Ground State Solutions for p-Fractional Choquard-Kirchhoff Equations Involving Electromagnetic Fields and Critical Nonlinearity

Xueqian Yan

School of Mathematics, Liaoning Normal University, Dalian, China.

DOI: 10.4236/oalib.1108837                

Abstract

This paper is concerned with the existence of ground state solutions for p-fractional Choquard-Kirchhoff equations involving electromagnetic fields and critical nonlinearity. Under assumptions on the nonlinear term, by applying the method of Nehari manifold, we obtain that the equation possesses a ground state solution.

Keywords

Fractional Choquard-Kirchhoff Equation, Magnetic Fractional p-Laplacian, Critical Growth

1. Introduction

We consider the existence of ground state solutions for following p-fractional Choquard-Kirchhoff equations with electromagnetic fields and critical growth

$\begin{array}{l}\left(a+b{\left[u\right]}_{s\mathrm{,}A}^p\right){\left(-\Delta \right)}_{p\mathrm{,}A}^{s}u+V\left(x\right){\left|u\right|}^{p-2}u\\ =\mu h\left(x\mathrm{,}{\left|u\right|}^2\right)u+\left[I_{\sigma }\left(x\right)\ast {\left|u\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\right]{\left|u\right|}^{p_{\sigma \mathrm{,}s}^{\ast }-2}u\mathrm{,}x\in {ℝ}^N\mathrm{,}\end{array}$ (1.1)

where $a,b>0$, $\mu >0$, $I_{\sigma }\left(x\right)={\left|x\right|}^{-\sigma }$ is the Riesz potential. ${\left(-\Delta \right)}_{p\mathrm{,}A}^s$ denotes the p-fractional magnetic operator with $0<s<1$, $2\le p<N/s$, $0<\sigma <2ps$

and $p_{\sigma \mathrm{,}s}^{\ast }=\frac{p\left(2N-\sigma \right)}{2\left(N-ps\right)}$. $A\in C\left({ℝ}^N\mathrm{,}{ℝ}^N\right)$ and $V\in C\left({ℝ}^N\mathrm{,}ℝ\right)$ are the electric and magnetic potentials, respectively. h is a continuous function satisfying some conditions.

When $p=2$, the fractional magnetic Laplacian ${\left(-\Delta \right)}_A^s$, up to normalization constants, which is defined on smooth functions u as

${\left(-\Delta \right)}_A^{s}u\left(x\right)=2\underset{\eta \to 0}{lim}{\displaystyle {\int }_{{ℝ}^N\backslash B_{\eta }\left(x\right)}}\frac{u\left(x\right)-{\text{e}}^{i\left(x-y\right)\cdot A\left(\frac{x+y}{2}\right)}u\left(y\right)}{{\left|x-y\right|}^{N+2s}}\text{d}y\mathrm{,}x\in {ℝ}^N\mathrm{.}$ (1.2)

Here, $B_{\eta }\left(x\right)$ denotes the ball of ${ℝ}^N$ centered at $x\in {ℝ}^N$ and of radius $\eta >0$. This operator was defined by d’Avenia and Squassina [1], and it can be considered as the fractional counterpart of the magnetic Laplacian

${\left(-\Delta \right)}_{A}u\mathrm{<mo>\; :\; </mo>}={\left(\frac{1}{i}\nabla -A\right)}^{2}u=-\Delta u-\frac{2}{i}A\left(x\right)\cdot \nabla u+{\left|A\left(x\right)\right|}^{2}u-\frac{1}{i}u\text{div}\left(A\left(x\right)\right)\mathrm{,}$ (1.3)

which plays a fundamental role in quantum mechanics in the description of the dynamics of the particle in a non-relativistic setting. In this context, the curl of A represents magnetic field acting on a charged particle. Motivated by this fact, many authors dealt with the existence of nontrivial solutions of the Schrödinger equations with magnetic fields.

For more details on fractional magnetic operators, we refer to d’Avenia and Squassina [1], and for the physical background, we can refer to previous studies [2] and [3]. This paper was inspired by previous works concerning the magnetic Schrödinger equations. Next, let us mention some enlightening works related to the problem (1.1). Recently, a great attention has been devoted to the study of the following fractional magnetic Schrödinger equation

${\epsilon }^{2s}{\left(-\Delta \right)}_A^{s}u+V\left(x\right)u=f\left(x,{\left|u\right|}^2\right)u,x\in {ℝ}^N.$ (1.4)

For instance, Ambrosio and d’Avenia established with the existence and multiplicity of solutions to (1.4) for small $\epsilon >0$, when f has a subcritical growth and the potential V satisfies some global conditions, by applying variational methods and Ljusternick-Schnirelmann theory in [4]. By employing the fractional version of the concentration compactness principle and variational methods, Liang et al., in [5], studied the existence and multiplicity of solutions for the fractional Schrödinger-Kirchhoff equations with external magnetic operator and critical nonlinearity

$(\begin{array}{l}{\epsilon }^{2s}M\left({\left[u\right]}_{s\mathrm{,}{A}_{\epsilon }}^2\right){\left(-\Delta \right)}_{A_{\epsilon }}^{s}u+V\left(x\right)u={\left|u\right|}^{2_s^{\ast }-2}u+h\left(x\mathrm{,}{\left|u\right|}^2\right)u\mathrm{,}x\in {ℝ}^N\mathrm{,}\\ u\left(x\right)\to \mathrm{0,}\text{as}\left|x\right|\to \infty \mathrm{.}\end{array}$ (1.5)

Others related fractional Schrödinger-Kirchhoff equations can be seen in [6] - [11]. Moreover, as mentioned above, if the magnetic field $A\equiv 0$, the operator ${\left(-\Delta \right)}_{p\mathrm{,}A}^s$ can be reduced to the p-fractional Laplacian operator ${\left(-\Delta \right)}_p^s$, up to normalization constants, which is defined as

${\left(-\Delta \right)}_p^{s}u\left(x\right)=2\underset{\eta \to 0}{\lim }{\displaystyle {\int }_{{ℝ}^N\backslash B_{\eta }\left(x\right)}}\frac{{\left|u\left(x\right)-u\left(y\right)\right|}^{p-2}\left(u\left(x\right)-u\left(y\right)\right)}{{\left|x-y\right|}^{N+ps}}\text{d}y\mathrm{,}x\in {ℝ}^N\mathrm{,}$ (1.6)

where $B_{\eta }\left(x\right):=\left\{y\in {ℝ}^N:\left|x-y\right|<\eta \right\}$. There are also some interesting results that are obtained by using some different approaches under various hypotheses on the potential and the nonlinearity. Xiang et al. [12] obtained weak solutions for the following Kirchhoff type problem involving the fractional p-Laplacian by using the mountain pass theorem

$(\begin{array}{ll}M\left({\displaystyle {\iint }_{{ℝ}^{2N}}}{\left|u\left(x\right)-u\left(y\right)\right|}^{p}K\left(x-y\right)\text{d}x\text{d}y\right)L_K^{p}u=f\left(x,u\right)\hfill & \text{in}\Omega ,\hfill \\ u=0\hfill & \text{in}{ℝ}^N\backslash \Omega .\hfill \end{array}$ (1.7)

Iannizzotto et al. [13] studied a class of quasilinear nonlocal problems involving the fractional p-Laplacian and obtained the existence and multiplicity of solutions by Morse theory. In [14], the authors investigated the existence of weak solutions for a perturbed nonlinear elliptic equation driven by the fractional p-Laplacian operator by variational methods.

For the Choquard equation, we refer to [15], Shen et al. considered the following Choquard equation, and proved that the existence of ground states for it by variational methods

${\left(-\Delta \right)}^{s}u+u=\left({\left|x\right|}^{-\mu }\ast F\left(u\right)\right)f\left(u\right)\mathrm{.}$ (1.8)

And in [16], by applying the variational methods, Ma and Zhang obtained the existence and multiplicity of weak solutions, considering the following fractional Choquard equation with critical nonlinearity

${\left(-\Delta \right)}^{\frac{\alpha }{2}}u+\left(\lambda V\left(x\right)-\beta \right)u=\left[{\left|x\right|}^{-\mu }\ast {\left|u\right|}^{2_{\mu }^{\ast }}\right]{\left|u\right|}^{2_{\mu }^{\ast }-2}u\mathrm{,}x\in {ℝ}^N\mathrm{.}$ (1.9)

And Li et al. [17] obtained a ground state solution for fractional Choquard equation involving upper critical exponent. For others related, we can see [18] - [24]. It is worth mentioning that Li et al., in [25], established that the following fractional equation has a ground state solution by the Nehari methods, when $\lambda$ is quite large

${\left(-\Delta \right)}^{s}u+V\left(x\right)u={\left|u\right|}^{2_s^{\ast }-2}u+\lambda f\left(x,u\right),x\in {ℝ}^N.$ (1.10)

We borrowed some brilliant ideas from them, while the structure of Choquard-Kirchhoff equations and appearance of the magnetic fields, such that our results are different from theirs and extend their results in some degree.

