\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 306, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/306\hfil Multiplicity of solutions]
{Multiplicity of solutions to a nonlocal Choquard equation
involving fractional magnetic operators and critical exponent}

\author[F. Wang, M. Xiang \hfil EJDE-2016/306\hfilneg]
{Fuliang Wang, Mingqi Xiang}

\address{Fuliang Wang \newline
College of Science,
Civil Aviation University of China,
Tianjin 300300, China}
\email{flwang@cauc.edu.cn}

\address{Mingqi Xiang (corresponding author) \newline
College of Science,
Civil Aviation University of China,
Tianjin 300300, China}
\email{xiangmingqi\_hit@163.com}

\thanks{Submitted October 27, 2016. Published November 30, 2016.}
\subjclass[2010]{49A50, 26A33, 35J60, 47G20}
\keywords{Choquard equation; fractional magnetic operator;
\hfill\break\indent variational method; critical exponent}

\begin{abstract}
 In this article, we study the multiplicity of solutions to a nonlocal
 fractional Choquard equation involving an external magnetic potential
 and critical exponent, namely,
 \begin{gather*}
 \begin{aligned}
 &(a+b[u]_{s,A}^2)(-\Delta)_A^su+V(x)u \\
 &=\int_{\mathbb{R}^N}\frac{|u(y)|^{2_{\mu,s}^*}}{|x-y|^{\mu}}dy|u|^{2_{\mu,s}^*-2}u
 +\lambda h(x)|u|^{p-2}u\quad \text{in }\mathbb{R}^N,
 \end{aligned}\\
 [u]_{s,A}=\Big(\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^N}
 \frac{|u(x)-e^{i(x-y)\cdot A(\frac{x+y}{2})}u(y)|^2}{|x-y|^{N+2s}}\,dx\,dy\Big)^{1/2}
 \end{gather*}
 where $a\geq 0, b>0$, $0<s<\min\{1,N/4\}$, $4s\leq \mu<N$,
 $V:\mathbb{R}^N\to \mathbb{R}$ is a sign-changing scalar potential,
 $A:\mathbb{R}^N\to \mathbb{R}^N$ is the magnetic potential,
 $(-\Delta )_A^s$ is the fractional magnetic operator, $\lambda>0$ is
 a parameter, $2_{\mu,s}^*=\frac{2N-\mu}{N-2s}$ is the critical exponent
 in the sense of the Hardy-Littlewood-Sobolev inequality and $2<p<2_s^*$.
 Under suitable assumptions on $a,b$ and $\lambda$, we obtain multiplicity
 of nontrivial solutions by using variational methods.
 In particular, we obtain the existence of infinitely many nontrivial solutions
 for the degenerate Kirchhoff case, that is, $a=0$, $b>0$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction and statement of main results}

