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Elliptic equation with van der Waals type potential

Part of: Elliptic equations and systems

Published online by Cambridge University Press:  18 October 2022

Yu Su  and
Senli Liu
Yu Su
Affiliation:
School of Mathematics and Big data, Anhui University of Science and Technology, Huainan, Anhui 232001, China (yusumath@aust.edu.cn)
Senli Liu
Affiliation:
School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China (mathliusl@csu.edu.cn)
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Abstract

In this paper, we study the Lieb's translation lemma in Coulomb–Sobolev space and then apply it to investigate the existence of Pohožaev type ground state solution for elliptic equation with van der Waals type potential.

Keywords

Coulomb–Sobolev spaceelliptic equationground stateLieb's translation lemmaPohožaev manifold

MSC classification

Primary: 35J20: Variational methods for second-order elliptic equations
Secondary: 35J60: Nonlinear elliptic equations

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 65 , Issue 4 , November 2022 , pp. 1048 - 1062
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

1. Introduction

Consider a elliptic problem

(ℰ)\begin{equation} -\Delta u = (W*|u|^{2})u,\quad x\in \mathbb{R}^{N},\quad \quad {\mathcal E}\end{equation}

where

\[ W(x)=\displaystyle\frac{a_{1}}{|x|^{N-\alpha}}+\frac{a_{2}}{|x|^{N-\beta}}, \quad x\in \mathbb{R}^{N}{\setminus}\{0\}, \]

and $N\geqslant 3$, $\alpha,\beta \in (0,N)$ and $a_{1},a_{2}\in \mathbb {R}$. Physically, if $N-\alpha >1$ and $N-\beta >1$, then $W$ is the van der Waals type potential, which is two-body potentials of short range, see [Reference Dzyaloshinskii, Lifshitz and Pitaevskii6, Reference Yang, Li and Tang14, Reference Zheng and Narayanaswamy16]. If $a_{1}=1$ and $a_{2}=0$, then Equation (ℰ) is the Choquard equation. If $a_{1}=-1$ and $a_{2}=0$, then equation (ℰ) is a Schrödinger–Poisson type equation.

In this paper, we take

\[ a_{1}={-}A_{\alpha} \text{ and } a_{2}=A_{\beta}, \]

where

\[ A_{\alpha}=\displaystyle\frac{\Gamma(\frac{N-\alpha}{2})}{2^{\alpha}\pi^{\frac{N}{2}}\Gamma(\frac{\alpha}{2})}. \]

Then equation (ℰ) becomes

(𝒮)\begin{equation} -\!\Delta u + (I_{\alpha}*|u|^{2})u = (I_{\beta}*|u|^{2})u,\quad x\in \mathbb{R}^{N},\quad \quad {\mathcal S} \end{equation}

where $N\geqslant 3$, $\alpha,\beta \in (0,N)$, and $I_{\alpha }$ is the Riesz potential

\[ I_{\alpha}:=\displaystyle\frac{A_{\alpha}}{|x|^{N-\alpha}}, \quad x\in \mathbb{R}^{N}{\setminus}\{0\}. \]

Cao-Jia-Luo [Reference Cao, Jia and Luo5] considered the normalized solution of the following equation

(1.1)\begin{equation} -\!\Delta u =\lambda u+\mu (I_{\alpha}*|u|^{2})u + (I_{\beta}*|u|^{2})u,\quad x\in \mathbb{R}^{N}, \end{equation}

where $\mu >0$, and $\lambda <0$ is unknown which appears as Lagrange multiplier. For $\mu <0$ and fixed $\lambda <0$, Yang-Yu [Reference Yang and Yu15] investigated the existence and non-existence of solution for equation (1.1). Specially, if $\lambda =0$ and $\mu =-1$, then equation (1.1) goes back to equation (𝒮).

The case of $\lambda <0$ has been studied in [Reference Cao, Jia and Luo5, Reference Yang and Yu15]. It is natural to ask whether there exists a solution for equation (1.1) with $\lambda =0$? In this paper, we studied the existence of non-trivial solution of equation (1.1) with $\lambda =0$ and $\mu =-1$.

Due to $\lambda =0$, the non-homogeneous Sobolev space $H^{1}(\mathbb {R}^{N})$ [Reference Willem13] is not a suitable working space for equation (𝒮). To overcome this difficulty, for $N\geqslant 3$, we need to introduce the Coulomb–Sobolev space [Reference Mercuri, Moroz and Van Schaftingen11]

\[ E^{\alpha}(\mathbb{R}^{N}) := \left\{ u \in D^{1,2}(\mathbb{R}^{N}) \left| \displaystyle\int_{\mathbb{R}^{N}} (I_{\alpha}*|u|^{2})|u|^{2} \mathrm{d}x <\infty \right\}\right. \]

with the norm

\[ \|u\|_{E^{\alpha}(\mathbb{R}^{N})}:=\Bigg( \|u\|_{D^{1,2}(\mathbb{R}^{N})}^{2} +\left(\displaystyle\int_{\mathbb{R}^{N}} (I_{\alpha}*|u|^{2})|u|^{2} \mathrm{d}x \right)^{\frac{1}{2}} \Bigg)^{\frac{1}{2}}, \]

where $D^{1,2}(\mathbb {R}^{N})$ is the homogeneous Sobolev space

\[ D^{1,2}(\mathbb{R}^{N}) := \Big\{ u\in L^{2^{*}}(\mathbb{R}^{N}) \left| \|u\|_{D^{1,2}(\mathbb{R}^{N})} <\infty \Big\} \right., \quad 2^{*}=\displaystyle\frac{2N}{N-2}, \]

with the semi-norm

\[ \|u\|_{D^{1,2}(\mathbb{R}^{N})}^{2} := \displaystyle\int_{\mathbb{R}^{N}} |\nabla u|^{2} {\rm d}x . \]

For more about Coulomb–Sobolev space, we refer to [Reference Bellazzini, Frank and Visciglia1–Reference Bellazzini, Ghimenti, Mercuri, Moroz and Van Schaftingen3, Reference Ianni and Ruiz8, Reference Ruiz12].

Proposition 1.1 Mercuri et al. [Reference Mercuri, Moroz and Van Schaftingen11, Theorem 1]

Let $N\geqslant 3$ and $\alpha \in (0,N)$. Then the following continuous embeddings hold

\[ E^{\alpha}(\mathbb{R}^{N})\hookrightarrow L^{r}(\mathbb{R}^{N}), \quad \begin{cases} p\in[2_{\alpha}^{\sharp},2^{*}],\quad \alpha>N-4,\\ p\in[2^{*},2_{\alpha}^{\sharp}],\quad\alpha< N-4,\\ \end{cases} \]

where $2_{\alpha }^{\sharp }=2+\frac {4}{2+\alpha }$ and $2^{*}=\frac {2N}{N-2}$ are the critical exponents.