Inspired by the above works, in this paper, we focus our attention on the existence of ground state solutions to (1.1). To our best knowledge, there are a few results in the literature to study the p-fractional Choquard-Kirchhoff equations with electromagnetic fields and critical growth. Some difficulties arise when dealing with this problem, the main difficulty origins from the strongly nonlocality in the sense that the leading operator takes care of the behavior of the solutions in the whole space. Indeed, the appearance of the magnetic fields and the existence of criticality also bring additional difficulties into the study of our problem, such as the effects of the magnetic fields on the linear spectral sets and on the structure of solutions, and the possible interactions between the magnetic fields and the linear potentials. Therefore, we need to take more considerations to overcome the difficulties induced by these new traits.

The main goal of this paper is to investigate the existence of ground state solutions for the problem (1.1), when $\mu >0$ is sufficiently large, $A\in C\left({ℝ}^N\mathrm{,}{ℝ}^N\right)$, under assumptions (V₁) - (V₂) on the potential V and h is a superlinear but subcritical function satisfying the following conditions. Let K be the class of functions $k\in L^{\infty }\left({ℝ}^N\right)$ such that for every $\delta >0$, the set $\left\{x\in {ℝ}^N\mathrm{<mo>\; :\; </mo>}\left|k\left(x\right)\right|\ge \delta \right\}$ has a finite Lebesgue measure. We shall assume that V satisfies

(V₁) $V\in L^{\infty }\left({ℝ}^N\right)$ and $V_0={\inf }_{x\in {ℝ}^N}V\left(x\right)>0$.

(V₂) There exists a function $V_{\alpha }\in L^{\infty }\left({ℝ}^N\right)$, which is 1-periodic in $x_i\left(i=1,\cdots ,N\right)$, such that $V\left(x\right)-V_{\alpha }\left(x\right)\in K$ and $V\left(x\right)\le V_{\alpha }\left(x\right)$ for all $x\in {ℝ}^N$.

And h satisfies the assumptions:

(h₁) $h\in C\left({ℝ}^N\times ℝ\mathrm{,}ℝ\right)$ and there exists $p<q<p_s^{\ast }$ such that

$\left|h\left(x\mathrm{,}t\right)\right|\le C\left(1+{\left|t\right|}^{\frac{q-2}{2}}\right)$

for all $\left(x\mathrm{,}t\right)\in {ℝ}^N\times ℝ$, where C is a positive constant.

(h₂) $h\left(x\mathrm{,}t\right)=o\left(1\right)$ uniformly in $x\in {ℝ}^N$ as $\left|t\right|\to 0$.

(h₃) $th\left(x\mathrm{,}t\right)-pH\left(x\mathrm{,}t\right)\ge \omega th\left(x\mathrm{,}\omega t\right)-pH\left(x\mathrm{,}\omega t\right)$ for all $x\in {ℝ}^N\times ℝ$ and $\omega \in \left[\mathrm{0,1}\right]$, where $H\left(x\mathrm{,}t\right)={\displaystyle {\int }_0^t}h\left(x\mathrm{,}\tau \right)\text{d}\tau$.

(h₄) $h\left(x,t\right)t>0$ for all $\left(x\mathrm{,}t\right)\in {ℝ}^N\times ℝ\backslash \left\{0\right\}$.

(h₅) There exists a function $h_{\alpha }\in C\left({ℝ}^N\times ℝ\mathrm{,}ℝ\right)$, which is 1-periodic in $x_i\left(i=1,\cdots ,N\right)$, such that

1) $\left|h_{\alpha }\left(x\mathrm{,}t\right)\right|\le \left|h\left(x\mathrm{,}t\right)\right|\mathrm{,}\forall \left(x\mathrm{,}t\right)\in {ℝ}^N\times ℝ$ ;

2) $\left|h_{\alpha }\left(x\mathrm{,}t\right)-h\left(x\mathrm{,}t\right)\right|\le \left|k\left(x\right)\right|\left(1+{\left|t\right|}^{\frac{q-2}{2}}\right)\mathrm{,}\forall \left(x\mathrm{,}t\right)\in {ℝ}^N\times ℝ$, where $k\in K$ and q is given by (h₁);

3) $t{h}_{\alpha }\left(x\mathrm{,}t\right)-p{H}_{\alpha }\left(x\mathrm{,}t\right)\ge \omega t{h}_{\alpha }\left(x\mathrm{,}\omega t\right)-p{H}_{\alpha }\left(x\mathrm{,}\omega t\right)$ for all $x\in {ℝ}^N\times ℝ$, and $\omega \in \left[\mathrm{0,1}\right]$, where $H_{\alpha }\left(x\mathrm{,}t\right)={\displaystyle {\int }_0^t}{h}_{\alpha }\left(x\mathrm{,}\tau \right)\text{d}\tau$ ;

4) $h_{\alpha }\left(x\mathrm{,}t\right)t\ge 0$ for all $\left(x\mathrm{,}t\right)\in {ℝ}^N\times ℝ$.

The main result of this paper can be summarized as follows:

Theorem 1.1. Let $0<s<1$, $2\le p<N/s$, $0<\sigma <2ps$. Assume that $A\in C\left({ℝ}^N\mathrm{,}{ℝ}^N\right)$, V satisfies (V₁) - (V₂) and h satisfies (h₁) - (h₅). Then there exists ${\mu }^{\ast }>0$ such that for each $\mu >{\mu }^{\ast }$, problem (1.1) possesses a positive ground state solution.

2. Preliminaries

Let $0<s<1$, $2\le p<N/s$. The magnetic Gagliardo seminorm is defined by

${\left[u\right]}_{s\mathrm{,}A}={\left({\displaystyle {\iint }_{{ℝ}^{2N}}}\frac{{\left|u\left(x\right)-{\text{e}}^{i\left(x-y\right)\cdot A\left(\frac{x+y}{2}\right)}u\left(y\right)\right|}^p}{{\left|x-y\right|}^{N+ps}}\text{d}x\text{d}y\right)}^{\frac{1}{p}}\mathrm{,}$

and $W_A^{s\mathrm{,}p}\left({ℝ}^N\mathrm{,}ℂ\right)$ is denoted by

$W_A^{s\mathrm{,}p}\left({ℝ}^N\mathrm{,}ℂ\right)=\left\{u\in L^p\left({ℝ}^N\mathrm{,}ℂ\right)\mathrm{<mo>\; :\; </mo>}{\left[u\right]}_{s\mathrm{,}A}^p<+\infty \right\}\mathrm{,}$

endowed with the norm

${\Vert u\Vert }_{W_A^{s\mathrm{,}p}\left({ℝ}^N\mathrm{,}ℂ\right)}={\left({\left[u\right]}_{s\mathrm{,}A}^p+{\left|u\right|}_p^p\right)}^{\frac{1}{p}}\mathrm{.}$

In view of the V, the subspace of $W_A^{s\mathrm{,}p}\left({ℝ}^N\mathrm{,}ℂ\right)$ is defined by

$X=\left\{u\in W_A^{s\mathrm{,}p}\left({ℝ}^N\mathrm{,}ℂ\right)\mathrm{<mo>\; :\; </mo>}{\displaystyle {\int }_{{ℝ}^N}}V\left(x\right){\left|u\right|}^p\text{d}x<+\infty \right\}\mathrm{,}$

where the norm

${\Vert u\Vert }_{s\mathrm{,}A}={\left(a{\left[u\right]}_{s\mathrm{,}A}^p+{\displaystyle {\int }_{{ℝ}^N}}V\left(x\right){\left|u\right|}^p\text{d}x\right)}^{\frac{1}{p}}\mathrm{.}$

On account of (V₁) - (V₂), we know that the norms ${\Vert u\Vert }_{s\mathrm{,}A}$ and

${\Vert u\Vert }_{s\mathrm{,}\alpha \mathrm{,}A}={\left(a{\left[u\right]}_{s\mathrm{,}A}^p+{\displaystyle {\int }_{{ℝ}^N}}{V}_{\alpha }\left(x\right){\left|u\right|}^p\text{d}x\right)}^{\frac{1}{p}}$

are equivalent. In addition, the best constant of Hardy-Littlehood-Sobolev inequality is

