Define a function the same as (63), then, for any positive sequence $\left\{s_k\right\}$ with $s_k\searrow 0$ as $k\to \infty$. Using the definitions of (51) and (63), it follows that

$${\int }_{{\mathbb{R}}^2}|w_{s_k,\beta }{|}^2=1\mathrm{and}{\int }_{{\mathbb{R}}^2}{|\nabla w_{s_k,\beta }|}^2=1.$$

(75)

It then shows that $\left\{w_{s_k,\beta }\right\}$ is a bounded sequence in $H^1\left({\mathbb{R}}^2\right)$. Passing to a subsequence if necessary (still denoted by $\left\{w_{s_k,\beta }\right\}$), there exists a $w_0\in H^1\left({\mathbb{R}}^2\right)$ satisfying, as $k\to \infty$,

$$w_{s_k,\beta }\rightharpoonup w_0\mathrm{weakly}\mathrm{in}{H}^1\left({\mathbb{R}}^2\right).$$

(76)

Since $w_{s_k,\beta }$ satisfies the elliptic equation (70), passing to the weak limit, it is deduced from (9), (54), (68) and (76) that $w_0$ satisfies (in the weak sense)

$$-\Delta w_0+w_0={\parallel Q\parallel }_{L^2}^2{w}_0^3,x\in {\mathbb{R}}^2.$$

(77)

In fact, since (72) holds, we always say that $w_0>0$ applyies the strong maximum principle to (77). Moreover, because (5) has a unique (up to translation) positive radially symmetric solution $Q\left(x\right)$, it is restricted to the fact that there exists a $y_0\in {\mathbb{R}}^2$ such that $w_0$ fulfills

$$w_0={\displaystyle \frac{Q\left(\right|x-y_0\left|\right)}{{\parallel Q\parallel }_{L^2}}}$$

(78)

which, together with (6), gives

$${\int }_{{\mathbb{R}}^2}|w_0{|}^2=1,{\int }_{{\mathbb{R}}^2}{|\nabla w_0|}^2=1.$$

The above results show that $w_{s_k,\beta }\to w_0$ strongly in $L^2\left({\mathbb{R}}^2\right)$ as $k\to \infty$. By applying the Hölder and Sobolev inequalities, one further derives that ${\parallel u\parallel }_{L^4\left({\mathbb{R}}^2\right)}\le {C\parallel u\parallel }_{L^2\left({\mathbb{R}}^2\right)}^{\gamma }{\parallel u\parallel }_{H^1\left({\mathbb{R}}^2\right)}^{1-\gamma }$ for any $u\in H^1\left({\mathbb{R}}^2\right)$ with $\gamma \in (0,1)$. This indicates that $w_{s_k,\beta }\to w_0$ strongly in $L^4\left({\mathbb{R}}^2\right)$ as $k\to \infty$. One then obtains from (75)–(77) that

$$\underset{k\to \infty }{lim}{\int }_{{\mathbb{R}}^2}|\nabla w_{s_k,\beta }{|}^{2}dx={\int }_{{\mathbb{R}}^2}{|\nabla w_0|}^{2}dx,$$

that is,

$$w_{s_k,\beta }\to w_0\mathrm{strongly}\mathrm{in}{H}^1\left({\mathbb{R}}^2\right)\mathrm{as}k\to \infty .$$

(79)

Under the assumption of $\left(V_2\right)$, using the technique of proving Theorem 1.2 in [7], one infers from (70) that $w_{s_k,\beta }\in C_{loc}^{2,{\alpha }_1}\left({\mathbb{R}}^2\right)$, ${\alpha }_1\in (0,1)$. Hence, we have $w_0\in C_{loc}^2\left({\mathbb{R}}^2\right)$ and

$$w_{s_k,\beta }\to w_0\mathrm{in}{C}_{loc}^2\left({\mathbb{R}}^2\right)\mathrm{as}k\to \infty .$$

(80)

Since the origin is a unique critical point (up to translation) of $Q\left(x\right)$, one then concludes from (78) that the origin is the unique critical point of $w_0$. Therefore, one obtains

$$w_0\left(x\right)={\displaystyle \frac{1}{{\parallel Q\parallel }_{L^2}}}Q\left(\left|x\right|\right).$$

(81)

At last, set $y_k$ as the local maximum point of $w_{s_k,\beta }$; then, the (72) shows that $w_{s_k,\beta }\left(y_k\right)\ge \eta >0$. Taking $\gamma$ small enough, one infers from [32] (Lemma 4.2) that the origin is the unique local maximum point of $w_{s_k,\beta }$ as $k\to \infty$. It then yields that $z_{s_k,\beta }$ is the unique maximum point of $u_{s_k,\beta }$. The proof of Theorem 3 is thereby completed. □