Proposition 1.2 Lieb and Loss [Reference Lieb and Loss10, Hardy–Littlewood–Sobolev inequality]

Let $t,r>1$ and $0<\beta < N$ with $\frac {1}{t}+\frac {1}{r}+\frac {N-\beta }{N}=2,$ $g\in L^{t}(\mathbb {R}^{N})$ and $h\in L^{r}(\mathbb {R}^{N})$. There exists a sharp constant $C(N,\beta,t,r)>0$ independent of $g,h$ such that

\[ \displaystyle\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}} \displaystyle\frac{|g(x)||h(y)|} {|x-y|^{N-\beta}} \mathrm{d}x\mathrm{d}y \leqslant C(N,\beta,t,r) \|g\|_{L^{t}(\mathbb{R}^{N})} \|h\|_{L^{r}(\mathbb{R}^{N})}. \]

If $t=r=\frac {2N}{N+\beta }$, then $C(N,\beta,t,r) = C(N,\beta ) = \pi ^{\frac {N-\beta }{2}} \frac {\Gamma (\frac {\beta }{2})}{\Gamma (N+\frac {\beta }{2})} ( \frac {\Gamma (\frac {N}{2})}{\Gamma (N)} )^{\frac {\beta }{N}}.$

From Proposition 1.2, we have

(1.2)\begin{equation} \displaystyle\int_{\mathbb{R}^{N}} (I_{\beta}*|u|^{2})|u|^{2} \mathrm{d}x \leqslant C(N,\beta) \|u\|_{L^{\frac{4N}{N+\beta}}(\mathbb{R}^{N})}^{4}. \end{equation}

If $2^{*} \leqslant \frac {4N}{N+\beta } \leqslant 2_{\alpha }^{\sharp }$ and $\alpha < N-4$, then

(1.3)\begin{equation} \alpha <\displaystyle\frac{N\alpha}{\alpha+4}\leqslant\beta\leqslant N-4. \end{equation}

By Propositions 1.1–1.2, if (1.3) holds, then equation (𝒮) has a variational structure, and its solutions are the critical points of the functional

\[ J(u):=\displaystyle\frac{1}{2} \|u\|_{D^{1,2}(\mathbb{R}^{N})}^{2} + \frac{1}{4} \displaystyle\int_{\mathbb{R}^{N}} (I_{\alpha}*|u|^{2})|u|^{2} \mathrm{d}x - \frac{1}{4} \int_{\mathbb{R}^{N}} (I_{\beta}*|u|^{2})|u|^{2} \mathrm{d}x. \]

Moreover, we can deduce that the functional $J\in C^{1}(E^{\alpha }(\mathbb {R}^{N}),\mathbb {R})$. It is easy to see that if $u\in E^{\alpha }(\mathbb {R}^{N})$ is a critical point of $J$, i.e.

\[ 0= \langle J'(u),\varphi\rangle = \displaystyle\int_{\mathbb{R}^{N}} \nabla u\nabla \varphi {\rm d}x + \int_{\mathbb{R}^{N}} (I_{\alpha}*|u|^{2})u\varphi \mathrm{d}x - \int_{\mathbb{R}^{N}} (I_{\beta}*|u|^{2})u\varphi \mathrm{d}x \]

for all $\varphi \in E^{\alpha }(\mathbb {R}^{N})$, then $u\in E^{\alpha }(\mathbb {R}^{N})$ is a weak solution of equation (𝒮). To find a weak solution, one may consider a constraint which is the Nehari manifold as follows

\[ \mathcal{N}= \big\{ u\in E^{\alpha}(\mathbb{R}^{N})\backslash\{0\} \big| \langle J'(u),u\rangle=0 \big\}. \]

For any $u\in \mathcal {N}$, we have

\[ J(u)=J(u)-\displaystyle\frac{1}{4}\langle J'(u),u\rangle =\frac{1}{4}\|u\|_{D^{1,2}(\mathbb{R}^{N})}^{2}. \]

Compare the right hand with norm $\|u\|_{E^{\alpha }(\mathbb {R}^{N})}$, it loses the term of $\int _{\mathbb {R}^{N}}(I_{\alpha }*|u|^{2})|u|^{2}\mathrm {d}x$. And the semi-norm $\|u\|_{D^{1,2}(\mathbb {R}^{N})}^{2}$ can not control the lost term. Hence, it is hard to show that there exists $C>0$ such that $\min \{J(u)| u\in \mathcal {N}\}>C$. Moreover, assume that there exists a minimizing sequence on Nehari manifold $\mathcal {N}$, but it is difficult to show the boundedness of it. To overcome these difficulties, we introduce the Pohožaev maniflod instead

\[ \mathcal{M}= \big\{ u\in E^{\alpha}(\mathbb{R}^{N})\backslash\{0\} \big| P(u)=0 \big\}, \]

where $P(\cdot )$ is the Pohožaev type identity

\[ P(u):= \displaystyle\frac{N-2}{2} \|u\|_{D^{1,2}(\mathbb{R}^{N})}^{2} + \frac{N+\alpha}{4} \displaystyle\int_{\mathbb{R}^{N}} (I_{\alpha}*|u|^{2})|u|^{2} \mathrm{d}x - \frac{N+\beta}{4} \int_{\mathbb{R}^{N}} (I_{\beta}*|u|^{2})|u|^{2} \mathrm{d}x. \]

A function $u$ is called the Pohožaev type ground state solution of equation (𝒮), if $u$ is a weak solution of the least energy problem

\[ c:=\min\{J(u)| u\in \mathcal{M}\}. \]

Our main result is the following.

Theorem 1.1 Let $N\geqslant 5$ and $\alpha,\beta \in (0,N)$. If

\[ \alpha<\displaystyle\frac{N\alpha}{\alpha+4}<\beta< N-4. \]

Then equation (𝒮) has a Pohožaev type ground state solution.

Remark 1.1 From Proposition 1.1, we know that the following embeddings are not compact

\[ E^{\alpha}(\mathbb{R}^{N})\hookrightarrow L^{r}(\mathbb{R}^{N}), \quad r\in[2^{*},2_{\alpha}^{\sharp}]. \]

Hence, it is hard to show that a bounded sequence has a convergence subsequence when we seek the weak solution of equation (𝒮). To overcome it, we establish a Lieb's translation lemma in Coulomb–Sobolev space, which is inspired by [Reference Lieb9, Reference Mercuri, Moroz and Van Schaftingen11] and different from their results, see Lemma 2.3.

This paper is organized as follows: In § 2, we present the Lieb's translation lemma. In § 3, we prove Theorem 1.1.