$S=\underset{u\in W^{s\mathrm{,}p}\left({ℝ}^N\right)\backslash \left\{0\right\}}{\inf }\frac{{\left[u\right]}_s^p}{{\left({\displaystyle {\int }_{{ℝ}^N}}\left[I_{\sigma }\left(x\right)\ast {\left|u\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\right]{\left|u\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\text{d}x\right)}^{\frac{p}{2p_{\sigma \mathrm{,}s}^{\ast }}}}\mathrm{,}$ (2.1)

where ${\left[u\right]}_s^p$ is Gagliardo seminorm defined in $W^{s\mathrm{,}p}\left({ℝ}^N\right)$. We will show the existence of ground solutions of (1.1) by searching for the critical points of energy functional associated to (1.1)

$\begin{array}{c}J\left(u\right)=\frac{1}{p}{\displaystyle {\int }_{{ℝ}^N}}\left(a{\left[u\right]}_{s\mathrm{,}A}^p+V\left(x\right){\left|u\right|}^p\right)\text{d}x+\frac{b}{2p}{\left[u\right]}_{s\mathrm{,}A}^{2p}-\frac{\mu }{2}{\displaystyle {\int }_{{ℝ}^N}}H\left(x\mathrm{,}{\left|u\right|}^2\right)\text{d}x\\ -\frac{1}{2p_{\sigma \mathrm{,}s}^{\ast }}{\displaystyle {\int }_{{ℝ}^N}}\left[I_{\sigma }\left(x\right)\ast {\left|u\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\right]{\left|u\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\text{d}x\mathrm{.}\end{array}$

The Nehari manifolds can be defined on X as follows:

$\mathcal{N}\mathrm{<mo>\; :\; </mo>}=\left\{u\in X\backslash \left\{0\right\}\mathrm{<mo>\; :\; </mo>}\langle J^{\prime }\left(u\right)\mathrm{,}u\rangle =0\right\}$

and

${\mathcal{N}}_{\alpha }\mathrm{<mo>\; :\; </mo>}=\left\{u\in X\backslash \left\{0\right\}\mathrm{<mo>\; :\; </mo>}\langle {J^{\prime }}_{\alpha }\left(u\right)\mathrm{,}u\rangle =0\right\}\mathrm{,}$

where

$\begin{array}{c}{J}_{\alpha }\left(u\right)=\frac{1}{p}{\displaystyle {\int }_{{ℝ}^N}}\left(a{\left[u\right]}_{s\mathrm{,}A}^p+V\left(x\right){\left|u\right|}^p\right)\text{d}x+\frac{b}{2p}{\left[u\right]}_{s\mathrm{,}A}^{2p}-\frac{\mu }{2}{\displaystyle {\int }_{{ℝ}^N}}{H}_{\alpha }\left(x\mathrm{,}{\left|u\right|}^2\right)\text{d}x\\ -\frac{1}{2p_{\sigma \mathrm{,}s}^{\ast }}{\displaystyle {\int }_{{ℝ}^N}}\left[I_{\sigma }\left(x\right)\ast {\left|u\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\right]{\left|u\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\text{d}x\mathrm{.}\end{array}$

Now we give the definition of weak solutions for problem (1.1).

Lemma 2.1. (Diamagnetic inequality) For every $u\in W_A^{s\mathrm{,}p}\left({ℝ}^N\mathrm{,}ℂ\right)$, it holds $\left|u\right|\in W^{s\mathrm{,}p}\left({ℝ}^N\right)$. More precisely,

${\Vert \left|u\right|\Vert }_{W^{s\mathrm{,}p}\left({ℝ}^N\right)}\le {\Vert u\Vert }_{W_A^{s\mathrm{,}p}\left({ℝ}^N\mathrm{,}ℂ\right)}\mathrm{,}\text{for}\text{each}u\in W_A^{s\mathrm{,}p}\left({ℝ}^N\mathrm{,}ℂ\right)\mathrm{.}$

Proof. It follows from Pointwise Diamagnetic inequality in [1] that

$\left|\left|u\left(x\right)\right|-\left|u\left(y\right)\right|\right|\le \left|u\left(x\right)-{\text{e}}^{i\left(x-y\right)\cdot A\left(\frac{x+y}{2}\right)}u\left(y\right)\right|\mathrm{,}$

which implies the conclusion holds.

Lemma 2.2. (Magnetic Sobolev embeddings) Let $s\in \left(\mathrm{0,1}\right)$ and $N>ps$, $p_s^{\ast }=\frac{Np}{N-ps}$, then the embedding $W_A^{s\mathrm{,}p}\left({ℝ}^N\mathrm{,}ℂ\right)$ ↪ $L^r\left({ℝ}^N\mathrm{,}ℂ\right)$ is continuous for $r\in \left[p\mathrm{,}{p}_s^{\ast }\right]$ and is locally compact for $r\in \left[p\mathrm{,}{p}_s^{\ast }\right)$.

Proof. In view of Theorem 6.7 in [24], we know that the embedding $W^{s\mathrm{,}p}\left({ℝ}^N\right)$ ↪ $L^r\left({ℝ}^N\right)$ is continuous for $r\in \left[p\mathrm{,}{p}_s^{\ast }\right]$, that is, there exists a constant $C_0$ such that

${\left|u\right|}_{L^r\left({ℝ}^N\right)}\le C_0{\Vert u\Vert }_{W^{s\mathrm{,}p}\left({ℝ}^N\right)}\mathrm{,}$

and similar to the argument of Lemma 3.3 in [1], since Pointwise Diamagnetic inequality, we have

${\left({\displaystyle {\int }_{{ℝ}^N}}\frac{{\left|u\left(x\right)-u\left(y\right)\right|}^p}{{\left|x-y\right|}^{N+ps}}\text{d}x\text{d}y\right)}^{\frac{1}{p}}\le C{\left({\displaystyle {\int }_{{ℝ}^N}}\frac{{\left|u\left(x\right)-{\text{e}}^{i\left(x-y\right)\cdot A\left(\frac{x+y}{2}\right)}u\left(y\right)\right|}^p}{{\left|x-y\right|}^{N+ps}}\text{d}x\text{d}y\right)}^{\frac{1}{p}}\mathrm{.}$

Consequently,

${\Vert u\Vert }_{L^{p_s^{\ast }}\left({ℝ}^N\right)}\le C{\left({\displaystyle {\int }_{{ℝ}^N}}\frac{{\left|u\left(x\right)-{\text{e}}^{i\left(x-y\right)\cdot A\left(\frac{x+y}{2}\right)}u\left(y\right)\right|}^p}{{\left|x-y\right|}^{N+ps}}\text{d}x\text{d}y\right)}^{\frac{1}{p}}\mathrm{,}\text{for}\text{all}u\in W_A^{s\mathrm{,}p}\left({ℝ}^N\mathrm{,}ℂ\right)\mathrm{.}$

Then, by interpolation the assertion immediately follows. For the compact embedding, note that the embedding $W_A^{s\mathrm{,}p}\left({ℝ}^N\mathrm{,}ℂ\right)$ ↪ $W^{s\mathrm{,}p}\left({ℝ}^N\right)$ is continuous, the assertion follows by the Corollary 7.2 [26].

Lemma 2.3. [27] Let $r,t>1$ and $0<\sigma <N$ with $\frac{1}{r}+\frac{1}{t}+\frac{N-\sigma }{N}=2$.

Assume that $f_1\in L^r\left({ℝ}^N\right)$ and $f_2\in L^t\left({ℝ}^N\right)$. Then there exists a sharp constant $C_{N\mathrm{,}\sigma \mathrm{,}r\mathrm{,}t}$ independent of $f_1$ and $f_2$ such that

$\left|{\displaystyle {\iint }_{{ℝ}^N\times {ℝ}^N}}\frac{f_1\left(x\right)f_2\left(y\right)}{{\left|x-y\right|}^{N-\sigma }}\text{d}x\text{d}y\right|\le C_{N\mathrm{,}\sigma \mathrm{,}r\mathrm{,}t}{\left|f_1\right|}_r{\left|f_2\right|}_t\mathrm{.}$

3. Proof of Main Results

Lemma 3.1. For each $\mu >0$, $u\in X\backslash \left\{0\right\}$, we have

1) Set $\Phi \left(t\right)=J\left(tu\right)$. Then there exists a unique $t_u>0$ such that

$\Phi \left(t_u\right)=\underset{t\ge 0}{\max }\Phi \left(t\right),$

${\Phi }^{\prime }\left(t\right)>0$ for $0<t<t_u$ and ${\Phi }^{\prime }\left(t\right)<0$ for $t_u<t$. Moreover, $tu\in \mathcal{N}$ if and only if $t=t_u$.

