On supercritical nonlinear Schr\"{o}dinger equations with ellipse-shaped potentials

TL;DR

This paper investigates the existence, concentration, and blow-up behavior of solutions to a supercritical nonlinear Schrödinger equation with ellipse-shaped potentials in two dimensions, especially as the nonlinearity approaches the critical exponent.

Contribution

It establishes the existence of excited solutions for nonlinearities near the critical value and describes their blow-up behavior as the nonlinearity approaches criticality.

Findings

Existence of excited solutions for q close to 2.

Precise description of blow-up formation of solutions.

Analysis of limiting behavior as q approaches 2.

Abstract

In this paper, we study the existence and concentration of normalized solutions to the supercritical nonlinear Schr\"{o}dinger equation \begin{equation*} \left\{ \begin{array}{l} -\Delta u + V(x) u = \mu_q u + a|u|^q u \quad {\rm in}\quad \mathbb{R}^2,\\ \int_{\mathbb{R}^2}|u|^2\,dx =1,\\ \end{array} \right. \end{equation*} where $\mu_q$ is the Lagrange multiplier. For ellipse-shaped potentials $V(x)$, we show that for $q>2$ close to $2$, the equation admits an excited solution $u_q$, and furthermore, we study the limiting behavior of $u_q$ when $q\to 2_+$. Particularly, we describe precisely the blow-up formation of the excited state $u_q$.

Equations

$$\left\{\begin{array}[]{l}-\Delta u+V(x)u=\mu_{q}u+a|u|^{q}u\quad{\rm in}\quad\mathbb{R}^{2},\\ \int_{\mathbb{R}^{2}}|u|^{2}\,dx=1,\\ \end{array}\right.$$

$$-\Delta u+V(x)u={\mu }_{q}u+a|u{|}^{q}u\mathrm{in}{\mathbb{R}}^2,$$

$$V(x)={\left(\sqrt{\frac{x_1^2}{b_1^2}+\frac{x_2^2}{b_2^2}}-A\right)}^2,$$

$$E_{a,q}(u)=\frac{1}{2}\int_{\mathbb{R}^{2}}\big{(}|\nabla u(x)|^{2}+V(x)|u(x)|^{2}\big{)}\,dx-\frac{a}{q+2}\int_{\mathbb{R}^{2}}|u|^{q+2}\,dx$$

$${\int }_{{\mathbb{R}}^2}|u(x){|}^{2}dx=1.$$

$$d_a(q):=\underset{u\in \mathcal{H},{\int }_{{\mathbb{R}}^2}|u{|}^{2}dx=1}{\inf }{E}_{a,q}(u)$$

$$\mathcal{H}:=\Big{\{}u\in H^{1}(\mathbb{R}^{2}):\int_{\mathbb{R}^{2}}V(x)|u(x)|^{2}\,dx<\infty\Big{\}}.$$

$$-\Delta u+u=u^3\mathrm{in}{\mathbb{R}}^2,u\in H^1({\mathbb{R}}^2).$$

$$a^{\ast }:=\Vert Q{\Vert }_{L^2({\mathbb{R}}^2)}^2.$$

$$-\Delta u+\frac{2}{q}u=\frac{2}{q}{u}^{q+1},q>0,u\in H^1({\mathbb{R}}^2)$$

$$a_q^{\ast }=\Vert {\phi }_q{\Vert }_2^q.$$

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

$${\tau_{q}}=\Big{(}\frac{2a_{q}^{*}}{qa}\Big{)}^{\frac{1}{q-2}}$$

$$\frac{1}{\|\nabla u_{q_{k}}\|_{L^{2}(\mathbb{R}^{2})}}u_{q_{k}}\Big{(}\frac{x}{\|\nabla u_{q_{k}}\|_{L^{2}(\mathbb{R}^{2})}}+x_{k}\Big{)}\rightarrow\frac{\beta}{\|Q\|_{L^{2}(\mathbb{R}^{2})}}Q(\beta x)$$

$$\frac{q-2}{2q}\tau_{q}^{2}\leq E_{a,q}(u_{q})\leq\frac{q-2}{2q}\tau_{q}^{2}+\tau_{q}^{-2}\Big{(}\frac{1}{2}\int_{\mathbb{R}^{2}}|x|^{2}\frac{|Q(x)|^{2}}{\|Q\|_{2}^{2}}\,dx+o(1)\Big{)}\ \ {\rm as}\ \ q\to 2_{+}.$$

$$x_k\to x_0\in \mathcal{Z}:=\{(b_{1}A,0),(-b_{1}A,0)\}$$

$${\tau }_{q_k}(|x_k{|}_b-A)\to 0.$$

$${\phi }_q(x)\le C{e}^{-\delta |x|}\mathrm{for}x\in {\mathbb{R}}^2.$$

$$E_{a,q}{|}_{V=0}(u)=\frac{1}{2}{\int }_{{\mathbb{R}}^2}|\nabla u{|}^{2}dx-\frac{a}{q+2}{\int }_{{\mathbb{R}}^2}|u{|}^{q+2}dx,u\in H^1({\mathbb{R}}^2).$$

$${\tilde{c}}_q=\underset{g\in \Gamma }{\inf }\underset{u\in g}{\max }{E}_{a,q}{|}_{V=0}(u),$$