2. Lieb's translation lemma

In this section, we present Lieb's translation lemma in Coulomb–Sobolev space. Let us define $\|\cdot \|_{M}: E^{\alpha }(\mathbb {R}^{N})\to \mathbb {R}$ as

\[ \|u\|_{M}^{2}:= \|u\|_{D^{1,2}(\mathbb{R}^{N})}^{2} + \displaystyle\int_{\mathbb{R}^{N}} (I_{\alpha}*|u|^{2})|u|^{2} \mathrm{d}x. \]

Lemma 2.1 Let $N\geqslant 3$ and $\alpha \in (0,N)$. There exists $S_{p}>0$ such that

\[ S_{p} \|u\|_{L^{p}(\mathbb{R}^{N})}^{p} \leqslant \|u\|_{M}^{\frac{p(2+\alpha)-2N}{4+\alpha-N}}, \quad \begin{cases} p\in[2_{\alpha}^{\sharp},2^{*}],\quad\alpha>N-4,\\ p\in[2^{*},2_{\alpha}^{\sharp}],\quad\alpha< N-4,\\ \end{cases} \]

for all $u\in E^{\alpha }(\mathbb {R}^{N})$.

Proof. For any $u\in E^{\alpha }(\mathbb {R}^{N})$, if $\|u\|_{E^{\alpha }(\mathbb {R}^{N})}\leqslant 1$ or $\|u\|_{M}^{2}\leqslant 1$, then

\[ \displaystyle\frac{1}{2}\|u\|_{E^{\alpha}(\mathbb{R}^{N})}^{4}\leqslant \|u\|_{M}^{2}\leqslant \|u\|_{E^{\alpha}(\mathbb{R}^{N})}^{2}. \]

For $\alpha < N-4$ and $t\in \mathbb {R}^{+}$, let $u_{t}(x):=t^{-\frac {2+\alpha }{2}}u(t^{-1}x)$. Then

\[ \|u_{t}\|_{M}^{2}=t^{N-4-\alpha}\|u\|_{M}^{2}. \]

Using the embeddings $E^{\alpha }(\mathbb {R}^{N})\hookrightarrow L^{p}(\mathbb {R}^{N})$, we have

(2.1)\begin{equation} \displaystyle\int_{\mathbb{R}^{N}} |u|^{p} \mathrm{d}x = t^{p\frac{2+\alpha}{2}-N} \int_{\mathbb{R}^{N}} |u_{t}|^{p} \mathrm{d}x \leqslant C_{p} t^{p\frac{2+\alpha}{2}-N} \|u_{t}\|_{E^{\alpha}(\mathbb{R}^{N})}^{p}. \end{equation}

We take

(2.2)\begin{equation} t:=\|u\|_{M}^{-\frac{2}{N-4-\alpha}}, \end{equation}

which further gives

\[ \|u_{t}\|_{M}^{2}=1 \]

and

(2.3)\begin{equation} \|u\|_{E^{\alpha}(\mathbb{R}^{N})}\leqslant 2^{\frac{1}{4}}. \end{equation}

Putting (2.2) and (2.3) into (2.1), we obtain

\[ \displaystyle\int_{\mathbb{R}^{N}} |u|^{p} \mathrm{d}x \leqslant C_{p} t^{p\frac{2+\alpha}{2}-N} \|u_{t}\|_{E^{\alpha}(\mathbb{R}^{N})}^{p} \leqslant C_{p} 2^{\frac{p}{4}} t^{p\frac{2+\alpha}{2}-N} = C_{p} 2^{\frac{p}{4}} \|u\|_{M}^{\frac{p(2+\alpha)-2N}{4+\alpha-N}}. \]

Similarly, for $\alpha >N-4$ and $t\in \mathbb {R}^{+}$, let $u_{t}(x):=t^{\frac {2+\alpha }{2}}u(tx)$, then we have the desired result.

We establish a Lions type refined Sobolev inequality.

Lemma 2.2 Let $N\geqslant 3$ and $\alpha \in (0,N)$. Then following statements are ture.

  1. (1) For $\alpha >N-4$ and $q\in (2_{\alpha }^{\sharp },2^{*}),$ the following inequality hold

    \[ \displaystyle\int_{\mathbb{R}^{N}} |u|^{q} \mathrm{d}x \leqslant C \left(\sup_{z\in \mathbb{R}^{N}} \int_{B(z,1)} |u|^{q} \mathrm{d}x\right)^{\frac{(2^{*}-2)(q-2_{\alpha}^{\sharp})}{2(2^{*}-2_{\alpha}^{\sharp})+(2^{*}-2)(q-2_{\alpha}^{\sharp})}} \|u\|_{M}^{2}, \]
    for all $u\in E^{\alpha }(\mathbb {R}^{N})$.

  2. (2) For $\alpha < N-4$ and $q\in (2^{*},2_{\alpha }^{\sharp }),$ the following inequality hold

    \[ \displaystyle\int_{\mathbb{R}^{N}} |u|^{q} \mathrm{d}x \leqslant C \left(\sup_{z\in \mathbb{R}^{N}} \int_{B(z,1)} |u|^{q} \mathrm{d}x\right)^{\frac {(2^{*}-2)(2_{\alpha}^{\sharp}-q)} {2(2_{\alpha}^{\sharp}-2^{*})+(2^{*}-2)(2_{\alpha}^{\sharp}-q)}} \|u\|_{M}^{2}, \]
    for all $u\in E^{\alpha }(\mathbb {R}^{N})$.

Proof. Let $u\in E^{\alpha }(\mathbb {R}^{N})$, $q\in (2_{\alpha }^{\sharp },2^{*})$ and $\alpha >N-4$. From Hölder's inequality and Lemma 2.1, we have

(2.4)\begin{align} & \displaystyle\int_{B(z,1)} |u|^{q}\mathrm{d}x\nonumber\\ & \quad\leqslant\left(\displaystyle\int_{B(z,1)} |u|^{2_{\alpha}^{\sharp}} \mathrm{d}x\right)^{\frac{2^{*}-q} {2^{*}-2_{\alpha}^{\sharp}}} \left(\int_{B(z,1)} |u|^{2^{*}} \mathrm{d}x\right)^{\frac{q-2_{\alpha}^{\sharp}} {2^{*}-2_{\alpha}^{\sharp}}}\nonumber\\ & \quad\leqslant C\left(\int_{B(z,1)} |\nabla u|^{2} \mathrm{d}x + \int_{B(z,1)} (I_{\alpha}*|u|^{2})|u|^{2} \mathrm{d}x\right)^{\frac{2^{*}-q} {2^{*}-2_{\alpha}^{\sharp}}} \left(\int_{B(z,1)} |\nabla u|^{2}\mathrm{d}x\right)^{\frac{2^{*}}{2}\cdot\frac{q-2_{\alpha}^{\sharp}} {2^{*}-2_{\alpha}^{\sharp}}}\nonumber\\ & \quad\leqslant C\left(\int_{B(z,1)} |\nabla u|^{2} \mathrm{d}x + \int_{B(z,1)} (I_{\alpha}*|u|^{2})|u|^{2} \mathrm{d}x\right)^{\frac{2(2^{*}-2_{\alpha}^{\sharp})+(2^{*}-2)(q-2_{\alpha}^{\sharp})} {2(2^{*}-2_{\alpha}^{\sharp})}}. \end{align}