2) Set $\Psi \left(t\right)=J_{\alpha }\left(tv\right)$. Then there exists a unique $t_v>0$ such that

$\Psi \left(t_v\right)=\underset{t\ge 0}{\max }\Psi \left(t\right),$

${{\Psi }^{\prime }}_v\left(t\right)>0$ for $0<t<t_v$ and ${{\Psi }^{\prime }}_v\left(t\right)<0$ for $t_v<t$. Moreover, $tv\in {\mathcal{N}}_{\alpha }$ if and only if $t=t_v$.

Proof. 1) For any $\epsilon >0$, by (h₁) and (h₂), there exists $C_{\epsilon }>0$ such that

$\left|h\left(x\mathrm{,}u\right)\right|\le \epsilon +C_{\epsilon }{\left|u\right|}^{q-2}$ (3.1)

and

$\left|H\left(x\mathrm{,}u\right)\right|\le \epsilon \left|u\right|+\frac{C_{\epsilon }}{q}{\left|u\right|}^q\mathrm{.}$ (3.2)

In view of Lemma 2.2 and Lemma 2.3, we get

$\left|{\displaystyle {\int }_{{ℝ}^N}}\left[I_{\sigma }\left(x\right)\ast {\left|u\right|}^{2_{\sigma \mathrm{,}s}^{\ast }}\right]{\left|u\right|}^{2_{\sigma \mathrm{,}s}^{\ast }}\text{d}x\right|\le C_1{\left|u\right|}_{p_s^{\ast }}^{2p_{\sigma \mathrm{,}s}^{\ast }}\le C_2{\Vert u\Vert }_{s\mathrm{,}A}^{2p_{\sigma \mathrm{,}s}^{\ast }}\mathrm{.}$ (3.3)

Hence, for small $\epsilon >0$ and $t>0$, it follows from (3.2) and (3.3) that

$\begin{array}{c}\Phi \left(t\right)=J\left(tu\right)\\ =\frac{1}{p}{t}^p{\Vert u\Vert }_{s\mathrm{,}A}^p+\frac{b{t}^{2p}}{2p}{\left[u\right]}_{s\mathrm{,}A}^{2p}-\frac{\mu }{2}{\displaystyle {\int }_{{ℝ}^N}}H\left(x\mathrm{,}{\left|tu\right|}^2\right)\text{d}x\\ -\frac{1}{2p_{\sigma \mathrm{,}s}^{\ast }}{t}^{2p_{\sigma \mathrm{,}s}^{\ast }}{\displaystyle {\int }_{{ℝ}^N}}\left[I_{\sigma }\left(x\right)\ast {\left|u\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\right]{\left|u\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\text{d}x\\ \ge \frac{1}{p}{t}^p{\Vert u\Vert }_{s\mathrm{,}A}^2-\frac{\mu }{2}\left(C_3\epsilon t^2{\Vert u\Vert }_{s\mathrm{,}A}^2+C_4{C}_{\epsilon }{t}^q{\Vert u\Vert }_{s\mathrm{,}A}^q\right)-C_1{t}^{2p_{\sigma \mathrm{,}s}^{\ast }}{\left|u\right|}_{p_s^{\ast }}^{2p_{\sigma \mathrm{,}s}^{\ast }}\\ >0,\end{array}$

and due to (3.1), we have

$\begin{array}{c}{\Phi }^{\prime }\left(t\right)=\langle J^{\prime }\left(tu\right)\mathrm{,}u\rangle \\ =t^{p-1}{\Vert u\Vert }_{s\mathrm{,}A}^p+b{t}^{2p-1}{\left[u\right]}_{s\mathrm{,}A}^{2p}-\mu {\displaystyle {\int }_{{ℝ}^N}}h\left(x\mathrm{,}{\left|tu\right|}^2\right)t{u}^2\text{d}x\\ -t^{2p_{\sigma \mathrm{,}s}^{\ast }-1}{\displaystyle {\int }_{{ℝ}^N}}\left[I_{\sigma }\left(x\right)\ast {\left|u\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\right]{\left|u\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\text{d}x\\ \ge t^{p-1}{\Vert u\Vert }_{s\mathrm{,}A}^p-\mu \left(C_3\epsilon t{\Vert u\Vert }_{s\mathrm{,}A}^2+C_4{C}_{\epsilon }{t}^{q-1}{\Vert u\Vert }_{s\mathrm{,}A}^q\right)-C_1{t}^{2p_{\sigma \mathrm{,}s}^{\ast }-1}{\left|u\right|}_{p_s^{\ast }}^{2p_{\sigma \mathrm{,}s}^{\ast }}\\ >0.\end{array}$

Furthermore, by means of (h₄), we obtain that

$\begin{array}{c}\Phi \left(t\right)=J\left(tu\right)\\ \le \frac{1}{p}{t}^p{\Vert u\Vert }_{s\mathrm{,}A}^p+\frac{b}{2p}{t}^{2p}{\left[u\right]}_{s\mathrm{,}A}^{2p}\\ -\frac{1}{2p_{\sigma \mathrm{,}s}^{\ast }}{t}^{2p_{\sigma \mathrm{,}s}^{\ast }}{\displaystyle {\int }_{{ℝ}^N}}\left[I_{\sigma }\left(x\right)\ast {\left|u\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\right]{\left|u\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\text{d}x\\ \to -\infty \mathrm{,}\end{array}$

as $t\to +\infty$. Therefore, $\Phi \left(t\right)$ has a maximum and then there exists $t_u>0$ such that ${\Phi }^{\prime }\left(t_u\right)=0$ and ${\Phi }^{\prime }\left(t\right)>0$ for $0<t<t_u$. We claim that ${\Phi }^{\prime }\left(t\right)\ne 0$ for all $t>t_u$. Indeed, if the conclusion is false, then, from the above arguments, there exists a $t_u<t_{\omega }<+\infty$ such that ${\Phi }^{\prime }\left(t_{\omega }\right)>0$ and $\Phi \left(t_u\right)\ge \Phi \left(t_{\omega }\right)$. Nevertheless, (h₃) implies that

$\begin{array}{c}\Phi \left(t_{\omega }\right)=\Phi \left(t_{\omega }\right)-\frac{1}{2p}\langle {\Phi }^{\prime }\left(t_{\omega }\right)\mathrm{,}{t}_{\omega }\rangle \\ =\frac{1}{2p}{t}_{\omega }^p{\Vert u\Vert }_{s\mathrm{,}A}^p+\frac{\mu }{2p}{\displaystyle {\int }_{{ℝ}^N}}\left[h\left(x\mathrm{,}{\left|t_{\omega }u\right|}^2\right){\left|t_{\omega }u\right|}^2-pH\left(x\mathrm{,}{\left|t_{\omega }u\right|}^2\right)\right]\text{d}x\\ +\left(\frac{1}{2p}-\frac{1}{2p_{\sigma \mathrm{,}s}^{\ast }}\right)t_{\omega }^{2p_{\sigma \mathrm{,}s}^{\ast }}{\displaystyle {\int }_{{ℝ}^N}}\left[I_{\sigma }\left(x\right)\ast {\left|u\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\right]{\left|u\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\text{d}x\\ \ge \frac{1}{2p}{t}_u^p{\Vert u\Vert }_{s\mathrm{,}A}^p+\frac{\mu }{2p}{\displaystyle {\int }_{{ℝ}^N}}\left[h\left(x\mathrm{,}{\left|t_{u}u\right|}^2\right){\left|t_{u}u\right|}^2-pH\left(x\mathrm{,}{\left|t_{u}u\right|}^2\right)\right]\text{d}x\\ +\left(\frac{1}{2p}-\frac{1}{2p_{\sigma \mathrm{,}s}^{\ast }}\right)t_u^{2p_{\sigma \mathrm{,}s}^{\ast }}{\displaystyle {\int }_{{ℝ}^N}}\left[I_{\sigma }\left(x\right)\ast {\left|u\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\right]{\left|u\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\text{d}x\\ =\Phi \left(t_u\right)-\frac{1}{2p}\langle {\Phi }^{\prime }\left(t_u\right)\mathrm{,}{t}_u\rangle =\Phi \left(t_u\right)\mathrm{,}\end{array}$

which is a contradiction. Thereupon, the one conclusion of (1) has been proved, we can obtain the other one by the fact that ${\Phi }^{\prime }\left(t\right)=t^{-1}\langle J^{\prime }\left(tu\right),tu\rangle .$ This completes the proof of (1).

2) Similar to the proof of (1), we can obtain that (2) holds.

Lemma 3.2. For each $\mu >0$, we have the following results.

1) There exists $t_{\delta }>0$ such that $t_u\ge t_{\delta }$ for each

$u\in S_1:=\left\{u\in X:{\Vert u\Vert }_{s,A}=1\right\}.$

Moreover, for each compact subset $\Lambda \subset S_1$, there exsits $C_{\Lambda }>0$ such that $t_u\le C_{\Lambda }$ for all $u\in \Lambda$.