$$\tilde{\Gamma }=\{g\in C([0,1],\tilde{S}(1))|g(0)=u_1,g(1)=u_2\}$$

$${\tilde{c}}_q=\frac{q-2}{2q}{\tau }_q^2.$$

$${\tilde{\phi }}_q(x)=\frac{{\tau }_q}{\Vert {\phi }_q{\Vert }_2}{\phi }_q({\tau }_{q}x).$$

$${\tilde{c}}_q=E_{a,q}{|}_{V=0}({\tilde{\phi }}_q),\Vert {\tilde{\phi }}_q{\Vert }_2^2=1$$

$$-\Delta {\tilde{\phi }}_q+{\mu }_q{\tilde{\phi }}_q=a{\tilde{\phi }}_q^{q+1},$$

$$s^2{\mu }_q=a{t}^{-q}{s}^2=\frac{2}{q}.$$

$$1={\int }_{{\mathbb{R}}^2}|{\tilde{\phi }}_q{|}^{2}dx=t^{-2}{s}^2{\int }_{{\mathbb{R}}^2}|w_q{|}^{2}dx=t^{-2}{s}^2(a_q^{\ast }{)}^{\frac{2}{q}}.$$

$$a{t}^{-q}{s}^2=\frac{2}{q}\mathrm{and}{t}^{-2}{s}^2(a_q^{\ast }{)}^{\frac{2}{q}}=1,$$

$$t=\Big{(}\frac{2a_{q}^{*}}{qa}\Big{)}^{\frac{1}{2-q}}(a_{q}^{*})^{\frac{1}{q}},\ \ s=\Big{(}\frac{2a_{q}^{*}}{qa}\Big{)}^{\frac{1}{2-q}}=\tau_{q}^{-1}.$$

$${\tilde{\phi }}_q(x)=\frac{{\tau }_q}{\Vert {\phi }_q{\Vert }_2}{\phi }_q({\tau }_{q}x).$$

Full text

On supercritical nonlinear Schrödinger equations with ellipse-shaped potentials

Jianfu Yang

Department of Mathematics, Jiangxi Normal University

Nanchang, Jiangxi 330022, P. R. China

email: Jianfu Yang: jfyang_2000@yahoo.com

Jinge Yang*

School of Sciences, Nanchang Institute of Technology

Nanchang 330099, P. R. China

email: Jinge Yang: jgyang2007@yeah.net

††Key words: Nonlinear elliptic equation, $L^{2}$ supercritical, Constrained minimization, Gross CPitaevskii functional, Ellipse-shaped potential.

Abstract.  In this paper, we study the existence and concentration of normalized solutions to the supercritical nonlinear Schrödinger equation

[TABLE]

where $\mu_{q}$ is the Lagrange multiplier. For ellipse-shaped potentials $V(x)$, we show that for $q>2$ close to $2$, the equation admits an excited solution $u_{q}$, and furthermore, we study the limiting behavior of $u_{q}$ when $q\to 2_{+}$. Particularly, we describe precisely the blow-up formation of the excited state $u_{q}$.

1. Introduction

In this paper, we study the following supercritical Gross-Pitaevskii (GP) type equation

[TABLE]

that is, $q>2$, with the ellipse-shaped potential

[TABLE]

where $b_{1}>b_{2}>0,\ \ A>0$ are constants, and $\mu_{q}$ is the Lagrange multiplier.

In the case $q=2$, equation (1.1) stems from the study of Bose-Einstein condensation. It was derived independently by Gross and Pitaevskii, and it is the main theoretical tool for investigating nonuniform dilute Bose gases at low temperatures. Especially, equation (1.1) is called the Gross-Pitaevskii equation.

In general, problem (1.1) is associated with critical points of the energy functional

[TABLE]

under the constraint

[TABLE]

In particular, in the critical or subcritical case, that is, $q=2$ or $0<q<2$, the functional $E_{a,q}$ is bounded below if $0<a<a^{*}$ for some $a^{*}>0$. Hence, the constrained minimization problem

[TABLE]

is well defined, where $\mathcal{H}$ is defined by

[TABLE]

In the critical case $q=2$, for $a>0$ the system of Bose-Einstein condensates collapses if the particle number increases beyond a critical value; see [7, 9, 15, 20] etc. Mathematically, it was proved in [11] that for a non-negative potential $V(x)$ with finite number zero points, there exists a threshold value $a^{*}>0$ such that $d_{a}(2)$ is achieved if $0<a<a^{*}$, and there is no minimizer for $d_{a}(2)$ if $a\geq a^{*}$. The threshold value $a^{*}$ is determined in terms of the solution of the nonlinear scalar field equation

[TABLE]

It is known from [16] that problem (1.6) admits a unique positive solution up to translations. Such a solution is radially symmetric and exponentially decaying at infinity, see for instance, [5]. Denote by $Q$ in the sequel the positive solution of (1.6). It was found in [11] that

[TABLE]

The similar symmetry breaking phenomenon was considered in [13] in the subcritical case, i.e. $0<q<2$, for the functional $E_{a,q}$. When $q$ approaching $2$, the limit behavior of the minimizer of $E_{a,q}$ constrained by (1.4) is described by the unique positive solution $\varphi_{q}$ of the nonlinear scalar field equation

[TABLE]

in terms of

[TABLE]

If the non-negative potential $V(x)$ has infinitely many minima, or particular $V(x)=(|x|-A)^{2}$, it was shown in [12] that the symmetry breaking occurs in the GP minimizers. That is, any non-negative minimizer of $E_{a,2}$ concentrates at a point on the ring $\{x\in\mathbb{R}^{2}:|x|=A\}$ as $a\to a^{*}_{-}$. Such a result was generalized to the case that the potential $V(x)$ is ellipse-shaped in [10].