By using (2.4), we know

\begin{align*} \displaystyle\int_{B(z,1)} |u|^{q}\mathrm{d}x & = \left(\int_{B(z,1)} |u|^{q}\mathrm{d}x\right)^{\frac{(2^{*}-2)(q-2_{\alpha}^{\sharp})}{2(2^{*}-2_{\alpha}^{\sharp})+(2^{*}-2)(q-2_{\alpha}^{\sharp})}} \left(\int_{B(z,1)} |u|^{q}\mathrm{d}x\right)^{\frac{2(2^{*}-2_{\alpha}^{\sharp})}{2(2^{*}-2_{\alpha}^{\sharp})+(2^{*}-2)(q-2_{\alpha}^{\sharp})}}\\ & \leqslant C\left(\displaystyle\int_{B(z,1)} |u|^{q} \mathrm{d}x\right)^{\frac{(2^{*}-2)(q-2_{\alpha}^{\sharp})}{2(2^{*}-2_{\alpha}^{\sharp})+(2^{*}-2)(q-2_{\alpha}^{\sharp})}}\\ & \quad\times\left(\int_{B(z,1)} |\nabla u|^{2}\mathrm{d}x + \int_{B(z,1)} (I_{\alpha}*|u|^{2})|u|^{2}\mathrm{d}x\right). \end{align*}

Covering $\mathbb {R}^{N}$ by balls of radius $1$, in such a way that each point of $\mathbb {R}^{N}$ is contained in at most $N + 1$ balls, we find

\[ \displaystyle\int_{\mathbb{R}^{N}} |u|^{q}\mathrm{d}x \leqslant C (N+1) \left(\sup_{z\in \mathbb{R}^{N}}\int_{B(z,1)} |u|^{q}\mathrm{d}x\right)^{\frac{(2^{*}-2)(q-2_{\alpha}^{\sharp})}{2(2^{*}-2_{\alpha}^{\sharp})+(2^{*}-2)(q-2_{\alpha}^{\sharp})}} \|u\|_{M}^{2}. \]

For $q\in (2^{*},2_{\alpha }^{\sharp })$, $\alpha < N-4$, from Hölder's inequality and Lemma 2.1, we have

(2.5)\begin{equation} \begin{aligned} & \displaystyle\int_{B(z,1)} |u|^{q}\mathrm{d}x\\ & \quad\leqslant\left(\displaystyle\int_{B(z,1)} |u|^{2_{\alpha}^{\sharp}} \mathrm{d}x\right)^{\frac{q-2^{*}} {2_{\alpha}^{\sharp}-2^{*}}} \left(\int_{B(z,1)} |u|^{2^{*}}\mathrm{d}x\right)^{\frac{2_{\alpha}^{\sharp}-q} {2_{\alpha}^{\sharp}-2^{*}}}\\ & \quad\leqslant C\left(\int_{B(z,1)} |\nabla u|^{2} \mathrm{d}x +\int_{B(z,1)} (I_{\alpha}*|u|^{2})|u|^{2}\mathrm{d}x\right)^{\frac{2(2_{\alpha}^{\sharp}-2^{*})+(2^{*}-2)(2_{\alpha}^{\sharp}-q)} {2(2_{\alpha}^{\sharp}-2^{*})}}. \end{aligned} \end{equation}

By using (2.5), we know

\begin{align*} \displaystyle\int_{B(z,1)} |u|^{q}\mathrm{d}x & =\left(\int_{B(z,1)} |u|^{q}\mathrm{d}x\right)^{\frac{(2^{*}-2)(2_{\alpha}^{\sharp}-q)} {2(2_{\alpha}^{\sharp}-2^{*})+(2^{*}-2)(2_{\alpha}^{\sharp}-q)}} \left(\int_{B(z,1)} |u|^{q}\mathrm{d}x\right)^{\frac {2(2_{\alpha}^{\sharp}-2^{*})} {2(2_{\alpha}^{\sharp}-2^{*})+(2^{*}-2)(2_{\alpha}^{\sharp}-q)}}\\ & \leqslant C\left(\displaystyle\int_{B(z,1)} |u|^{q} \mathrm{d}x\right)^{\frac {(2^{*}-2)(2_{\alpha}^{\sharp}-q)} {2(2_{\alpha}^{\sharp}-2^{*})+(2^{*}-2)(2_{\alpha}^{\sharp}-q)}}\\ & \quad\times\left(\int_{B(z,1)} |\nabla u|^{2}\mathrm{d}x +\int_{B(z,1)} (I_{\alpha}*|u|^{2})|u|^{2}\mathrm{d}x \right). \end{align*}

Then,

\[ \displaystyle\int_{\mathbb{R}^{N}} |u|^{q} \mathrm{d}x\leqslant C\left(\sup_{z\in \mathbb{R}^{N}}\int_{B(z,1)} |u|^{q}\mathrm{d}x \right)^{\frac {(2^{*}-2)(2_{\alpha}^{\sharp}-q)} {2(2_{\alpha}^{\sharp}-2^{*})+(2^{*}-2)(2_{\alpha}^{\sharp}-q)}} \|u\|_{M}^{2}. \]

□

As the application of Lemma 2.2, we present the Lieb's translation lemma.

Lemma 2.3 Let $N\geqslant 3,$ $\alpha \in (0,N)$ and $\{u_{n}\}$ be a bounded sequence in $E^{\alpha }(\mathbb {R}^{N})$ satisfying

Condition A $\lim \limits _{n\to \infty }\int _{\mathbb {R}^{N}} |u_{n}|^{q} \mathrm {d}x>0,$ where

\[ \begin{cases} q\in(2_{\alpha}^{\sharp},2^{*}),\quad\alpha>N-4,\\ q\in(2^{*},2_{\alpha}^{\sharp}),\quad\alpha< N-4. \end{cases} \]

Then there exists $\{x_{n}\}\subset \mathbb {R}^{N}$ such that $\{\bar {u}_{n}:=u_{n}(x+x_{n})\}$ convergence strongly and a.e. to $\bar {u}\not \equiv 0$ in $L^{q}_{loc}(\mathbb {R}^{N})$.

Proof. By applying the so-called $p,q,r$ theorem [Reference Fröhlich, Lieb and Loss7, p. 258, Lemma 2.1], [Reference Mercuri, Moroz and Van Schaftingen11, Lemma 5.3] and [Reference Mercuri, Moroz and Van Schaftingen11, Proposition 2.5], we get the desired result.