2) There exists $\rho >0$ such that

$c_{\mu }=\underset{u\in \mathcal{N}}{\inf }J\left(u\right)\ge \underset{u\in S_{\rho }}{\inf }J\left(u\right)>0,$

where $S_{\rho }\mathrm{<mo>\; :\; </mo>}=\left\{u\in X\mathrm{<mo>\; :\; </mo>}{\Vert u\Vert }_{s\mathrm{,}A}=\rho \right\}$.

Proof. 1) For $u\in S_1$, owing to Lemma 3.1 (1), there exists $t_u>0$ such that $t_{u}u\in \mathcal{N}$. Also, by (3.1) and (3.3), we have

$\begin{array}{c}0=\langle J^{\prime }\left(t_{u}u\right)\mathrm{,}{t}_{u}u\rangle \\ =t_u^p{\Vert u\Vert }_{s\mathrm{,}A}^p+b{t}_u^{2p}{\left[u\right]}_{s\mathrm{,}A}^{2p}-\mu {\displaystyle {\int }_{{ℝ}^N}}h\left(x\mathrm{,}{\left|t_{u}u\right|}^2\right){\left|t_{u}u\right|}^2\text{d}x\\ -t_u^{2p_{\sigma \mathrm{,}s}^{\ast }}{\displaystyle {\int }_{{ℝ}^N}}\left[I_{\sigma }\left(x\right)\ast {\left|u\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\right]{\left|u\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\text{d}x\\ \ge t_u^p{\Vert u\Vert }_{s\mathrm{,}A}^p-\mu \left(C_3\epsilon t_u^2{\Vert u\Vert }_{s\mathrm{,}A}^2+C_4{C}_{\epsilon }{t}_u^q{\Vert u\Vert }_{s\mathrm{,}A}^q\right)-C_2{t}_u^{2p_{\sigma \mathrm{,}s}^{\ast }}{\Vert u\Vert }_{s\mathrm{,}A}^{2p_{\sigma \mathrm{,}s}^{\ast }}\\ \ge C_5{t}_u^p-\mu C_4{C}_{\epsilon }{t}_u^q-C_2{t}_u^{2p_{\sigma \mathrm{,}s}^{\ast }}\mathrm{,}\end{array}$

which implies that there exists $t_{\delta }>0$ such that $t_u\ge t_{\delta }$ for all $u\in S_1$. Assume that there exists $\left\{u_n\right\}\subset \Lambda \subset S_1$ such that $t_n=t_{u_n}\to +\infty$ as $n\to \infty$. Since $\Lambda$ is compact, there exists $u\in \Lambda$ such that $u_n\to u$ in X. Set

$\beta \left(u\right)=\frac{1}{2p_{\sigma \mathrm{,}s}^{\ast }}{\displaystyle {\int }_{{ℝ}^N}}\left[I_{\sigma }\left(x\right)\ast {\left|u\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\right]{\left|u\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\text{d}x\mathrm{,}\forall u\in X\mathrm{.}$

Indeed, we have

$\beta \left(t_n{u}_n\right)\ge C_6{t}_n^{2p_{\sigma \mathrm{,}s}^{\ast }}{\Vert u_n\Vert }_{s\mathrm{,}A}^{2p_{\sigma \mathrm{,}s}^{\ast }}\mathrm{.}$ (3.4)

It follows from (3.4) and (h₄) that

$\begin{array}{c}J\left(t_n{u}_n\right)\le \frac{1}{p}{t}_n^p{\Vert u_n\Vert }_{s\mathrm{,}A}^p+\frac{b}{2p}{t}_n^{2p}{\left[u_n\right]}_{s\mathrm{,}A}^{2p}-\beta \left(t_n{u}_n\right)\\ \le \frac{1}{p}{t}_n^p{\Vert u_n\Vert }_{s\mathrm{,}A}^p+\frac{b}{2p}{t}_n^{2p}{\left[u_n\right]}_{s\mathrm{,}A}^{2p}-C_6{t}_n^{2p_{\sigma \mathrm{,}s}^{\ast }}{\Vert u_n\Vert }_{s\mathrm{,}A}^{2p_{\sigma \mathrm{,}s}^{\ast }}\\ \to -\infty \end{array}$

as $n\to \infty$. However, by (h₃), we have

$\begin{array}{c}J\left(t_n{u}_n\right)=J\left(t_n{u}_n\right)-\frac{1}{2p}\langle J^{\prime }\left(t_n{u}_n\right)\mathrm{,}{t}_n{u}_n\rangle \\ =\frac{1}{2p}{t}_n^p{\Vert u_n\Vert }_{s\mathrm{,}A}^p+\frac{\mu }{2p}{\displaystyle {\int }_{{ℝ}^N}}\left[h\left(x\mathrm{,}{t}_n^2{\left|u_n\right|}^2\right)t_n^2{\left|u_n\right|}^2-pH\left(x\mathrm{,}{t}_n^2{\left|u_n\right|}^2\right)\right]\text{d}x\\ +\left(\frac{1}{2p}-\frac{1}{2p_{\sigma \mathrm{,}s}^{\ast }}\right)t_n^{2p_{\sigma \mathrm{,}s}^{\ast }}{\displaystyle {\int }_{{ℝ}^N}}\left[I_{\sigma }\left(x\right)\ast {\left|u_n\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\right]{\left|u_n\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\text{d}x\\ \ge \mathrm{0,}\end{array}$

a contradiction. Hence the conclusion holds.

2) For $u\in S_{\rho }$, and small $\epsilon >0$, it follows from (3.2) and (3.3) that

$\begin{array}{c}J\left(u\right)\ge \frac{1}{p}{\Vert u\Vert }_{s\mathrm{,}A}^p-\mu \left(C_3\epsilon {\Vert u\Vert }_{s\mathrm{,}A}^2+C_4{C}_{\epsilon }{\Vert u\Vert }_{s\mathrm{,}A}^q\right)-C_2{\Vert u\Vert }_{s\mathrm{,}A}^{2p_{\sigma \mathrm{,}s}^{\ast }}\\ \ge C_7{\Vert u\Vert }_{s,A}^2=C_7{\rho }^2>0\end{array}$

for small $\rho >0$. Furthermore, for each $u\in \mathcal{N}$, there exists $t_{\epsilon }>0$ such that $t_{\epsilon }u\in S_{\rho }$. Then we have

$0<C_6{\rho }^2\le \underset{u\in S_{\rho }}{\inf }J\left(u\right)\le J\left(t_{\epsilon }u\right)\le \underset{t>0}{\max }J\left(tu\right)=J\left(u\right),$

which implies that

$c_{\mu }=\underset{u\in \mathcal{N}}{\inf }J\left(u\right)\ge \underset{u\in S_{\rho }}{\inf }J\left(u\right)>0.$

The proof is completed.

It follows from [28] that we have the following lemma.

Lemma 3.3. The mapping $I\mathrm{<mo>\; :\; </mo>}{S}_1\to \mathcal{N}$ is a homeomorphism between $S_1$ and $\mathcal{N}$, and the inverse of I is given by $I^{-1}\left(u\right)=\frac{u}{{\Vert u\Vert }_{s\mathrm{,}A}}$. Considering the functional ${\varphi }_{\mu }\mathrm{<mo>\; :\; </mo>}{S}_1\to ℝ$ given by

${\varphi }_{\mu }\left(w\right)=J\left(I\left(w\right)\right)\mathrm{,}$

then the lemma follows.

Lemma 3.4. 1) If $\left\{w_n\right\}$ is a Palais-Smale sequence for ${\varphi }_{\mu }$, then $\left\{I\left(w_n\right)\right\}$ is a Palais-Smale sequence for J. If $\left\{u_n\right\}\subset \mathcal{N}$ is a bounded Palais-Smale sequence for J, then $\left\{I^{-1}\left(u_n\right)\right\}$ is a Palais-Smale sequence for ${\varphi }_{\mu }$.

2) $w\in S_1$ is a critical point of ${\varphi }_{\mu }$ if and only if $I\left(w\right)$ is a nontrivial critical point of J. Moreover, the corresponding values of ${\varphi }_{\mu }$ and J coincide and $\underset{S_1}{\inf }{\varphi }_{\mu }=\underset{\mathcal{N}}{\inf }J$.

3) A minimizer of J on $\mathcal{N}$ is a ground state solution of (1.1).