The situation becomes different if we turn to the supercritical case $q>2$. Arguments for critical and subcritical cases can not be carried through for the supercritical case, since in the supercritical case the functional $E_{a,q}(u)$ is not bounded below on the manifold

[TABLE]

and the minimization problem (1.5) is not well defined.

In this paper, we focus on the supercritical problem (1.1) with ellipse-shaped potentials. Although in this case, there is no minimizer for the minimization problem (1.5), or no ground state solution for (1.1), we can find critical points of $E_{a,q}$ constrained on the manifold $S(1)$. Such a critical point is an excited state solution of (1.1). Actually, for the supercritical case, the functional $E_{a,q}$ has the mountain pass geometry on $S(1)$. This was revealed in [3, 14], and developed to be applied to various problems, see [2, 4] etc. We first look for critical points of $E_{a,q}$ on $S(1)$ by the variant mountain pass theorem, then we investigate the asymptotic behavior of the critical points when $q$ tends to 2.

In the sequel, we denote $|x|_{b}=\sqrt{b_{1}^{-2}x_{1}^{2}+b_{2}^{-2}x_{2}^{2}}$. Choose $a\in(0,a^{*})$ and denote

[TABLE]

in the sequel. Our existence results are stated as follows.

Theorem 1.1**.**

Let $0<a<a^{*}$. There exists an $\varepsilon_{0}>0$ such that, for any $q\in(2,2+\varepsilon_{0})$, $E_{a,q}(u)$ admits a nonnegative critical point $u_{q}$ at mountain level on $S(1)$.

One may verify that the functional $E_{a,q}(u)$ has the mountain pass geometry on $S(1)$. Then it is standard to show that $E_{a,q}(u)$ has a Palais-Smale $(PS)$ sequence at the mountain pass level. However, such a $(PS)$ sequence may fail to be bounded. In order to bound $(PS)$ sequence, we use a variant mountain pass theorem inspired of [14].

Next, we study the limit behavior of mountain pass point $u_{q}$ as $q\rightarrow 2_{+}$.

Theorem 1.2**.**

Suppose $0<a<a^{*}$. Then, for any sequence $\{q_{k}\}$ with $q_{k}\to 2_{+}$ as $k\rightarrow\infty$, there exist $\{x_{k}\}\subset\mathbb{R}^{2}$, $\beta>0$ and a subsequence of $\{q_{k}\}$, still denoted by $\{q_{k}\}$, such that

[TABLE]

strongly in $L^{2}(\mathbb{R}^{2})$.

The proof of Theorem 1.2 is delicate. We commence with the estimate $E_{a,q}(u_{q})$ for the mountain pass solution $u_{q}$. It is shown in section 4 that

[TABLE]

In contrast with subcritical or critical cases, such an estimate can not established for the supercritical case by simply choosing a suitable trail function. Instead, we need to construct a suitable path, and estimate energy of $E_{a,q}(u)$ on it. Then the proof of Theorem 1.2 is completed by the blow up analysis. This is different and more difficult than subcritical and critical cases because there is essentially no compactness for the sequence $\{u_{q}\}$.

Finally, we study the asymptotic behavior of $x_{k}$.

Theorem 1.3**.**

Let $\{x_{k}\}$ be in Theorem 1.2. Then either $\liminf_{k\to\infty}|x_{k}|\rightarrow+\infty$; or there exists a subsequence of $\{x_{k}\}$, still denoted by $\{x_{k}\}$ such that

[TABLE]

with

[TABLE]

Moreover, in this case equation (1.11) holds with $\beta=1$, and $\tau_{q_{k}}^{-2}\|\nabla u_{q_{k}}\|_{L^{2}(\mathbb{R}^{2})}^{2}\rightarrow 1$ as $k\rightarrow\infty$.

This paper is organized as follows. In section 2, we collect and prove some relevant results for future reference. Then, in section 3, we establish the existence of critical points of $E_{a,q}$. Finally, we analyze the limiting behavior of these critical points in section 4.

2. Preliminaries

In this section, we collect and prove some relevant results for future reference. The following result is used frequently in literatures, we state it explicitly.

Lemma 2.1**.**

Let $\varphi_{q}\geq 0$ be the unique radially symmetric positive solution of (1.8) with $2<q\leq 3$. Then, $\varphi_{q}\rightarrow Q$ strongly in $H^{1}(\mathbb{R}^{2})$ as $q\to 2_{+}$, and there exist positive constants $C$ and $\delta$ independent of $q$ such that

[TABLE]

In particular, $a_{q}^{*}=\|\varphi\|_{2}^{q}\to a^{*}=\|Q\|_{2}^{2}$ as $q\to 2_{+}$.