We now give another proof. According to Lemma 2.2 and Condition A, there exists $C>0$ such that

\[ \sup_{x\in \mathbb{R}^{N}} \displaystyle\int_{B(x,1)} |u_{n}(y)|^{q} \mathrm{d}y\geqslant C>0. \]

Then there exists $x_{n}\in \mathbb {R}^{N}$ such that

\[ \displaystyle\int_{B(x_{n},1)} |u_{n}(y)|^{q} \mathrm{d}y \geqslant \sup_{x\in \mathbb{R}^{N}} \int_{B(x,1)} |u_{n}(y)|^{q} \mathrm{d}y - \displaystyle\frac{C}{2n} \geqslant C_{1}>0. \]

Set $\bar {u}_{n}:=u_{n}(x+x_{n})$. Then $\|\bar {u}_{n}\|_{M}=\|u_{n}\|_{M}$ and

\[ \displaystyle\int_{B(0,1)} |\bar{u}_{n}(y)|^{q} \mathrm{d}y \geqslant C_{1}>0. \]

Up to a subsequence, there exists $\bar {u}$ such that

\[ \bar{u}_{n}\rightharpoonup \bar{u} \; \mathrm{in} ~ E^{\alpha}(\mathbb{R}^{N}), \quad \bar{u}_{n}\rightarrow \bar{u} ~ \mathrm{a.e. ~in} ~ \mathbb{R}^{N}. \]

Applying the embedding $E^{\alpha }(\mathbb {R}^{N}) \hookrightarrow L^{q}_{loc}(\mathbb {R}^{N})$ is compact, we deduce that $\bar {u}\not \equiv 0$.

3. Proof of Theorem 1.1

In this section, we assume that the assumptions of Theorem 1.1 hold.

Lemma 3.1 Let $C_{1},C_{2},C_{3}>0$ and $g_{1}:[0,\infty )\to \mathbb {R}$ as

\[ g_{1}(t) = C_{1}t^{N-2} + C_{2}t^{N+\alpha} - C_{3}t^{N+\beta}. \]

Then, $g_{1}$ has a unique critical point which corresponds to its maximum.

Proof. We compute

\[ g_{1}'(t) = C_{1}(N-2)t^{N-3} + C_{2}(N+\alpha)t^{N+\alpha-1} - C_{3}(N+\beta)t^{N+\beta-1}. \]

Clearly, $g_{1}'(t)>0$ for $t>0$ small, $g_{1}'(t)<0$ for $t>0$ large. Then there exists $\bar {t}>0$ such that $g_{1}'(\bar {t})=0$.

To prove the uniqueness of $\bar {t}$, we suppose that $0<\bar {t}<\bar {\bar {t}}$ satisfies $g_{1}'(\bar {t})=g_{1}'(\bar {\bar {t}})=0$. Then

\[ \displaystyle\frac{g_{1}'(\bar{t})}{\bar{t}^{N+\alpha-1}} = \frac{g_{1}'(\bar{\bar{t}})}{\bar{\bar{t}}^{N+\alpha-1}}=0, \]

which shows

\[ C_{1}(N-2) (\bar{t}^{N-3}-\bar{\bar{t}}^{N-3}) = C_{3}(N+\beta) (\bar{t}^{N+\beta-1}-\bar{\bar{t}}^{N+\beta-1}). \]

This yields a contradiction.

Lemma 3.2 For all $u\in E^{\alpha }(\mathbb {R}^{N}){\setminus} \{0\},$ there exists a unique $t_{u}> 0$ such that $u(\tfrac {x}{t_{u}})\in \mathcal {M}$.

Proof. Let $u\in E^{\alpha }(\mathbb {R}^{N}){\setminus} \{0\}$ and $t>0$. Set

\begin{align*} g_{2}(t)& =J(u(\displaystyle\frac{x}{t})) \\ & =\displaystyle\frac{t^{N-2}}{2} \|u\|_{D^{1,2}(\mathbb{R}^{N})}^{2} + \frac{t^{N+\alpha}}{4} \displaystyle\int_{\mathbb{R}^{N}} (I_{\alpha}*|u|^{2})|u|^{2} \mathrm{d}x -\frac{t^{N+\beta}}{4} \int_{\mathbb{R}^{N}} (I_{\beta}*|u|^{2})|u|^{2} \mathrm{d}x. \end{align*}

Clearly,

\begin{align*} g_{2}'(t)& =\displaystyle\frac{N-2}{2}t^{N-3} \|u\|_{D^{1,2}(\mathbb{R}^{N})}^{2} + \frac{N+\alpha}{4}t^{N+\alpha-1}\displaystyle\int_{\mathbb{R}^{N}} (I_{\alpha}*|u|^{2})|u|^{2} \mathrm{d}x\\ & \quad-\frac{N+\beta}{4}t^{N+\beta-1} \int_{\mathbb{R}^{N}} (I_{\beta}*|u|^{2})|u|^{2}\mathrm{d}x \end{align*}

and

(3.1)\begin{equation} tg_{2}'(t)=P\left(u\left(\displaystyle\frac{x}{t}\right)\right). \end{equation}

Combining (3.1) and Lemma 3.1, the result is valid.

Lemma 3.3

  1. (i) $\mathcal {M}\not =\emptyset$ and for all $u\in \mathcal {M},$ $J(u)=\max \limits _{t>0}J(u(\frac {x}{t}))$.

  2. (ii) $c=\inf \limits _{u\in \mathcal {M}}J(u)>0$.

  3. (iii) $c= \inf \limits _{u\in E^{\alpha }(\mathbb {R}^{N}){\setminus} \{0\}}\max \limits _{t\geqslant 0}J(u(\frac {x}{t}))$.

Proof. (i) From Lemmas 3.1 and 3.2, we have this result. (ii) For $u\in \mathcal {M}$, by Lemma 2.1, we have

\begin{align*} 0& = P(u) \!=\!\displaystyle\frac{N\!-\!2}{2} \|u\|_{D^{1,2}(\mathbb{R}^{N})}^{2} + \frac{N\!+\!\alpha}{4}\displaystyle\int_{\mathbb{R}^{N}} (I_{\alpha}*|u|^{2})|u|^{2} \mathrm{d}x -\frac{N\!+\!\beta}{4}\int_{\mathbb{R}^{N}} (I_{\beta}*|u|^{2})|u|^{2} \mathrm{d}x\\ & \geqslant\displaystyle\frac{1}{4} \|u\|_{M}^{2} - \frac{1}{4} C(N,\beta) \|u\|_{L^{\frac{4N}{N\!+\!\beta}}(\mathbb{R}^{N})}^{4}\\ & \geqslant\frac{1}{4} \|u\|_{M}^{2} - C \|u\|_{M}^{\frac{4(\tfrac{4N}{N+\beta}(2+\alpha)-2N)}{\tfrac{4N}{N+\beta}(4+\alpha-N)}}. \end{align*}

From $\frac {4(\frac {4N}{N+\beta }(2+\alpha )-2N)}{\frac {4N}{N+\beta }(4+\alpha -N)}=4+\frac {2N-2\beta -8}{4+\alpha -N}>2$, we get

\[ \|u\|_{M} >C>0. \]

Then

\begin{align*} J(u)& = J(u) -\displaystyle\frac{1}{N+\beta}P(u) \\ & =\displaystyle\frac{1}{2} \left(1-\frac{N-2}{N+\beta}\right) \|u\|_{D^{1,2}(\mathbb{R}^{N})}^{2} + \frac{1}{4}\left(1-\frac{N+\alpha}{N+\beta}\right) \displaystyle\int_{\mathbb{R}^{N}} (I_{\alpha}*|u|^{2})|u|^{2}\mathrm{d}x \\ & \geqslant\frac{1}{4} \left(1-\frac{N+\alpha}{N+\beta}\right) \|u\|_{M}^{2} \geqslant C>0. \end{align*}

This gives $c=\inf \limits _{u\in \mathcal {M}}J(u)>0$.