Similar to the argument of Lemma 2.6 in [8], the results as follows

Lemma 3.5. If $\left\{u_n\right\}\subset X$ satisfies $u_n\rightharpoonup 0$ in X and ${\phi }_n\in X$ is bounded. Then

${\displaystyle {\int }_{{ℝ}^N}}\left[V\left(x\right)-V_{\alpha }\left(x\right)\right]u_n\overline{{\phi }_n}\text{d}x\to 0$ (3.5)

and

${\displaystyle {\int }_{{ℝ}^N}}\left[h\left(x\mathrm{,}{\left|u_n\right|}^2\right)-h_{\alpha }\left(x\mathrm{,}{\left|u_n\right|}^2\right)\right]u_n\overline{{\phi }_n}\text{d}x\to \mathrm{0,}$ (3.6)

also

${\displaystyle {\int }_{{ℝ}^N}}\left[H\left(x\mathrm{,}{\left|u_n\right|}^2\right)-H_{\alpha }\left(x\mathrm{,}{\left|u_n\right|}^2\right)\right]\text{d}x\to 0.$ (3.7)

Lemma 3.6. There exists ${\mu }^{\ast }>0$ such that $0<c_{\mu }<\frac{2ps-\sigma }{2p\left(2N-\sigma \right)}{S}^{\frac{p\left(2N-\sigma \right)}{2N\left(p-1\right)+p\left(2s-\sigma \right)}}$ for all $\mu >{\mu }^{\ast }$.

Proof. Assume that the conclusion is not true. Then there exists a sequence ${\mu }_n$ with ${\mu }_n\to +\infty$ such that $c_{{\mu }_n}\ge \frac{2ps-\sigma }{2p\left(2N-\sigma \right)}{S}^{\frac{p\left(2N-\sigma \right)}{2N\left(p-1\right)+p\left(2s-\sigma \right)}}$. Take $u\in X\backslash \left\{0\right\}$, by Lemma 3.1 (1), there exists a unique $t_{{\mu }_n}>0$ such that $\underset{t\mathrm{<mo>\; >\; </mo>0}}{\max }J\left(tu\right)=J\left(t_{{\mu }_n}u\right)\mathrm{.}$ Since (h₄), we have

$\begin{array}{l}{t}_{{\mu }_n}^p{\Vert u\Vert }_{s\mathrm{,}A}^p+b{t}_{{\mu }_n}^{2p}{\left[u\right]}_{s\mathrm{,}A}^{2p}\\ =t_{{\mu }_n}^{2p_{\sigma \mathrm{,}s}^{\ast }}{\displaystyle {\int }_{{ℝ}^N}}\left[I_{\sigma }\left(x\right)\ast {\left|u\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\right]{\left|u\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\text{d}x+{\mu }_n{\displaystyle {\int }_{{ℝ}^N}}h\left(x\mathrm{,}{\left|t_{{\mu }_n}u\right|}^2\right){\left|t_{{\mu }_n}u\right|}^2\text{d}x\\ \ge t_{{\mu }_n}^{2p_{\sigma \mathrm{,}s}^{\ast }}{\displaystyle {\int }_{{ℝ}^N}}\left[I_{\sigma }\left(x\right)\ast {\left|u\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\right]{\left|u\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\text{d}x\mathrm{,}\end{array}$

which means that $\left\{t_{{\mu }_n}\right\}$ is bounded. Therefore, up to a subsequence, and there exists $t_{\kappa }\ge 0$ such that $t_{{\mu }_n}\to t_{\kappa }$ as $n\to \infty$. Suppose $t_{\kappa }>0$. In view of (h₄), one has

$\underset{n\to \infty }{\lim }\left[{\mu }_n{\displaystyle {\int }_{{ℝ}^N}}h\left(x\mathrm{,}{\left|t_{{\mu }_n}u\right|}^2\right){\left|t_{{\mu }_n}u\right|}^2\text{d}x+t_{{\mu }_n}^{2p_{\sigma \mathrm{,}s}^{\ast }}{\displaystyle {\int }_{{ℝ}^N}}\left[I_{\sigma }\left(x\right)\ast {\left|u\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\right]{\left|u\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\text{d}x\right]=+\infty \mathrm{.}$

However, we know that

$t_{{\mu }_n}^p{\Vert u\Vert }_{s\mathrm{,}A}^p\to t_{\kappa }^p{\Vert u\Vert }_{s\mathrm{,}A}^p\mathrm{,}$

which is a contradiction. Thereupon, we get $t_{\kappa }=0$. And it follows from (h₄) that

$\underset{t>0}{\max }J\left(tu\right)=J\left(t_{{\mu }_n}u\right)\le \frac{1}{p}{t}_{{\mu }_n}^p{\Vert u\Vert }_{s\mathrm{,}A}^2-\frac{1}{2p_{\sigma \mathrm{,}s}^{\ast }}{t}_{{\mu }_n}^{2p_{\sigma \mathrm{,}s}^{\ast }}{\displaystyle {\int }_{{ℝ}^N}}\left[I_{\sigma }\left(x\right)\ast {\left|u\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\right]{\left|u\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\text{d}x\to \mathrm{0,}$

as $n\to \infty$. Hence,

$0<\frac{2ps-\sigma }{2p\left(2N-\sigma \right)}{S}^{\frac{p\left(2N-\sigma \right)}{2N\left(p-1\right)+p\left(2s-\sigma \right)}}\le \underset{u\in X\backslash \left\{0\right\}}{\inf }\underset{t>0}{\sup }J\left(tu\right)\le \underset{t>0}{\max }J\left(tu\right)\to 0,$

a contradiction. As a result, there exists ${\mu }^{\ast }>0$ such that $0<c_{\mu }<\frac{2ps-\sigma }{2p\left(2N-\sigma \right)}{S}^{\frac{p\left(2N-\sigma \right)}{2N\left(p-1\right)+p\left(2s-\sigma \right)}}$ for all $\mu >{\mu }^{\ast }$. The proof is completed.

Proof of Theorem 1.1. In virtue of Lemma 3.4 (3), we know that $c_{\mu }$ is achieved. For $\mu >{\mu }^{\ast }$. let $\left\{w_n\right\}\subset S_1$ be a minimizing sequence satisfying

${\varphi }_{\mu }\left(w_n\right)\to c_{\mu }=\underset{S_1}{\inf }{\varphi }_{\mu }.$

Thanks to the Ekeland variational principle, we assume that ${{\varphi }^{\prime }}_{\mu }\left(w_n\right)\to 0$ in $X^{\prime }$. Set $u_n=I\left(w_n\right)\in \mathcal{N}$. By Lemma 3.4 (1), we have

$J\left(u_n\right)={\varphi }_{\mu }\left(w_n\right)\to c_{\mu }\mathrm{,}$

and $J^{\prime }\left(u_n\right)\to 0$ in $X^{\prime }$. Thus, by virtue of (h₃), we get

$\begin{array}{c}{c}_{\mu }+o_n\left(1\right){\Vert u_n\Vert }_{s\mathrm{,}A}=J\left(u_n\right)-\frac{1}{2p}\langle J^{\prime }\left(u_n\right)\mathrm{,}{u}_n\rangle \\ =\frac{1}{2p}{\Vert u_n\Vert }_{s\mathrm{,}A}^p+\frac{\mu }{2p}{\displaystyle {\int }_{{ℝ}^N}}\left[h\left(x\mathrm{,}{\left|u_n\right|}^2\right){\left|u_n\right|}^2-pH\left(x\mathrm{,}{\left|u_n\right|}^2\right)\right]\text{d}x\\ +\left(\frac{1}{2p}-\frac{1}{2p_{\sigma \mathrm{,}s}^{\ast }}\right){\displaystyle {\int }_{{ℝ}^N}}\left[I_{\sigma }\left(x\right)\ast {\left|u_n\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\right]{\left|u_n\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\text{d}x\\ \ge \frac{1}{2p}{\Vert u_n\Vert }_{s\mathrm{,}A}^p\mathrm{,}\end{array}$

which implies $\left\{u_n\right\}$ is bounded in X. Hence, there exists a subsequence, still denoted by $\left\{u_n\right\}$, and $u\in X$. Then we have

$\begin{array}{l}{u}_n\rightharpoonup u\text{in}X\mathrm{,}\\ u_n\to u\text{in}{L}_{\text{loc}}^q\left({ℝ}^N\mathrm{,}ℂ\right)\text{for}p\le q<p_s^{\ast }\mathrm{.}\end{array}$

Thereupon, $J^{\prime }\left(u\right)=0$. The next, we prove it by case.