Let $\tilde{S}(1)=\{u\in H^{1}(\mathbb{R}^{2}):\|u\|_{2}^{2}=1\}$. Denote by $E_{a,q}|_{V=0}(u)$ the energy functional without the trapping potential

[TABLE]

We define

[TABLE]

where

[TABLE]

for $u_{1},u_{2}\in\tilde{S}(1)$ such that $\tilde{c}_{q}>\max\{E_{a,q}|_{V=0}(g(0)),E_{a,q}|_{V=0}(g(1))\}$. We know from Theorem 2.1 in [14] that for every $a>0$, there exists a unique positive critical point $\tilde{\varphi}_{q}$ of $E_{a,q}|_{V=0}(u)$ constrained on $S(1)$ at the energy level $\tilde{c}_{q}$. Now, we give an explicit description of $\tilde{c}_{q}$ and $\tilde{\varphi}_{q}$ in terms of $a,a^{*}_{q}$ and $\varphi_{q}$.

Lemma 2.2**.**

Let $q>2$. Then

[TABLE]

and

[TABLE]

Proof.

It is know from [14] that

[TABLE]

and $\tilde{\varphi}_{q}$ satisfies

[TABLE]

where $\mu_{q}$ is the Lagrange multiplier. Let $w_{q}=t\tilde{\varphi}_{q}(sx)$, where $s$ and $t$ satisfies

[TABLE]

Then, $w_{q}$ is a positive solution of (1.8). By the uniqueness of positive solutions of (1.8), we have $w_{q}=\varphi_{q}$. Since $\|\tilde{\varphi}_{q}\|_{2}^{2}=1$,

[TABLE]

Solving

[TABLE]

we obtain

[TABLE]

Hence, we get

[TABLE]

Substituting $\tilde{\varphi}_{q}$ into $E_{a,q}|_{V=0}(u)$, we obtain

[TABLE]

∎

Remark 2.1**.**

It is known from Lemma 2.10 in [14] that $\tilde{c}_{q}$ can also be described by

[TABLE]

where

[TABLE]

and

[TABLE]

3. Existence

In this section, we show the existence of critical points of the functional $E_{a,q}(u)$ on the sphere

[TABLE]

that is, we will prove Theorem 1.1. This will be done by a variant mountain pass theorem. Before we proceed further, we recall the following compactness lemma, which can be proved as that in [23].

Lemma 3.1**.**

Suppose $V\in L_{loc}(\mathbb{R}^{2})$ and $\lim_{x\to\infty}V(x)=\infty$. Then the embedding $\mathcal{H}\hookrightarrow L^{q}(\mathbb{R}^{2})$ is compact for $q\in[2,\infty)$.

Proof of Theorem 1.1. Obviously, $E_{a,q}(u)\geq E_{a,q}|_{V=0}(u)$ for every $u\in\mathcal{H}$. We claim that

[TABLE]

where

[TABLE]

which implies

[TABLE]

Indeed, by the Gagliardo-Nirenberg inequality (see [21]), we get

[TABLE]

Hence

[TABLE]

Let $\varphi\in C_{c}(\mathbb{R}^{2})$ be a nonnegative function such that $\|\varphi\|_{2}^{2}=1$, and denote $\varphi^{t}(x)=t\varphi(tx)$. Then,

[TABLE]

Let $t_{1}=2^{(q-2)^{-2}}$. Since

[TABLE]

and

[TABLE]

we have

[TABLE]

Hence

[TABLE]

It implies that there exists $\varepsilon_{0}>0$ such that

[TABLE]

if $q\in(2,2+\varepsilon_{0})$. Let

[TABLE]

where

[TABLE]

Since $\lim_{q\rightarrow 2_{+}}\frac{q-2}{2q}\tau_{q}^{2}=+\infty,$ we deduce

[TABLE]

that is,

[TABLE]

Hence, by (3.2),(3.5) for any $g\in\Gamma_{q}$, if $q>2$ and $q$ is close to $2$, there holds

[TABLE]

As a result,

[TABLE]

Equations (3.5) and (3.6) indicate that the functional $E_{a,q}$ has the mountain pass geometry, and then there exists a $(PS)$ sequence of $E_{a,q}$ at the mountain pass level. However, such a $(PS)$ sequence may fail to be bounded. In order to bound (PS) sequence, we use the following variant mountain pass theorem.

Let $H(u,s)=e^{s}u(e^{s}x)$. The functional $\mathcal{E}_{a,q}:\mathcal{H}\times\mathbb{R}\rightarrow\mathbb{R}$ is defined as

[TABLE]

Denote the set of paths by

[TABLE]

and define

[TABLE]

Since $\Gamma_{q}=\{H(\gamma):\gamma\in\mathcal{P}_{q}\}$, we get

[TABLE]

By (3.6), $\mathcal{E}_{a,q}(\gamma(0))=E_{a,q}(\varphi)$ and $\mathcal{E}_{a,q}(\gamma(1))=E_{a,q}(\varphi^{t_{1}})$, we have

[TABLE]

Let $Y=\mathcal{H}\times\mathbb{R}$ be the space with the norm

[TABLE]

Then, $S(1)\times\mathbb{R}$ is a submanifold of $Y$ of codimension $1$ and its tangent subspace at a given point $(u,s)\in S(1)\times\mathbb{R}$ is

[TABLE]