In this article we consider the multiplicity of solutions to the
 Choquard-Kirchhoff type problem
\begin{equation} \label{eq1}
\begin{aligned}
&(a+b\|u\|_{s,A}^2)(-\Delta)_A^su+V(x)u \\
&=\int_{\mathbb{R}^N}|u|^{2_{\mu,s}^*}\mathcal{K}_\mu(x-y)dy|u|^{2_{\mu,s}^*-2}u
+\lambda h(x)|u|^{p-2}u\quad \text{in }\mathbb{R}^N
\end{aligned}
\end{equation}
where $a\geq 0, b> 0$, $s\in(0,1)$, $N>\mu\geq4s$,
$2_{\mu,s}^*=\frac{2N-\mu}{N-2s}$, $2_s^*=\frac{2N}{N-2s}$,
 $V:\mathbb{R}^N\to\mathbb{R}$ is the scalar
potential, $\mathcal{K}_\mu(x)=|x|^{-\mu}$, $A:\mathbb{R}^N\to \mathbb{R}^N$
is the magnetic potential, $h:\mathbb{R}^N\to\mathbb{R}_0^+$, $\lambda>0$ and
 $(-\Delta )_A^s$ is the fractional magnetic operator which, up to normalization,
 defined as
\begin{equation*}
(-\Delta)_A^su(x)=2
\lim_{\varepsilon\to 0^+}\int_{\mathbb{R}^N\setminus B_\varepsilon(x)}
\frac{u(x)-e^{{\rm i}(x-y)\cdot A(\frac{x+y}{2})}u(y)}{|x-y|^{N+2s}}\,dy,\quad
 \forall x\in\mathbb{R}^{N},
\end{equation*}
along any $\varphi\in C_0^\infty(\mathbb{R}^N,\mathbb{C})$, see \cite{PDMS}
 and the references therein for further details on this kinds of operators.
Here $B_\varepsilon(x)$ denotes the ball in $\mathbb{R}^N$ with radius
$\varepsilon>0$ centered at $x\in\mathbb{R}^N$. As showed in \cite{MSBV},
up to correcting the operator with factor $(1-s)$ it follows that
$(-\Delta)_A^s u$ converges to $-(\nabla u-{\rm i} A)^2u$ in the limit
$s\uparrow1$, where
\begin{align*}
-(\nabla u-{\rm i} A)^2u=-\Delta u+2{\rm i}A(x)\cdot\nabla u+|A(x)|^2u
+{\rm i}u\operatorname{div}A(x).
\end{align*}
Thus, up to normalization, we may think the nonlocal case as an approximation
of the local case. In recent years, the following magnetic Schr\"{o}dinger
equations like
\begin{align*}
-(\nabla u-{\rm i} A)^2u+V(x)u=f(x,u)
\end{align*}
have been extensively studied; see \cite{GAAS,JDJV,AFY,MS}.
We also collect some recent results on the fractional magnetic operators;
see \cite{MPSZ,ZSZ,PSV1,PSV2} and the references cited there.

Clearly, the operator $(-\Delta)_A^s$ is consistent with the definition of
fractional Laplacian $(-\Delta )^s$ if $A\equiv0$. For more details on
 the fractional Laplacian, we refer to \cite{r28}. The fractional Laplacian
operator $(-\Delta)^s$ can be seen as the infinitesimal generators of
L\'{e}vy stable diffusion processes (see \cite{AFY}). This type of operators arises
in a quite natural way in many different applications, such as, continuum mechanics,
phase transition phenomena, population dynamics
and game theory, as they are the typical outcome of stochastically stabilization
of L\'{e}vy processes, see \cite{r8, r3,r5,r6,r7}. In the context of fractional
quantum mechanics, non-linear fractional Schr\"{o}dinger equation has been
 proposed by Laskin \cite{laskin1,laskin2} as a result of expanding the Feynman
path integral, from the Brownian-like to the L\'{e}vy-like
quantum mechanical paths. The literature on non-local operators and on
their applications is very interesting and quite large, we refer the interested
readers to see \cite{r14,r15,r21,r16,r17,r18,r20} and the references therein.


 Equation \eqref{eq1} is a nonlocal elliptic type equation and covers in particular
for $s=1, 2_{\mu,s}^*=2, A\equiv 0$ the Choquard-Pekar equation, which appears
as a model in quantum theory of a polaron at rest, see \cite{SP}.
The time-dependent form of \eqref{eq1} also describes the self-gravitational
collapse of a quantum mechanical wave function, in which context it is called
Hartree equation or the Newton-Schrodinger eqution \cite{MPT}.
In recent years, the Choquard and related equations
have been studied by many authors, see \cite{Lieb, Lions, GPM, VMJS} and the
references therein. Very recently, D'Avenia, Siciliano and Squassina studied
the existence, regularity and asymptotic of the solutions for the following
fractional Choquard equation
\begin{equation} \label{eq2}
(-\Delta)^s u+\omega u=(\mathcal{K}_\alpha*|u|^p)|u|^{p-2}u,\quad
 u\in H^s(\mathbb{R}^N),
\end{equation}
where $s\in (0,1)$, $\omega>0$, $N\geq 3$,
$1+\frac{\alpha}{N}<p<\frac{N+\alpha}{N-2s}$, $\alpha\in (0,N)$ and
$\mathcal{K}_\alpha(x)=|x|^{\alpha-N}$. The existence of groundstates for
 fractional Choquard equations with general nonlinearities was obtained by Shen,
Gao and Yang \cite{SGY} using variational methods.
In \cite{PXZ}, Pucci, Xiang and Zhang extended equation \eqref{eq2} to the
 fractional $p$-Laplacian and obtained several existence results by using
variational methods.