If $u\ne 0$. we know that $u\in \mathcal{N}$ and $c_{\mu }\le J\left(u\right)$. It follows from Fatou’s Lemma, the weakly lower semi-continuity of the norm and (h₃) that

$\begin{array}{c}{c}_{\mu }=\underset{n\to \infty }{\lim }\left[J\left(u_n\right)-\frac{1}{2p}\langle J^{\prime }\left(u_n\right)\mathrm{,}{u}_n\rangle \right]\\ \ge \frac{1}{2p}\underset{n\to \infty }{\lim \inf }{\Vert u_n\Vert }_{s\mathrm{,}A}^p+\frac{\mu }{2p}\underset{n\to \infty }{\lim \inf }{\displaystyle {\int }_{{ℝ}^N}}\left[h\left(x\mathrm{,}{\left|u_n\right|}^2\right){\left|u_n\right|}^2-pH\left(x\mathrm{,}{\left|u_n\right|}^2\right)\right]\text{d}x\\ +\underset{n\to \infty }{\lim \inf }\left(\frac{1}{2p}-\frac{1}{2p_{\sigma \mathrm{,}s}^{\ast }}\right){\displaystyle {\int }_{{ℝ}^N}}\left[I_{\sigma }\left(x\right)\ast {\left|u_n\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\right]{\left|u_n\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\text{d}x\\ \ge \frac{1}{2p}{\Vert u\Vert }_{s\mathrm{,}A}^p+\frac{\mu }{2p}{\displaystyle {\int }_{{ℝ}^N}}\left[h\left(x\mathrm{,}{\left|u\right|}^2\right){\left|u\right|}^2-pH\left(x\mathrm{,}{\left|u\right|}^2\right)\right]\text{d}x\\ +\left(\frac{1}{2p}-\frac{1}{2p_{\sigma \mathrm{,}s}^{\ast }}\right){\displaystyle {\int }_{{ℝ}^N}}\left[I_{\sigma }\left(x\right)\ast {\left|u\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\right]{\left|u\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\text{d}x\\ =J\left(u\right)-\frac{1}{2p}\langle J^{\prime }\left(u\right)\mathrm{,}u\rangle =J\left(u\right)\ge c_{\mu }\mathrm{.}\end{array}$

Consequently, we get $J\left(u\right)=c_{\mu }$.

In the following, we consider the case for $u=0$. On account of the concentration-compactness principle by Lions, we know that two cases may happen:

1): Vanishing, that is, $\underset{n\to \infty }{\lim }\underset{y\in {ℝ}^N}{\sup }{\displaystyle {\int }_{B_1\left(y\right)}}{\left|u_n\left(x\right)\right|}^2\text{d}x=0$.

2): Nonvanishing, that is, there exists a sequence $\left\{y_n\right\}\subset {ℝ}^N$ and a constant $d>0$ such that

$\underset{n\to \infty }{\lim \inf }{\displaystyle {\int }_{B_1\left(y_n\right)}}{\left|u_n\left(x\right)\right|}^2\text{d}x\ge d\mathrm{.}$ (3.8)

Assume that (1) occurs. In view of Lemma 1.21 in [29], we get $\left|u_n\right|\to 0$ in $L^q\left({ℝ}^N\right)$ for $p<q<p_s^{\ast }$. Thus, by means of (3.1) and (3.2), we have

${\displaystyle {\int }_{{ℝ}^N}}H\left(x\mathrm{,}{\left|u_n\right|}^2\right)\text{d}x\to 0\text{and}{\displaystyle {\int }_{{ℝ}^N}}h\left(x\mathrm{,}{\left|u_n\right|}^2\right){\left|u_n\right|}^2\text{d}x\to 0.$ (3.9)

Consequently,

$o_n\left(1\right)={\Vert u_n\Vert }_{s\mathrm{,}A}^p+b{\left[u_n\right]}_{s\mathrm{,}A}^{2p}-{\displaystyle {\int }_{{ℝ}^N}}\left[I_{\sigma }\left(x\right)\ast {\left|u_n\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\right]{\left|u_n\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\text{d}x\mathrm{.}$

Suppose ${\Vert u_n\Vert }_{s\mathrm{,}A}^p+b{\left[u_n\right]}_{s\mathrm{,}A}^{2p}\to m$. Then ${\displaystyle {\int }_{{ℝ}^N}}\left[I_{\sigma }\left(x\right)\ast {\left|u_n\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\right]{\left|u_n\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\text{d}x\to m$. If $m>0$, in virtue of (2.1), we get

$S{\left({\displaystyle {\int }_{{ℝ}^N}}\left[I_{\sigma }\left(x\right)\ast {\left|u_n\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\right]{\left|u_n\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\text{d}x\right)}^{\frac{1}{2p_{\sigma \mathrm{,}s}^{\ast }}}\le {\left[u_n\right]}_{s\mathrm{,}A}^p\le {\Vert u_n\Vert }_{s\mathrm{,}A}^p\le {\Vert u_n\Vert }_{s\mathrm{,}A}^p+b{\left[u_n\right]}_{s\mathrm{,}A}^{2p}\mathrm{.}$

Hence, $S{m}^{\frac{2\left(N-ps\right)}{p\left(2N-\sigma \right)}}\le m$. Then we have either $m=0$ or $m\ge S^{\frac{p\left(2N-\sigma \right)}{2N\left(p-1\right)+p\left(2s-\sigma \right)}}$.

If $m=0$, we have $c_{\mu }=0$, which contradicts with Lemma 3.6. When $m\ge S^{\frac{p\left(2N-\sigma \right)}{2N\left(p-1\right)+p\left(2s-\sigma \right)}}$, it follows from $J\left(u_n\right)\to c_{\mu }$ and (3.9) that

$\begin{array}{c}{c}_{\mu }=\underset{n\to \infty }{\lim }[\frac{1}{p}{\Vert u_n\Vert }_{s\mathrm{,}A}^p+\frac{b}{2p}{\left[u_n\right]}_{s\mathrm{,}A}^{2p}-\frac{\mu }{2}{\displaystyle {\int }_{{ℝ}^N}}H\left(x\mathrm{,}{\left|u_n\right|}^2\right)\text{d}x\\ -\frac{1}{2p_{\sigma \mathrm{,}s}^{\ast }}{\displaystyle {\int }_{{ℝ}^N}}\left[I_{\sigma }\left(x\right)\ast {\left|u_n\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\right]{\left|u_n\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\text{d}x]\\ \ge \frac{m}{2p}-\frac{m}{2p_{\sigma \mathrm{,}s}^{\ast }}=\frac{2ps-\sigma }{2p\left(2N-\sigma \right)}m\\ \ge \frac{2ps-\sigma }{2p\left(2N-\sigma \right)}{S}^{\frac{p\left(2N-\sigma \right)}{2N\left(p-1\right)+p\left(2s-\sigma \right)}}\mathrm{,}\end{array}$

which also contradicts with Lemma 3.6. Therefore, nonvanishing occurs. Without loss of generality, we may suppose $y_n\in {\mathbb{Z}}^N$. Let ${\tilde{u}}_n(\cdot )=u_n\left(\cdot +y_n\right)$. Up to a subsequence, then there exists $\tilde{u}\in X$ such that ${\tilde{u}}_n\rightharpoonup \tilde{u}$ in X, ${\tilde{u}}_n\to \tilde{u}$ in $L_{\text{loc}}^q\left({ℝ}^N\mathrm{,}ℂ\right)$ for $p\le q<p_s^{\ast }$, and $u_n\to u$ a.e. on ${ℝ}^N$. Due to (3.8), we know that $\tilde{u}\ne 0$.

The next job is to prove that ${J^{\prime }}_{\alpha }\left(\tilde{u}\right)=0$. For all $\phi \in X$, set ${\phi }_n(\cdot )={\phi }_n\left(\cdot -y_n\right)$. Owing to Lemma 3.5, we know that

${\displaystyle {\int }_{{ℝ}^N}}\left[V\left(x\right)-V_{\alpha }\left(x\right)\right]u_n\overline{{\phi }_n}\text{d}x\to 0$

and

${\displaystyle {\int }_{{ℝ}^N}}\left[h\left(x\mathrm{,}{\left|u_n\right|}^2\right)-h_{\alpha }\left(x\mathrm{,}{\left|u_n\right|}^2\right)\right]u_n\overline{{\phi }_n}\text{d}x\to 0.$

Hence we have

$\begin{array}{l}\langle J^{\prime }\left(u_n\right)\mathrm{,}{\phi }_n\rangle -\langle {J^{\prime }}_{\alpha }\left(u_n\right)\mathrm{,}{\phi }_n\rangle \\ =\mathcal{R}\left\{{\displaystyle {\int }_{{ℝ}^N}}\left[V\left(x\right)-V_{\alpha }\left(x\right)\right]u_n\overline{{\phi }_n}\text{d}x-\mu {\displaystyle {\int }_{{ℝ}^N}}\left[h\left(x\mathrm{,}{\left|u_n\right|}^2\right)-h_{\alpha }\left(x\mathrm{,}{\left|u_n\right|}^2\right)\right]u_n\overline{{\phi }_n}\text{d}x\right\}\\ \to 0.\end{array}$

Consequently, $\langle {J^{\prime }}_{\alpha }\left(u_n\right)\mathrm{,}{\phi }_n\rangle \to 0$. In addition, it follows from the periodicity of $V_{\alpha }$ and $h_{\alpha }$ with regard to the variable x and $y_n\in {\mathbb{Z}}^N$ that

$\langle {J^{\prime }}_{\alpha }\left({\tilde{u}}_n\right)\mathrm{,}\phi \rangle =\langle {J^{\prime }}_{\alpha }\left(u_n\right)\mathrm{,}{\phi }_n\rangle \mathrm{,}$

which means that $\langle {J^{\prime }}_{\alpha }\left({\tilde{u}}_n\right)\mathrm{,}\phi \rangle \to 0$. Therefore, as previous arguments we can conclude that ${J^{\prime }}_{\alpha }\left(\tilde{u}\right)=0$.