Denote by $\mathcal{E}_{a,q}|_{S(1)\times\mathbb{R}}$ the trace of $\mathcal{E}_{a,q}$ on $S(1)\times\mathbb{R}$. Then, $\mathcal{E}_{a,q}|_{S(1)\times\mathbb{R}}$ is a $C^{1}$ functional on $S(1)\times\mathbb{R}$, and for any $(u,s)\in S(1)\times\mathbb{R}$,

[TABLE]

Moreover,

[TABLE]

By (3.8), there exists $\{(g_{n},0)\}\subset\mathcal{P}_{q}$ such that

[TABLE]

Because $E_{a,q}(|g_{n}|)\leq E_{a,q}(g_{n})$, let $\gamma_{n}=(|g_{n}|,0)$, we have

[TABLE]

By Theorem 3.2 in [11], there exists $\{(w_{n},s_{n})\}\subset S(1)\times\mathbb{R}$ such that

[TABLE]

[TABLE]

and

[TABLE]

Let $u_{n}=H(w_{n},s_{n})$. Choose $v_{n}\in|g_{n}|$ such that

[TABLE]

Then

[TABLE]

By (3.11),

[TABLE]

For any (\varphi,\tau)\in T_{(u,s)}\big{(}S(1)\times\mathbb{R}\big{)} with $(u,s)\in S(1)\times\mathbb{R}$, we claim that

[TABLE]

where

[TABLE]

Indeed,

[TABLE]

We deduce that

[TABLE]

and

[TABLE]

Since $\langle\varphi,u\rangle_{L^{2}(\mathbb{R}^{2})}=0$, we have

[TABLE]

and

[TABLE]

As a result, the claim (3.15) holds.

For any (\varphi,\tau)\in T_{(w_{n},\tau_{n})}\big{(}S(1)\times\mathbb{R}\big{)}, we infer from (3.12) and (3.15) that,

[TABLE]

Choosing particularly $(\varphi,\tau)=(0,1)$ in (3.17), we obtain

[TABLE]

On the other hand, for any $\psi\in T_{u_{n}}S(1)$, we may choose $\varphi$ such that $\psi(x)=e^{s_{n}}\varphi(e^{s_{n}}x)$. Therefore,

[TABLE]

Taking $\tau=0$ in (3.17), we have

[TABLE]

By (3.13),

[TABLE]

as $n\rightarrow\infty$ and $\|\psi\|_{\mathcal{H}}\leq 2\|\varphi\|_{\mathcal{H}}$ . Consequently,

[TABLE]

that is

[TABLE]

The identity

[TABLE]

and $\|u_{n}\|_{2}^{2}=1$ yield

[TABLE]

This together with (3.14) and (3.18) implies that $\{u_{n}\}$ is bounded in $\mathcal{H}$. By Lemma 3.1, we may assume that

[TABLE]

and

[TABLE]

We know from (3.17) and Lemma 3 in [5] that there exists a $\mu_{q}^{n}$ such that

[TABLE]

and then,

[TABLE]

which is bounded. Without loss of generality, we assume that $\mu_{q}^{n}\to\mu_{q}$ as $n\to\infty$. Passing the limit $n\to\infty$ in (3.24), we obtain

[TABLE]

By (3.25) and (3.26), we have

[TABLE]

namely,

[TABLE]

By (3.13) and (3.19), we obtain $v_{n}\to u_{q}$ in $\mathcal{H}$. Note that $v_{n}$ is positive, we get $u_{q}\geq 0$ and $u_{q}$ is a solution of (3.26). Then the strong maximum principle implies $u_{q}>0$.

Since $u_{n}\to u_{q}$ in $\mathcal{H}$, by (3.14) and (3.18),

[TABLE]

and

[TABLE]

The proof is complete. $\Box$

4. Energy estimates and asymptotic behavior

In this section, we first establish an asymptotic expansion of the energy $E_{a,q}(u_{q})$ at the mountain pass point $u_{q}$. This relies on, among other things, a carefully choosing a path. Next, we study asymptotic behavior of $u_{q}$ as $q\to 2_{+}$ and prove Theorem 1.2 and Theorem 1.3.

Proposition 4.1**.**

There holds

[TABLE]

Proof.

By (3.6) and $c_{q}=E_{a,q}(u_{q})$, it suffices to prove that

[TABLE]

To this purpose, we will construct a path $g\in\Gamma_{q}$ linking $\varphi$ and $\varphi^{t_{1}}$ so that

[TABLE]

The path $g$ is constructed in three parts.

First, we construct a path $g_{1}$ linking $\varphi$ to some $w_{q}^{\tilde{t}_{0}}$ and estimate $E_{a,q}(g_{1}(s))$. Let $\varphi_{q}$ be the positive solution of (1.8). Denote

[TABLE]

with $x_{0}\in\mathcal{Z}$. By Lemma 8.12 in [6], $\varphi_{q}$ satisfies

[TABLE]

Note that $a_{q}^{*}=\|\varphi_{q}\|_{2}^{q}$, we obtain

[TABLE]

and

[TABLE]

Choosing $\tilde{t}_{0}=\sqrt{\frac{q-2}{12q}}$ such that $w_{q}^{\tilde{t}_{0}}(x)=\tilde{t}_{0}w_{q}(\tilde{t}_{0}x)$ satisfies

[TABLE]

we define

[TABLE]

where $\varphi\in C_{c}(\mathbb{R}^{2})$ is a nonnegative function with $\|\varphi\|_{2}^{2}=1$ given in $\Gamma_{q}$.