Fiscella and Valdinoci \cite{fiscella} proposed a stationary Kirchhoff variational
model, in bounded regular domains of $\mathbb{R}^N$, which takes into account
the nonlocal aspect of the tension arising from nonlocal measurements of the
fractional length of the string, and obtained the existence and multiplicity
of solutions for fractional Kirchhoff problems by using variational method
and the concentration-compactness principle. Nyamoradi \cite{nyamoradi}
studied a class of fractional Kirchhoff type equation in a bounded domain
$\Omega$ and obtained three solutions by using three critical point theorem.
For more details about stationary Kirchhoff problems involving the
fractional Laplacian, we refer the interested readers to
\cite{PDMS,FMS,XZF1,XZR1,XZR2}.

Inspired by the above cited papers, we consider the critical case of \eqref{eq2}
and prove multiplicity results depending on $\lambda,a$ $b$ and $N$.
In particular, when $N>\mu=4s, a=0,b>0, V\equiv0$, we obtain
infinitely many solutions for \eqref{eq1} by applying critical point theory.
Since equation \eqref{eq1} contains a critical nonlinearity, it is
difficult to get the global $(PS)$ condition. To overcome this difficulty,
we borrow some tricks from articles \cite{NM,LLT}.


\begin{definition}\label{def1.1} \rm
We say that $u\in D_{A}^s(\mathbb{R}^N,\mathbb{C})$ is a weak solution of \eqref{eq1},
if \begin{align*}
&(a+b\|u\|_{s,A}^2)\\
&\times \Re\int_{\mathbb{R}^N}\int_{\mathbb{R}^{N}}\frac{(u(x)
 -e^{{\rm i}(x-y)\cdot A(\frac{x+y}{2})}u(y))(\overline{\varphi(x)
 -e^{{\rm i}(x-y)\cdot A(\frac{x+y}{2})}\varphi(y)})}{|x-y|^{N+2s}} \,dx\,dy \\
&+\Re\int_{\mathbb{R}^N}V(x)u\overline{\varphi} dx \\
&=\Re\int_{\mathbb{R}^N}(\mathcal{K}_\mu*|u|^{2_{\mu,s}^*})
|u|^{2_{\mu,s}^*-2}u\overline{\varphi} dx
+\lambda\Re\int_{\mathbb{R}^N}h(x)|u|^{p-2}u\overline{\varphi} dx,
\end{align*}
for any $\varphi\in D^s_{A}(\mathbb{R}^N,\mathbb{C})$.
\end{definition}

The best constant of Hardy-Littlehood-Sobolev inequality is
\begin{equation} \label{embed1}
S_{H,L}:=\inf_{u\in D^{s}_A(\mathbb{R}^N,\mathbb{C})\setminus\{0\}}
\frac{[u]_{s,A}^2}{(\int_{\mathbb{R}^N}
\int_{\mathbb{R}^N}\frac{|u(x)|^{2_{\mu,s}^*}|u(y)|^{2_{\mu,s}^*}}{|x-y|^\mu}\,dx\,dy)^{\frac{1}{2_{\mu,s}^*}}}.
\end{equation}