(iii) It follows from (ii) that

\[ \inf\limits_{u\in E^{\alpha}(\mathbb{R}^{N}){\setminus}\{0\}}\max \limits_{t\geqslant 0} J\left(u\left(\displaystyle\frac{x}{t}\right)\right) \leqslant \inf\limits_{u\in \mathcal{M}}\max \limits_{t\geqslant 0} J\left(u\left(\frac{x}{t}\right)\right) = \inf\limits_{u\in \mathcal{M}}J(u) = c. \]

Moreover, for any $u\in E^{\alpha }(\mathbb {R}^{N}){\setminus} \{0\}$, by $(\mathrm {ii})$ we have

\[ \max \limits_{t\geqslant 0} J\left(u\left(\displaystyle\frac{x}{t}\right)\right) \geqslant J\left(u\left(\frac{x}{t_{u}}\right)\right) \geqslant\inf\limits_{u\in \mathcal{M}}J(u) = c. \]

This implies

\[ \inf\limits_{u\in E^{\alpha}(\mathbb{R}^{N}){\setminus}\{0\}}\max \limits_{t\geqslant 0} J\left(u\left(\displaystyle\frac{x}{t}\right)\right) \geqslant \inf\limits_{u\in \mathcal{M}}J(u) = c. \]

Thus, the conclusion holds.

Lemma 3.4 For every $u\in E^{\alpha }(\mathbb {R}^{N}),$ $t>0,$

\begin{align*} & J(u) -\displaystyle\frac{1}{N+\beta}P(u) - J\left(u\left(\frac{x}{t}\right)\right) + \frac{t^{N+\beta}}{N+\beta}P(u) \\ & \quad=g_{3}(t) \|u\|_{D^{1,2}(\mathbb{R}^{N})}^{2} + g_{4}(t) \displaystyle\int_{\mathbb{R}^{N}} (I_{\alpha}*|u|^{2})|u|^{2} \mathrm{d}x, \end{align*}

where

\begin{align*} g_{3}(t)& =\displaystyle\frac{1}{2} \left(1-\frac{N-2}{N+\beta}- t^{N-2}+\frac{N-2}{N+\beta}t^{N+\beta}\right) > g_{3}(1)=0, \quad t\in(0,1)\cup (1,\infty), \\ g_{4}(t)& = \displaystyle\frac{1}{4} \left(1-\frac{N+\alpha}{N+\beta} - t^{N+\alpha} + \frac{N+\alpha}{N+\beta}t^{N+\beta} \right)> g_{4}(1)=0, \quad t\in(0,1)\cup(1,\infty). \end{align*}

Proof. Clearly,

\begin{align*} & J(u) -\displaystyle\frac{1}{N+\beta}P(u) \\ & \quad=\frac{1}{2} \left(1-\frac{N-2}{N+\beta}\right) \|u\|_{D^{1,2}(\mathbb{R}^{N})}^{2} + \frac{1}{4}\left(1-\frac{N+\alpha}{N+\beta}\right) \displaystyle\int_{\mathbb{R}^{N}} (I_{\alpha}*|u|^{2})|u|^{2} \mathrm{d}x \end{align*}

and

\begin{align*} & \displaystyle\frac{t^{N+\beta}}{N+\beta} P(u) - J\left(u\left(\frac{x}{t}\right)\right)\\ & \quad=\frac{1}{2}\left(\frac{N-2}{N+\beta}t^{N+\beta} - t^{N-2}\right) \|u\|_{D^{1,2}(\mathbb{R}^{N})}^{2}\\ & \quad\quad + \frac{1}{4}\left(\frac{N+\alpha}{N+\beta}t^{N+\beta} - t^{N+\alpha}\right) \displaystyle\int_{\mathbb{R}^{N}} (I_{\alpha}*|u|^{2})|u|^{2} \mathrm{d}x. \end{align*}

Combining the above two expressions, we have the desired results.

Lemma 3.5 Let $\{u_{n}\}\subset \mathcal {M}$ be a bounded minimizing sequence of $c>0$ as follows:

\[ J(u_{n})\to c\quad\text{and } P(u_{n})=0\text{ as } n\to\infty. \]

If $u_{n}\rightharpoonup u$ in $E^{\alpha }(\mathbb {R}^{N}),$ then $P(u)=0$. Moreover, if $u\not \equiv 0,$ then $J(u)=c$.

Proof. Let $\{u_{n}\}\subset \mathcal {M}$ be a bounded minimizing sequence of $c>0$. Up to a subsequence, there exists $u$ such that

\[ u_{n}\rightharpoonup u \mathrm{in} ~ E^{\alpha}(\mathbb{R}^{N}), \quad u_{n}\rightarrow u ~ \mathrm{a.e. ~in} ~\mathbb{R}^{N} \ \text{and } u_{n}\rightarrow u ~\mathrm{in} ~ L^{r}(\mathbb{R}^{N}) \text{ for all } r\in(2^{*},2_{\alpha}^{\sharp}). \]

For $u\equiv 0$, we have $P(u)=0$.

From now on, we suppose $u\not \equiv 0$. Firstly, we show $P(u)\geqslant 0$. By the way of contradiction, we suppose that $P(u)<0$. In terms of Lemma 3.2, there exists a unique $\bar {t}>0$ such that

\[ P\left(u\left(\displaystyle\frac{x}{\bar{t}}\right)\right)=0. \]

In view of (3.1) and $P(u)<0$, we obtain

\[ g_{2}'(1)<0. \]

By the expression of (3.1), one can choose $t_{0}>0$ small such that

\[ g_{2}'(t_{0})>0. \]

Using the Intermediate value theorem, we know there exists $\bar {\bar {t}}\in (t_{0},1)$ such that

\[ g_{2}'(\bar{\bar{t}})=0. \]