By (4.4) and the fact that $\|w_{q}^{\tilde{t}_{0}}\|_{2}^{2}=\|\varphi\|_{2}^{2}=1$, we have

[TABLE]

since $\lim_{q\to 2^{+}}\frac{q-2}{q}\tau_{q}^{2}=+\infty$. On the other hand,

[TABLE]

By Lemma 2.1,

[TABLE]

where $C>0,\ \ \delta>0$ independent of $q$, and Lebesgue dominated theorem,

[TABLE]

Since $\tilde{t}_{0}^{2}=\frac{q-2}{12q}$, (4.5) and (4.7) yield

[TABLE]

for $q>2$ and $q$ close to $2$.

Next, we construct the second part $g_{2}$ of the path $g$. Choose $\tilde{t}_{1}=\tau_{q}^{-1}t_{1}:=\tau_{q}^{-1}2^{(q-2)^{-2}}$. Let

[TABLE]

By (4.2), (4.3) and $\|w_{q}^{\tilde{t}_{1}}\|_{2}^{2}=\|\varphi^{t_{1}}\|_{2}^{2}=1$, we have

[TABLE]

and

[TABLE]

If $0\leq s\leq\frac{1}{2}$, then

[TABLE]

Similarly, the inequality (4.11) holds for $\frac{1}{2}\leq s\leq 1$. We may show as (4.7) that

[TABLE]

Therefore, we deduce from from (4.11) and (4.12) that

[TABLE]

for $q>2$ and close to $2$.

Finally, we construct a path linking $w_{q}^{\tilde{t}_{0}}$ and $w_{q}^{\tilde{t}_{1}}$. Since

[TABLE]

there exists a $\tilde{t}_{2}\in(\tilde{t}_{0},\tilde{t}_{1})$ such that $\|\nabla w_{q}^{\tilde{t}_{2}}\|_{2}^{2}=\tau_{q}^{2}$. By (3.2), we have

[TABLE]

Let

[TABLE]

Now, we define a path $g$ by $g_{1}$, $g_{2}$ and $g_{3}$, that is,

[TABLE]

Obviously, $g\in\Gamma_{q}$.

It follows from (4.8), (4.13) and (4.14) that

[TABLE]

Let $f(t)=E_{a,q}(w_{q}^{t})$. It implies by (4.2) and (4.3) that

[TABLE]

By (4.8), (4.13) and (4.14), there exists $t_{q}\in(\tilde{t}_{0},\tilde{t}_{1})$ such that

[TABLE]

and

[TABLE]

By Lemma 2.1 and the Lebegue dominated theorem, we have

[TABLE]

and

[TABLE]

as $q\to 2_{+}$. Since $t_{q}>\tilde{t}_{0}=\sqrt{\frac{q-2}{12q}}$, we see from (4.17)–(4.19) that

[TABLE]

Now, we claim that

[TABLE]

Indeed, if this is not the case, there would exist $q_{n}\to 2_{+}$ such that either

[TABLE]

If $t_{1q_{n}}\geq 1+\tau_{q_{n}}^{-\frac{3}{2}}$, then

[TABLE]

which contradicts (4.20). The case $t_{1q_{n}}\leq 1-\tau_{q_{n}}^{-\frac{3}{2}}$ can be ruled out in the same way.

Next, we deal with

[TABLE]

It holds

[TABLE]

By (4.21) and $|x_{0}|_{b}=A$, we get

[TABLE]

and

[TABLE]

as $q\to 2$. Since $Q$ is radially symmetric about the origin, by Lemma 2.1, (4.23), (4.24) and the Lebesgue dominated theorem, we have

[TABLE]

We derive from (4.2), (4.3), (4.24) and (4.25) that

[TABLE]

which and (4.15) immediately yield the conclusion. ∎

Proposition 4.2**.**

There holds

[TABLE]

with $C_{1}>0$, $C_{2}>0$.

Proof.

We deduce from Proposition 4.1, (3.21) and (3.27) that

[TABLE]

which implies

[TABLE]

since $\|u_{q}\|_{2}^{2}=1$.

Now, we derive the lower bound in (4.26).

Suppose on the contrary that there exists $q_{k}\to 2_{+}$ such that

[TABLE]

By (4.27), we have

[TABLE]

Observing that

[TABLE]

and $\|u_{q}\|_{2}^{2}=1$, we have

[TABLE]

Therefore,

[TABLE]

By (3.27) we obtain

[TABLE]

a contradiction. ∎

Proof of Theorem 1.2. It is known from (4.27) that

[TABLE]

Then, the fact that

[TABLE]

and $\|u_{q}\|_{2}^{2}=1$ yield

[TABLE]

Therefore,

[TABLE]

and

[TABLE]

Since $-\Delta+V$ is a compact operator, its first eigenvalue $\lambda_{1}$ and corresponding eigenfunction $\varphi_{1}$ are positive, which satisfy

[TABLE]

Hence, by (3.26),

[TABLE]

implying

[TABLE]

Using (3.26) again, we get

[TABLE]