\begin{theorem}\label{thm1}
Assume that $s\in(0,1)$, $N>\mu\geq4s$,
$ V\in L^{\frac{N}{2s}}(\mathbb{R}^N)$, $2<p<2_s^*$, $h\geq0$,
$h\not\equiv 0, h\in L^{\frac{2_s^*}{2_s^*-p}}(\mathbb{R}^N)$ and
$A\in C(\mathbb{R}^N,\mathbb{R}^N)$.
If $\mu=4s$, $a\geq 0$ and $b>S_{H,L}^{-2_{\mu,s}^*}$
or $\mu>4s$, $a>0,\ b>0$ and
\begin{align}\label{as1}
a>(2-2_{\mu,s}^*)
\Big(\frac{b}{2_{\mu,s}^*-1}\Big)^{-\frac{2_{\mu,s}^*-1}{2-2_{\mu,s}^*}}
S_{H,L}^{-\frac{2_{\mu,s}^*}{2-2_{\mu,s}^*}},
\end{align}
then there exists $\lambda^*>0$ such that \eqref{eq1} admits at least two
nontrivial solutions in $D^s_A(\mathbb{R}^N,\mathbb{C})$ for all
$\lambda>\lambda^*$.
\end{theorem}

\begin{theorem}\label{thm2}
Assume that $s\in(0,1)$, $N>\mu=4s$, $a=0$, $b>S_{H,L}^{-2_{\mu,s}^*}$,
$ V\equiv 0$, $2<p<2_s^*$, $h\geq0$, $h\not\equiv 0$,
$h\in L^{\frac{2_s^*}{2_s^*-p}}(\mathbb{R}^N)$ and
 $A\in C(\mathbb{R}^N,\mathbb{R}^N)$.
Then \eqref{eq1} has infinitely many pairs of solutions in
$D^s_A(\mathbb{R}^N,\mathbb{C})$ for all $\lambda>0$.
Moreover, any nontrivial solution $u\in D^s_A(\mathbb{R}^N,\mathbb{C})\setminus\{0\}$
satisfies
$$
[u]_{s,A}\leq \Bigg[\frac{\lambda\|h\|_{L^{\frac{2_s^*}{2_s^*-p}}
(\mathbb{R}^N)}}{S^{p/2}(b-S_{H,L}^{-2_{\mu,s}^*})}
\bigg]^{\frac{1}{4-p}},
$$
 where $S$ is the best constant of the embedding
$D^s_A(\mathbb{R}^N,\mathbb{C})\hookrightarrow L^{2_s^*}(\mathbb{R}^N,\mathbb{C})$
defined by
\begin{equation} \label{embed2}
S:= \inf_{u\in D^{s}_A(\mathbb{R}^N,\mathbb{C})\setminus\{0\}}
\frac{[u]_{s,A}^2}{\|u\|_{L^{2_s^*}(\mathbb{R}^N,\mathbb{C})}^2}\,.
\end{equation}
\end{theorem}

\begin{remark} \label{rmk1.1} \rm
We say that equation \eqref{eq1} is non-degenerate if $a>0,b\geq0$;
and degenerate if $a=0,b>0$. To the best of our knowledge, this article
 is the first to deal with the multiplicity of solutions for fractional
Choquard-Kirchhoff type equations with external magnetic operator and
 critical exponent.
\end{remark}

This article is organized as follows.
In Section 2, we recall some necessary definitions and properties of spaces
$D^s(\mathbb{R}^N)$ and $D_A^s(\mathbb{R}^N,\mathbb{C})$.
In Section 3, the multiplicity of solutions of \eqref{eq1} is obtained
by using variational methods.