By the uniqueness of $\bar {t}$, one has

(3.2)\begin{equation} \bar{t} =\bar{\bar{t}}\in(t_{0},1). \end{equation}

Making use of (3.2), there holds

\begin{align*} c& \leqslant J\left(u\left(\displaystyle\frac{x}{\bar{t}}\right)\right) = J\left(u\left(\frac{x}{\bar{t}}\right)\right) - \frac{1}{N+\beta} P\left(u\left(\frac{x}{\bar{t}}\right)\right)\\ & =\frac{1}{2}\left(1-\frac{N-2}{N+\beta}\right) \bar{t}^{N-2} \|u\|_{D^{1,2}(\mathbb{R}^{N})}^{2} + \frac{1}{4}\left(1-\frac{N+\alpha}{N+\beta}\right) \bar{t}^{N+\alpha}\displaystyle\int_{\mathbb{R}^{N}} (I_{\alpha}*|u|^{2})|u|^{2} \mathrm{d}x\\ & <\frac{1}{2}\left(1-\frac{N-2}{N+\beta}\right) \|u\|_{D^{1,2}(\mathbb{R}^{N})}^{2} + \frac{1}{4} \left(1-\frac{N+\alpha}{N+\beta}\right) \int_{\mathbb{R}^{N}} (I_{\alpha}*|u|^{2})|u|^{2} \mathrm{d}x\\ & \leqslant\displaystyle\frac{1}{2} \left(1-\frac{N-2}{N+\beta}\right) \lim_{n\rightarrow\infty} \|u_{n}\|_{D^{1,2}(\mathbb{R}^{N})}^{2} + \frac{1}{4} \left(1-\frac{N+\alpha}{N+\beta}\right) \lim_{n\rightarrow\infty} \displaystyle\int_{\mathbb{R}^{N}} (I_{\alpha}*|u_{n}|^{2})|u_{n}|^{2} \mathrm{d}x\\ & =\lim_{n\rightarrow\infty} \left[J(u_{n}) -\frac{1}{N+\beta} P(u_{n})\right] \\ & =\lim_{n\rightarrow\infty} J(u_{n}) = c. \end{align*}

This is a contradiction. Hence, $P(u)\geqslant 0$.

We prove $P(u)=0$. Suppose that $P(u)>0$. From $u\not \equiv 0$, we know

(3.3)\begin{equation} J(u) -\displaystyle\frac{1}{N+\beta} P(u) >0. \end{equation}

Set $v_{n}=u_{n}-u$. According to Brézis-Lieb lemma [Reference Brézis and Lieb4], we know

\[ c = J(u_{n})=J(v_{n})+J(u) \]

and

\[ 0=P(u_{n})=P(v_{n})+P(u), \]

which further implies

(3.4)\begin{equation} \begin{aligned} c & = J(u_{n})-\displaystyle\frac{1}{N+\beta} P(u_{n}) \\ & =\left[J(v_{n}) -\frac{1}{N+\beta} P(v_{n})\right] + \left[J(u)-\frac{1}{N+\beta}P(u)\right] \end{aligned} \end{equation}

and

\[ P(v_{n})={-}P(u)<0. \]

As discussed in (3.2), there exists a unique $\bar {t}_{n}$ such that

(3.5)\begin{equation} P\left(v_{n}\left(\displaystyle\frac{x}{\bar{t}_{n}}\right)\right)=0 \text{ and }\bar{t}_{n}\in(0,1). \end{equation}

From (3.3) to (3.5), we obtain

\begin{align*} c& \leqslant J\left(v_{n}\left(\displaystyle\frac{x}{\bar{t}_{n}}\right)\right) = J\left(v_{n}\left(\frac{x}{\bar{t}_{n}}\right)\right) - \frac{1}{N+\beta} P\left(v_{n}\left(\frac{x}{\bar{t}_{n}}\right)\right)\\ & < J(v_{n}) -\frac{1}{N+\beta} P(v_{n})\\ & <\left[J(v_{n}) -\frac{1}{N+\beta} P(v_{n})\right] + \left[J(u)-\frac{1}{N+\beta}P(u)\right]\\ & = J(u_{n}) -\displaystyle\frac{1}{N+\beta} P(u_{n}) = c+o_{n}(1). \end{align*}

This is a contradiction. Hence, $P(u)=0$.

Moreover, if $u\not \equiv 0$, then

\[ c\leqslant J(u) = J(u)- \displaystyle\frac{1}{N+\beta} P(u) \leqslant J(u_{n})- \frac{1}{N+\beta} P(u_{n}) = c+o_{n}(1), \]

which implies $J(u)=c$.

Proof Proof of Theorem 1.1

We process the proof into two steps.

Step 1. Let $\{u_{n}\}\subset \mathcal {M}$ be a minimizing sequence of $c\in (0,c^{*})$. Clearly, it is bounded in $E^{\alpha }(\mathbb {R}^{N})$. From $\alpha <\beta < N-4$, we know

\[ 2^{*} <\displaystyle\frac{4N}{N+\beta} < 2_{\alpha}^{\sharp}. \]

If $\lim \limits _{n\to \infty }\int _{\mathbb {R}^{N}} |u_{n}|^{\frac {4N}{N+\beta }} \mathrm {d}x=0$, then using (1.1) and the definition of minimizing sequence $\{u_{n}\}$, we have

\[ c+o(1)=\displaystyle\frac{1}{2}\|u_{n}\|_{D^{1,2}(\mathbb{R}^{N})}^{2} + \frac{1}{4} \displaystyle\int_{\mathbb{R}^{N}} (I_{\alpha}*|u_{n}|^{2})|u_{n}|^{2} \mathrm{d}x \]

and

\[ 0=\displaystyle\frac{N-2}{2}\|u_{n}\|_{D^{1,2}(\mathbb{R}^{N})}^{2} + \frac{N+\alpha}{4} \displaystyle\int_{\mathbb{R}^{N}} (I_{\alpha}*|u_{n}|^{2})|u_{n}|^{2} \mathrm{d}x, \]

which further gives

\[ c=0. \]

This is a contradiction. Hence, $\lim \limits _{n\to \infty }\int _{\mathbb {R}^{N}} |u_{n}|^{\frac {4N}{N+\beta }} \mathrm {d}x>0$.

By Lemma 2.3 and $\lim \limits _{n\to \infty }\int _{\mathbb {R}^{N}} |u_{n}|^{\frac {4N}{N+\beta }} \mathrm {d}x>0$, we know that there exists $\{x_{n}\}\subset \mathbb {R}^{N}$ such that $\{\bar {u}_{n}:=u_{n}(x+x_{n})\}$ convergence strongly and a.e. to $\bar {u}\not \equiv 0$ in $L^{q}_{loc}(\mathbb {R}^{N})$.

Clearly, $\{\bar {u}_{n}\}$ satisfies

\[ J(\bar{u}_{n})\to c\text{ and } P(\bar{u}_{n})=0\text{ as } n\to\infty. \]

Moreover, from Lemma 3.5, we get

\[ P(\bar{u})=0~\text{ and } J(\bar{u})=c. \]

Step 2. We show

\[ \|J'(\bar{u})\|_{(E^{\alpha}(\mathbb{R}^{N}))^{{-}1}}=0. \]

Otherwise, there exist $\delta,\mu >0$ such that

\[ \|J'(w)\|_{(E^{\alpha}(\mathbb{R}^{N}))^{{-}1}}\geqslant\mu, \]

for all $w\in B_{3\delta }(\bar {u}) := \big \{w\in E^{\alpha }(\mathbb {R}^{N}) \big | \|w-\bar {u}\|_{E^{\alpha }(\mathbb {R}^{N})} \leqslant 3\delta \big \}$.