Equation (4.30) and Proposition 4.2 yield for $q>2$ and close to $2$ that

[TABLE]

By (3.27), Proposition 4.2 and (4.29),

[TABLE]

Let $\varepsilon_{q}=\|\nabla u_{q}\|_{2}^{-1}$ and $\tilde{w}_{q}(x)=\varepsilon_{q}u_{q}(\varepsilon_{q}x)$. Then, $\|\nabla\tilde{w}_{q}\|_{2}^{2}=\|\tilde{w}_{q}\|_{2}^{2}=1$. We know from Lemma 2.1 that $a_{q}^{*}\to a^{*}$ as $q\to 2_{+}$, and then

[TABLE]

and

[TABLE]

Thus

[TABLE]

It follows from (4.32)–(4.34) that there exist $C_{1}>0$ and $C_{2}>0$ such that

[TABLE]

Now we claim that there exist $\{y_{q}\}\subset\mathbb{R}^{2}$, $R_{0}>0$ and $\eta>0$ such that

[TABLE]

Indeed, if this is not the case, for any $R>0$, there exists a sequence $\{\tilde{w}_{q_{k}}\}$ with $q_{k}\to 2_{+}$ as $k\to\infty$ such that

[TABLE]

By Lemma 1.21 in [22], we get $\tilde{w}_{q_{k}}\to 0$ strongly in $L^{\gamma}(\mathbb{R}^{2})$ for any $\gamma>2$, which contradicts to (4.35).

Denote $w_{q}(x)=\tilde{w}_{q}(x+y_{q})$. Then

[TABLE]

[TABLE]

[TABLE]

Hence, there exist a sequence $\{q_{k}\}$, $q_{k}\to 2_{+}$ as $k\to\infty$, and $w\in H^{1}(\mathbb{R}^{2})$ such that $w_{q_{k}}:=w_{k}\rightharpoonup w$ weakly in $\mathcal{H}$ and $w_{k}\to w\neq 0$ strongly in $L^{\gamma}_{loc}(\mathbb{R}^{2})$ for any $\gamma>2$. Denote $\varepsilon_{q_{k}}:=\varepsilon_{k}$ and $y_{q_{k}}=y_{k}$ for simiplity. By (3.26), we find that $w_{k}$ solves

[TABLE]

Now, we study the asymptotic behavior of $w_{k}$. By (4.30), the definition of $\varepsilon_{k}$ and Proposition 4.2, we have

[TABLE]

It follows from (4.31) and (4.32) that

[TABLE]

So we may assume

[TABLE]

Next, we study the asymptotic behavior of $\varepsilon_{k}^{2}(|\varepsilon_{k}(x+y_{k})|_{b}-A)^{2}$.

If $\liminf_{k\to\infty}|\varepsilon_{k}y_{k}|_{b}\leq C$, there exists a subsequence of $\{q_{k}\}$, still denoted by $\{q_{k}\}$, such that $|\varepsilon_{k}y_{k}|_{b}\leq C$, so for any $M>0$, we have

[TABLE]

uniformly for $x\in B_{M}(0)$ as $k\to\infty$.

If $\liminf_{k\to\infty}|\varepsilon_{k}y_{k}|_{b}=\infty$, there exists a subsequence of $\{q_{k}\}$, still denoted by $\{q_{k}\}$, such that $|\varepsilon_{k}y_{k}|_{b}\to\infty$, as $k\to\infty$. For any $M>0$, if $|x|_{b}\leq M$ and $k$ is large enough, we obtain

[TABLE]

Equation (4.28) then implies that

[TABLE]

Consequently, by (4.37), we have

[TABLE]

Let $\varepsilon_{q}=\|\nabla u_{q}\|_{2}^{-1}$. By Proposition 4.2,

[TABLE]

We may assume that $\varepsilon_{k}^{2}|\varepsilon_{k}y_{k}|_{b}^{2}\to\mu\geq 0$. It is easy to show that for any $M>0$ and $x\in B_{M}(0)$,

[TABLE]

In summary of (4.34) and (4.41)–(4.44), we have the following cases.

Case (i): $\varepsilon_{k}\mu_{q_{k}}\to 0$,  and $\varepsilon_{k}^{2}(|\varepsilon_{k}(x+y_{k})|_{b}-A)^{2}\to 0$;

Case (ii): $\varepsilon_{k}\mu_{q_{k}}\to-\beta^{2}<0$,  and $\varepsilon_{k}^{2}(|\varepsilon_{k}(x+y_{k})|_{b}-A)^{2}\to 0$;

Case (iii): $\varepsilon_{k}\mu_{q_{k}}\to 0$,  and $\varepsilon_{k}^{2}(|\varepsilon_{k}(x+y_{k})|_{b}-A)^{2}\to\mu>0$;

Case (iv): $\varepsilon_{k}\mu_{q_{k}}\to-\beta^{2}<0$,  and $\varepsilon_{k}^{2}(|\varepsilon_{k}(x+y_{k})|_{b}-A)^{2}\to\mu>0$;

Taking the limit as $k\to\infty$ in (4.40), we obtain that $w$ satisfies correspondingly in the case (i) that

[TABLE]

in the case (ii) that

[TABLE]

in the case (iii) that

[TABLE]

in the case (iv) that

[TABLE]

The case (i) can not happen. Indeed, if it would happen on the contrary, the fact that $w\geq 0$ and $w\neq 0$ would imply that the equation (4.44) admits a positive solution, which contradicts the Liouville theorem.