\section{Preliminaries}

In this section, we first give some basic results of fractional Sobolev
spaces that will be used later.
Let $N>1$, $0<s<1$ be real number satisfying $2s<N$ and the fractional
critical exponent $2^*_s$ be defined as
$2_s^*=\frac{2N}{N-2s}$.
The fractional Sobolev space $D^{s}(\mathbb{R}^N)$ is defined as the closure of
$C_0^\infty(\mathbb{R}^N)$ with respect to the norm
\begin{align*}
[u]_{s}=\Big(\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}\,dx\,dy
\Big)^{1/2},
\end{align*}
The embedding $D^{s}(\mathbb{R}^N)\hookrightarrow L^{2_s^*}(\mathbb{R}^N)$
is continuous by \cite[Theorem 6.7]{r28}.
Suppose that $A:\mathbb{R}^N\to\mathbb{R}^N$ is a continuous function.
Consider the magnetic Gagliardo semi-norm defined by
\begin{align*}
[u]_{s,A}:=\Big(\iint_{\mathbb{R}^{2N}}\frac{|u(x)-e^{{\rm i}(x-y)\cdot
A(\frac{x+y}{2})}u(y)|^2}{|x-y|^{N+2s}}\,dx\,dy\Big)^{1/2},
\end{align*}
and define $D^{s}_A(\mathbb{R}^N,\mathbb{C})$ as the closure of
 $C_0^\infty(\mathbb{R}^N,\mathbb{C})$ with respect to $[\cdot]_{s,A}$.

\begin{lemma}\label{lem2.1}
For each $u\in D^s_{A}(\mathbb{R}^N,\mathbb{C})$ it holds
$|u|\in D^s(\mathbb{R}^N)$. More precisely,
\[
\big[|u|\big]_{s}\leq [u]_{s,A}, \quad \text{for all }
 u\in D^s_{A}(\mathbb{R}^N,\mathbb{C}).
\]
\end{lemma}

\begin{proof}
The proof follows by using the pointwise diamagnetic inequality
\[
\big||u(x)|-|u(y)|\big|\leq \big|u(x)-e^{{\rm i}(x-y)\cdot
A(\frac{x+y}{2})}u(y)\big|,
\]
 for a.e. $x,y\in\mathbb{R}^N$, see \cite[Lemma 3.1, Remark 3.2]{PDMS}.
\end{proof}

Finally, we introduce the well-known
Hardy-Littlewood-Sobolev inequality, see \cite{LL}.

\begin{lemma}
Assume $1<r,t<\infty$ and $0<\mu<N$ with $\frac{1}{r}+\frac{1}{t}+\frac{\mu}{N}=2$.
If $u\in L^r(\mathbb{R}^N)$ and $v\in L^t(\mathbb{R}^N)$, then
there exists $C(N,\alpha,r,t)>0$ such that
\[
\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{|u(x)||v(y)|}{|x-y|^\mu}\,dx\,dy
\leq C(N,\mu,r,t)\|u\|_{L^r(\mathbb{R}^N)}\|v\|_{L^t(\mathbb{R}^N)}.
\]
\end{lemma}

\section{Proof of Theorem \ref{thm1}}

The functional associated with \eqref{eq1} is defined as
\begin{align*}
\mathcal{I}(u)
&=\frac{a}{2}[u]^2_{s,A}+\frac{b}{4}[u]_{s,A}^{4}
 +\frac{1}{2}\int_{\mathbb{R}^N}V(x)|u|^2dx\\
&\quad -\frac{1}{22_{\mu,s}}
\int_{\mathbb{R}^N}\int_{\mathbb{R}^{N}}
 \frac{|u(x)|^{2_{\mu,s}^*} |u(y)|^{2_{\mu,s}^*}}{|x-y|^\mu}\,dx\,dy
-\frac{\lambda}{p}\int_{\mathbb{R}^N}h(x)|u|^{p}dx.
\end{align*}
for all $u\in D_{A}^s(\mathbb{R}^N,\mathbb{C})$.