Using the fact that $C^{\infty }_{0}(\mathbb {R}^{N})$ is dense in $E^{\alpha }(\mathbb {R}^{N})$, one has

\[ \lim_{t\to 1}\left\|\bar{u}(\displaystyle\frac{\cdot }{t})-\bar{u}\right\|_{E^{\alpha}(\mathbb{R}^{N})} =0. \]

Then, there exists $\delta _{1}\in (0,\frac {1}{4})$ such that

(3.6)\begin{equation} |t-1|<\delta_{1}\Rightarrow\left\|\bar{u}\left(\displaystyle\frac{\cdot }{t}\right)-\bar{u}\right\|_{E^{\alpha}(\mathbb{R}^{N})}<\delta. \end{equation}

According to Lemma 3.4 and $P(\bar {u})=0$, for any $t>0$, we get

(3.7)\begin{equation} \begin{aligned} J\left(\bar{u}\left(\displaystyle\frac{x}{t}\right)\right) & =J(\bar{u}) - \frac{1}{N+\beta}P(\bar{u}) +\frac{t^{N+\beta}}{N+\beta}P(\bar{u})\\ & \quad- g_{3}(t) \|\bar{u}\|_{D^{1,2}(\mathbb{R}^{N})}^{2} - g_{4}(t) \displaystyle\int_{\mathbb{R}^{N}} (I_{\alpha}*|\bar{u}|^{2})|\bar{u}|^{2}\mathrm{d}x\\ & = J(\bar{u}) - g_{3}(t) \|\bar{u}\|_{D^{1,2}(\mathbb{R}^{N})}^{2} - g_{4}(t) \int_{\mathbb{R}^{N}} (I_{\alpha}*|\bar{u}|^{2})|\bar{u}|^{2} \mathrm{d}x\\ & \leqslant J(\bar{u}) - g_{3}(t) \|\bar{u}\|_{D^{1,2}(\mathbb{R}^{N})}^{2}. \end{aligned} \end{equation}

Let

(3.8)\begin{equation} 0<\varepsilon <\min\left\{\displaystyle\frac{g_{3}(0.5)}{2} \|\bar{u}\|_{D^{1,2}(\mathbb{R}^{N})}^{2} , \frac{g_{3}(1.5)}{2} \|\bar{u}\|_{D^{1,2}(\mathbb{R}^{N})}^{2} ,1,\frac{\mu\delta}{8} \right\}, \end{equation}

By [Reference Willem13], there exists a deformation $\eta \in C([0,1]\times E^{\alpha }(\mathbb {R}^{N}), E^{\alpha }(\mathbb {R}^{N}) )$ such that

\begin{align*} (a)\quad& \eta(1,w)=w, \ \, w\not\in J^{{-}1}([ c-2\varepsilon , c+2\varepsilon]) \cap B_{2\delta}(\bar{u}); \\ (b)\quad& \eta(1,J^{c+\varepsilon}\cap B_{\delta}(\bar{u})) =J^{c-\varepsilon}; \\ (c)\quad& J(\eta(1,w))\leqslant J(w), \ \, w\in E^{\alpha}(\mathbb{R}^{N}); \\ (d)\quad& \|\eta(1,w)-w\|_{E^{\alpha}(\mathbb{R}^{N})}\leqslant \delta, \quad w\in E^{\alpha}(\mathbb{R}^{N}). \end{align*}

In view of Lemma 3.2, $J(\bar {u}(\frac {x}{t}))\leqslant J(\bar {u})=c$ for $t>0$, then it follows from (3.6) and $(b)$ that

(3.9)\begin{equation} J(\eta(1,\bar{u}(\displaystyle\frac{x}{t}))) \leqslant c-\varepsilon, \quad t>0, \quad |t-1|<\delta_{1}. \end{equation}

On the other hand, by using (3.7) and $(c)$, we obtain

(3.10)\begin{equation} \begin{aligned} J\left(\eta\left(1,\bar{u}\left(\displaystyle\frac{x}{t}\right)\right)\right) & \leqslant J\left(\bar{u}\left(\frac{x}{t}\right)\right)\\ & \leqslant J(\bar{u}) - g_{3}(t) \|\bar{u}\|_{D^{1,2}(\mathbb{R}^{N})}^{2}\\ & = c - g_{3}(t) \|\bar{u}\|_{D^{1,2}(\mathbb{R}^{N})}^{2}\\ & \leqslant c -\delta_{2} \|\bar{u}\|_{D^{1,2}(\mathbb{R}^{N})}^{2}, \quad t>0, \quad |t-1| \geqslant\delta_{1}, \end{aligned} \end{equation}

where $\delta _{2} = \min \{ g_{3}(1-\delta _{1}) , g_{3}(1+\delta _{1}) \} >0$.

Combined (3.9) with (3.10), we have

\[ \max_{t\in[0.5,1.5]} J\left(\eta\left(1,\bar{u}\left(\displaystyle\frac{x}{t}\right)\right)\right)<\tilde{c}. \]

Then we prove $\eta (1,\bar {u}(\tfrac {x}{t}))\cap \mathcal {M} \not =\emptyset$ for some $t\in [0.5,1.5]$. Define

\[ \Psi_{0}(t) = P\left(\bar{u}\left(\displaystyle\frac{x}{t}\right)\right) \text{ and } \Psi_{1}(t)=P\left(\eta\left(1,\bar{u}\left(\frac{x}{t}\right)\right)\right),\quad t>0. \]

From Lemma 3.2 and the degree theory, we get

\[ \mathrm{deg}(\Psi_{0},(0.5, 1.5),0) \not=0. \]

Moreover, we can derive from (3.7), (3.8), and $(a)$ that $\eta (1,\bar {u}(\tfrac {x}{t}))=\bar {u}(\tfrac {x}{t})$ for $t =0.5$ and $t =1.5$. Thus, we have

\[ \mathrm{deg}(\Psi_{1},(0.5, 1.5),0) = \mathrm{deg}(\Psi_{0},(0.5, 1.5),0) \not= 0. \]

Hence, $\Psi _{1}(t_{0})=0$ for some $t_{0}\in (0.5,1.5)$, which implies $\eta (1,\bar {u}(\frac {x}{t_{0}}))\in \mathcal {M}$. This is a contradiction. Hence,

\[ J(\bar{u})=c, ~P(\bar{u})=0~\text{ and }\langle J'(\bar{u}),\bar{u}\rangle=0. \]

Acknowledgements

Y. Su is supported by the National Natural Science Foundation of China (Grant No. 12101006).

Conflict interest

We declare that we have no conflict of interests.

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