In the case (ii), by the uniqueness of positive solution of (4.45) and $w\neq 0$, there exists $y_{0}\in\mathbb{R}^{2}$ such that

[TABLE]

which implies $\|w\|_{2}^{2}=1$. Hence $w_{q_{k}}\to w$ strongly in $L^{2}(\mathbb{R}^{2})$, that is

[TABLE]

Let $x_{k}=\varepsilon_{k}(y_{k}+y_{0})$. Then

[TABLE]

The case (iii) and (iv) can be treated in the same way as the case (ii), we omit the detail. The proof is complete.   $\Box$

Finally, we study the limiting behavior of $x_{k}$.

Proof of Theorem 1.3 It is obvious that either $\liminf_{k\to\infty}|x_{k}|=+\infty$ or $\liminf_{k\to\infty}|x_{k}|$ is bounded.

If $\liminf_{k\to\infty}|x_{k}|=+\infty$, there is nothing to prove.

If $\liminf_{k\to\infty}|x_{k}|$ is bounded, there exist $x_{0}\in\mathbb{R}^{2}$ and a subsequence of $\{x_{k}\}$, still denoted by $\{x_{k}\}$ such that $x_{k}\to x_{0}$, as $k\to\infty$. We may show as the proof of Theorem 1.2 that only the case (ii) may happen, that is, $\varepsilon_{k}^{2}\mu_{q_{k}}\to-\beta^{2}<0$ and $\varepsilon_{k}^{2-q_{k}}a\to a^{*}$. By (4.40), we have

[TABLE]

Using the De Diorgi-Nash-Moser estimate, see Theorem 4.1 in [17], we obtain

[TABLE]

Since $w_{q_{k}}\to w$ strongly in $L^{2}(\mathbb{R}^{2})$, $w_{k}$ is uniformly bounded in $L^{\infty}(\mathbb{R}^{2})$. Hence, there exists $R>0$ such that if $k$ is large enough and $|x|\geq R$,

[TABLE]

and

[TABLE]

By the comparison principle,

[TABLE]

Since $w_{k}$ is uniformly bounded in $L^{\infty}(\mathbb{R}^{2})$, we have

[TABLE]

Noting $x_{k}=\varepsilon_{k}(y_{k}+y_{0})$ is bounded in $\mathbb{R}^{2}$, we have

[TABLE]

Similarly,

[TABLE]

By (4.27), (4.48) and (4.49), we obtain

[TABLE]

Let

[TABLE]

Then

[TABLE]

[TABLE]

and

[TABLE]

It is easy to deduce that there exist $t_{1}>0$ and $t_{2}>0$ such that

$(i)$ $g(t)$ is decreasing in $(0,t_{1})$ and $(t_{2},+\infty)$, while $g(t)$ is increasing in $(t_{1},t_{2})$;

$(ii)$ $g^{\prime}(t_{1})=g^{\prime}(t_{2})=0$ and $\lim_{t\to 0}g(t)=\infty$.

In order to estimate $t_{1}$ and $t_{2}$, we rewrite $g^{\prime}(t)$ as

[TABLE]

By the Pohozaev identity (3.27), we have $g^{\prime}(1)=0$.

Now, we claim that $t_{2}=1$. Indeed, we choose $t_{3}=C(q_{k}-2)^{\frac{1}{2}}$ such that

[TABLE]

Then by (4.49), we have

[TABLE]

By Proposition 4.2, for $q>2$ and $q$ close to $2$ we have,

[TABLE]

Hence, $t_{2}=1$ and

[TABLE]

Choose $t_{k}$ such that

[TABLE]

By (3.27), (4.49) and (4.50), we obtain

[TABLE]

Therefore,

[TABLE]

and $t_{k}>t_{3}$ for $k$ large enough.

By Lemma 2.2 and Remark 2.1, we get

[TABLE]

We derive from (4.51) and (4.53) that

[TABLE]

Then, Proposition 4.1 yields

[TABLE]

Next we claim that

[TABLE]

We argue by contradiction. Suppose this is not the case, there would exist a subsequence of $\{q_{k}\}$, still denoted by $\{q_{k}\}$, such that

[TABLE]

By the definition $\varepsilon_{k}$, (4.39) and (4.50), for any $C\geq 0$ and $k$ large enough,

[TABLE]

which contradicts (4.54). Hence, we may assume that

[TABLE]

By the Fatou Lemma, (4.47) and (4.50), we have

[TABLE]

Since $Q$ is symmetric, we deduce that

[TABLE]

By (4.54), this implies $x_{0}^{2}=0$ and $C_{0}=0$, and by (4.56), $x_{0}\in\mathcal{Z}$ and

[TABLE]

Finally, we further study the limiting behavior of $x_{k}$. Noting $\varepsilon_{q}=\|\nabla u_{q}\|_{2}^{-1}$, by (4.50), (4.52), (4.57) and $x_{k}\to x_{0}$, we get

[TABLE]

The proof is complete.  $\Box$

Acknowledgment The first author is supported by NNSF of China, No:11671179 and 11771300. The second author is supported by NNSF of China, No:11701260.