From $ V\in L^{\frac{N}{2s}}(\mathbb{R}^N)$ and
$h\in L^{\frac{2_s^*}{2_s^*-p}}(\mathbb{R}^N)$,
the Hardy-Littlehood-Sobolev inequality and the fractional Sobolev inequality,
 one can show that $\mathcal{I}$ is well-defined, of class $C^1$ and
\begin{align*}
&\langle \mathcal{I}'(u),v\rangle\\
&=(a+b[u]_{s,A}^2) \\
&\times \Re\int_{\mathbb{R}^N}\int_{\mathbb{R}^{N}}
\frac{[u(x)-e^{i(x-y)\cdot A(\frac{x+y}{2})}u(y)][\overline{v(x)-e^{i(x-y)
 \cdot A(\frac{x+y}{2})}v(y)}]}{|x-y|^{N+2s}}\,dx\,dy\\
&\quad +\Re\int_{\mathbb{R}^N}Vu\overline{v}dx
 -\Re\int_{\mathbb{R}^N}(\mathcal{K}*|u|^{2_{\mu,s}^*})
|u|^{2_{\mu,s}^*-2}u\overline{v}dx
-\lambda\Re\int_{\mathbb{R}^N}h|u|^{p-2}u\overline{v}dx,
\end{align*}
for all $u,v\in D_{A}^s(\mathbb{R}^N,\mathbb{C})$. Hence a critical
point of $\mathcal{I}$ is a (weak) solution of \eqref{eq1}.

\begin{definition} \label{def3.1} \rm
For any $c\in\mathbb{R}$, $\{u_n\}$ is called a $(PS)_c$ sequence of
$\mathcal{I}$ in $D^{s}_A(\mathbb{R}^N,\mathbb{C})$, if
$\mathcal{I}(u_n)\to c$ and
$\mathcal{I}'(u_n)\to 0$ as $n\to\infty$.
We say that $\mathcal{I}$ satisfies $(PS)_c$ condition if any
$(PS)_c$ sequence of $\mathcal{I}$ admits a convergent subsequence
 in $D^{s}_A(\mathbb{R}^N,\mathbb{C})$.
\end{definition}


Now we give a key lemma for proving the main results.

\begin{lemma}\label{lem3.1}
Under the conditions of Theorem \ref{thm1}, functional $\mathcal{I}$
satisfies the $(PS)_c$ conditions in $D_{A}^s(\mathbb{R}^N,\mathbb{C})$
for all $\lambda>0$.
\end{lemma}

\begin{proof}
Suppose that $\{u_n\}\subset D_{A}^s(\mathbb{R}^N,\mathbb{C})$
is a $(PS)_c$ sequence of functional $\mathcal{I}$, i.e.
$$\mathcal{I}(u_n)\to c,\quad
\mathcal{I}'(u_n)\to 0
$$
as $n\to\infty$.