What follows is to prove $J_{\alpha }\left(\tilde{u}\right)\le c_{\mu }$. In fact, it follows from the boundedness of $\left\{{\Vert u_n\Vert }_{s\mathrm{,}A}\right\}$ and Lemma 3.5 that

${\displaystyle {\int }_{{ℝ}^N}}\left[V\left(x\right)-V_{\alpha }\left(x\right)\right]{\left|u_n\right|}^2\text{d}x\to 0$

and

${\displaystyle {\int }_{{ℝ}^N}}\left[h\left(x\mathrm{,}{\left|u_n\right|}^2\right)-h_{\alpha }\left(x\mathrm{,}{\left|u_n\right|}^2\right)\right]{\left|u_n\right|}^2\text{d}x\to \mathrm{0,}$

also

${\displaystyle {\int }_{{ℝ}^N}}\left[H\left(x\mathrm{,}{\left|u_n\right|}^2\right)-H_{\alpha }\left(x\mathrm{,}{\left|u_n\right|}^2\right)\right]\text{d}x\to 0.$

Thereupon,

${\displaystyle {\int }_{{ℝ}^N}}\left[h\left(x\mathrm{,}{\left|u_n\right|}^2\right)-H\left(x\mathrm{,}{\left|u_n\right|}^2\right)\right]\text{d}x={\displaystyle {\int }_{{ℝ}^N}}\left[h_{\alpha }\left(x\mathrm{,}{\left|u_n\right|}^2\right)-H_{\alpha }\left(x\mathrm{,}{\left|u_n\right|}^2\right)\right]\text{d}x+o_n\left(1\right)\mathrm{.}$

By the periodicity of $V_{\alpha }$ and $h_{\alpha }$ in the variable x again, (3) in (h₅), and it follows from the weakly lower semi-continuity of the norm and Fatou’s Lemma that

$\begin{array}{c}{c}_{\mu }=\underset{n\to \infty }{\lim }\left[J\left(u_n\right)-\frac{1}{2p}\langle J^{\prime }\left(u_n\right)\mathrm{,}{u}_n\rangle \right]\\ \ge \frac{1}{2p}\underset{n\to \infty }{\lim \inf }{\Vert u_n\Vert }_{s\mathrm{,}A}^p+\frac{\mu }{2p}\underset{n\to \infty }{\lim \inf }{\displaystyle {\int }_{{ℝ}^N}}\left[h\left(x\mathrm{,}{\left|u_n\right|}^2\right){\left|u_n\right|}^2-pH\left(x\mathrm{,}{\left|u_n\right|}^2\right)\right]\text{d}x\\ +\underset{n\to \infty }{\lim \inf }\left(\frac{1}{2p}-\frac{1}{2p_{\sigma \mathrm{,}s}^{\ast }}\right){\displaystyle {\int }_{{ℝ}^N}}\left[I_{\sigma }\left(x\right)\ast {\left|u_n\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\right]{\left|u_n\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\text{d}x\\ =\frac{1}{2p}\underset{n\to \infty }{\lim \inf }{\Vert u_n\Vert }_{s\mathrm{,}\alpha \mathrm{,}A}^p+\frac{\mu }{2p}\underset{n\to \infty }{\lim \inf }{\displaystyle {\int }_{{ℝ}^N}}\left[h_{\alpha }\left(x\mathrm{,}{\left|u_n\right|}^2\right){\left|u_n\right|}^2-p{H}_{\alpha }\left(x\mathrm{,}{\left|u_n\right|}^2\right)\right]\text{d}x\\ +\underset{n\to \infty }{\lim \inf }\left(\frac{1}{2p}-\frac{1}{2p_{\sigma \mathrm{,}s}^{\ast }}\right){\displaystyle {\int }_{{ℝ}^N}}\left[I_{\sigma }\left(x\right)\ast {\left|u_n\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\right]{\left|u_n\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\text{d}x\end{array}$

$\begin{array}{c}=\frac{1}{2p}\underset{n\to \infty }{\lim \inf }{\Vert {\tilde{u}}_n\Vert }_{s\mathrm{,}\alpha \mathrm{,}A}^p+\frac{\mu }{2p}\underset{n\to \infty }{\lim \inf }{\displaystyle {\int }_{{ℝ}^N}}\left[h_{\alpha }\left(x\mathrm{,}{\left|{\tilde{u}}_n\right|}^2\right){\left|{\tilde{u}}_n\right|}^2-p{H}_{\alpha }\left(x\mathrm{,}{\left|{\tilde{u}}_n\right|}^2\right)\right]\text{d}x\\ +\underset{n\to \infty }{\lim \inf }\left(\frac{1}{2p}-\frac{1}{2p_{\sigma \mathrm{,}s}^{\ast }}\right){\displaystyle {\int }_{{ℝ}^N}}\left[I_{\sigma }\left(x\right)\ast {\left|{\tilde{u}}_n\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\right]{\left|{\tilde{u}}_n\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\text{d}x\\ \ge \frac{1}{2p}{\Vert \tilde{u}\Vert }_{s\mathrm{,}\alpha \mathrm{,}A}^p+\frac{\mu }{2p}{\displaystyle {\int }_{{ℝ}^N}}\left[h_{\alpha }\left(x\mathrm{,}{\left|\tilde{u}\right|}^2\right){\left|\tilde{u}\right|}^2-p{H}_{\alpha }\left(x\mathrm{,}{\left|\tilde{u}\right|}^2\right)\right]\text{d}x\\ +\left(\frac{1}{2p}-\frac{1}{2p_{\sigma \mathrm{,}s}^{\ast }}\right){\displaystyle {\int }_{{ℝ}^N}}\left[I_{\sigma }\left(x\right)\ast {\left|\tilde{u}\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\right]{\left|\tilde{u}\right|}^{p_{\sigma \mathrm{,}s}^{\ast }}\text{d}x\\ =J_{\alpha }\left(\tilde{u}\right)-\frac{1}{2p}\langle {J^{\prime }}_{\alpha }\left(\tilde{u}\right)\mathrm{,}\tilde{u}\rangle =J_{\alpha }\left(\tilde{u}\right)\mathrm{.}\end{array}$

Finally, we argue that $\underset{t>0}{\max }{J}_{\alpha }\left(t\tilde{u}\right)=J_{\alpha }\left(\tilde{u}\right)$. In virtue of $\tilde{u}\ne 0$ and ${J^{\prime }}_{\alpha }\left(\tilde{u}\right)=0$, we get $\tilde{u}\in {\mathcal{N}}_{\alpha }$. Therefore, we can deduce that the conclusion holds from Lemma 3.1 (2). It follows from $\tilde{u}\ne 0$ and Lemma 3.1 (1) that there exists $t_{\tilde{u}}>0$ such that $t_{\tilde{u}}\tilde{u}\in \mathcal{N}$. Then, we have

$c_{\mu }=\underset{\mathcal{N}}{\inf }J\le J\left(t_{\tilde{u}}\tilde{u}\right)\le J_{\alpha }\left(t_{\tilde{u}}\tilde{u}\right)\le \underset{t>0}{\max }{J}_{\alpha }\left(t\tilde{u}\right)=J_{\alpha }\left(\tilde{u}\right)\le c_{\mu },$

which means that $J\left(t_{\tilde{u}}\tilde{u}\right)=c_{\mu }$.

In summary, $c_{\mu }$ is achieved. Moreover, by Lemma 3.4 (3), the corresponding minimizer is a ground state solution of (1.1). Then, we complete the proof of Theorem 1.1.

Conflicts of Interest

The author declares no conflicts of interest.

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