By H\"{o}lder's inequality, \eqref{embed1} and \eqref{embed2}, we deuce
\begin{align}\label{ineq}
\mathcal{I}(u)
&\geq\frac{a}{2}[u]_{s,A}^2+\frac{b}{4}[u]_{s,A}^4
-\frac{1}{2}S^{-1}\|V\|_{L^{\frac{N}{2s}}(\mathbb{R}^N)} [u]_{s,A}^2\\
&\quad -\frac{1}{22_{\mu,s}^*}S_{H,L}^{-2_{\mu,s}^*}[u]_{s,A}^{22_{\mu,s}^*}-
\frac{1}{p}S^{-\frac{p}{2}}\lambda\|h\|_{L^{\frac{2_s^*}{2_s^*-p}}
(\mathbb{R}^N)}[u]_{s,A}^{p},
\end{align}
for all $u\in D_{A}^s(\mathbb{R}^N,\mathbb{C})$.
When $\mu=4s$, since $\frac{2}{2_{\mu,s}^*}S_{H,L}^{-4}<b$, $2_{\mu,s}^*=2$
and $2<p<2_s^*<4$ by $N>4s$, it follows that $\mathcal{I}$ is
coercive and bounded from below on
$D_{A}^s(\mathbb{R}^N,\mathbb{C})$.
When $N>\mu\geq 4s$, since $a> 0,b>0$, $2_{\mu,s}^*<2$ and
$2_s^*<4$, it follows that $\mathcal{I}$ is
coercive and bounded from below on
$D_{A}^s(\mathbb{R}^N,\mathbb{C})$.
Hence, $\{u_n\}$ is bounded in $D_{A}^s(\mathbb{R}^N,\mathbb{C})$.
Then there exists $u\in D_{A}^s(\mathbb{R}^N,\mathbb{C})$ such that,
up to a subsequence, it follows that
\begin{equation} \label{eq3.1}
\begin{aligned}
u_n\rightharpoonup u\quad\text{in } D_{A}^s(\mathbb{R}^N,\mathbb{C})\text{ and in }
 L^{2_s^*}(\mathbb{R}^N,\mathbb{C}),\\
u_n\to u\quad\text{a.e. in } \mathbb{R}^N\text{ and in }
 L^{p}_{\rm loc}(\mathbb{R}^N),\; 1\leq p<2_s^*,\\
|u_n|^{2_s^*-2}u_n\rightharpoonup |u|^{2_s^*-2}u\quad\text{weakly in }
 L^{\frac{2_s^*}{2_s^*-1}}(\mathbb{R}^N,\mathbb{C}),
\end{aligned}
\end{equation}
as $n\to\infty$. We first show that
\begin{align}\label{eq3.2}
\lim_{n\to\infty}\int_{\mathbb{R}^N}V(x)|u_n|^2dx
=\int_{\mathbb{R}^N}V(x)|u|^2dx.
\end{align}
Since $V\in L^{\frac{N}{2s}}(\mathbb{R}^N)$, for any $\varepsilon>0$
there exists $R_\varepsilon>0$ such that
\[
\Big(\int_{\mathbb{R}^N\setminus B_{R_\varepsilon}(0)}|V(x)|^{\frac{N}{2s}}dx
\Big)^{2s/N}<\varepsilon.
\]
By H\"{o}lder's inequality, we deduce
\begin{equation} \label{eq3.3}
\begin{aligned}
&\big|\int_{\mathbb{R}^N\setminus B_{R_\varepsilon}(0)}
V(x)(|u_n|^2-|u|^2)dx\big| \\
&\leq \Big(\int_{\mathbb{R}^N\setminus B_{R_\varepsilon}(0)}
 |V(x)|^{\frac{N}{2s}}dx\Big)^{2s/N}
\|u_n\|_{L^{2_s^*}(\mathbb{R}^N)}^2\\
&\quad +\Big(\int_{\mathbb{R}^N\setminus B_{R_\varepsilon}(0)}|V(x)|^{\frac{N}{2s}}dx
\Big)^{2s/N}
\|u\|_{L^{2_s^*}(\mathbb{R}^N)}^2\\
&\leq C\Big(\int_{\mathbb{R}^N\setminus B_{R_\varepsilon}(0)}|V(x)|^{\frac{N}{2s}}
 dx\Big)^{2s/N}
\leq C\varepsilon.
\end{aligned}
\end{equation}
On the other hand, by the boundedness of $\{u_n\}$, for any measurable
non-empt subset $\Omega\subset B_{R_\varepsilon}$, we have
\[
\big|\int_{\Omega} V(x)(|u_n|^2+|u|^2)dx\big|
\leq C\Big(\int_{\Omega}|V(x)|^{\frac{N}{2s}}dx\Big)^{2s/N}.
\]
It follows from $V\in L^{\frac{N}{2s}}(\mathbb{R}^N)$ that the sequence
$\{V(x)(|u_n|^2-|u|^2)\}$ is equi-integrable in $L^1(B_{R_\varepsilon}(0))$.
Thus the Vitali convergence theorem implies
\begin{align}\label{eq3.4}
\lim_{n\to\infty}
\int_{B_{R_\varepsilon}(0)}V(x)|u_n|^2dx=\int_{B_{R_\varepsilon}(0)}V(x)|u|^2dx.
\end{align}
Combining \eqref{eq3.3} with \eqref{eq3.4}, we obtain the desired result
\eqref{eq3.2}.
By using a similar discussion, we can deduce from $h\in L^{2_s^*}(\mathbb{R}^N)$
that
 \begin{align}\label{eq3.44}
\lim_{n\to\infty}
\int_{\mathbb{R}^N}h(x)|u_n|^pdx=\int_{\mathbb{R}^N}h(x)|u|^pdx.
\end{align}

Let $w_n=u_n-u$. Then by \eqref{eq3.1}, we obtain
\begin{equation} \label{eq3.5}
\begin{gathered}
[u_n]_{s,A}^2=[w_n]_{s,A}^2+[u]_{s,A}^2+o(1),\\
[u_n]_{s,A}^4=[w_n]_{s,A}^4+[u]_{s,A}^4+2[u_n]_{s,A}^2[u]_{s,A}^2+o(1).
\end{gathered}
\end{equation}
By the Brezis-Lieb type lemma (see \cite{GY}), one has
\begin{equation} \label{eq3.7}
\begin{aligned}
&\int_{\mathbb{R}^N}(\mathcal{K}_\mu*|w_n|^{2_{\mu,s}^*})|w_n|^{2_{\mu,s}^*}dx\\
&=\int_{\mathbb{R}^N}(\mathcal{K}_\mu*|u_n|^{2_{\mu,s}^*})|u_n|^{2_{\mu,s}^*}dx
-\int_{\mathbb{R}^N}(\mathcal{K}_\mu*|u|^{2_{\mu,s}^*})|u|^{2_{\mu,s}^*}dx
+o(1).
\end{aligned}
\end{equation}

Without loss of generality, we assume that $\lim_{n\to\infty}[w_n]_{s,A}=\eta$.
From $\{u_n\}$ is a $(PS)_c$ sequence and the boundedness of $\{u_n\}$, we have
\begin{equation} \label{eq3.9}
\begin{aligned}
\langle\mathcal{I}'(u_n),u_n\rangle
&= a[u_n]_{s,A}^2+b[u_n]_{s,A}^4+\int_{\mathbb{R}^N}V(x)|u_n|^2dx
\\&-\int_{\mathbb{R}^N}(\mathcal{K}_\mu*|u_n|^{2_{\mu,s}^*})|u_n|^{2_{\mu,s}^*}dx
-\lambda\int_{\mathbb{R}^N}h(x)|u_n|^{p}dx=o(1)
\end{aligned}
\end{equation}
and
\begin{equation} \label{eq3.10}
\begin{aligned}
\lim_{n\to\infty}\langle \mathcal{I}'(u_n),u\rangle
&=a[u]_{s,A}^2+b[u]_{s,A}^4+b\eta^2[u]_{s,A}^2+\int_{\mathbb{R}^N}V(x)|u|^2dx\\
&\quad -\int_{\mathbb{R}^N}
(\mathcal{K}_\mu*|u|^{2_{\mu,s}^*})|u|^{2_{\mu,s}^*}dx
-\lambda\int_{\mathbb{R}^N}h(x)|u|^{p}dx=0.
\end{aligned}
\end{equation}
Here we have used that
\begin{equation} \label{hb}
\lim_{n\to\infty}\int_{\mathbb{R}^N}(\mathcal{K}_\mu*|u_n|^{2_{\mu,s}^*})|u_n|^{2_{\mu,s}^*-2}u_nudx
=\int_{\mathbb{R}^N}(\mathcal{K}_\mu*|u_n|^{2_{\mu,s}^*})|u_n|^{2_{\mu,s}^*}dx.
\end{equation}
Indeed, by the Hardy-Littlewood-Sobolev inequality, the Riesz potential
defines a linear continuous map from $L^{\frac{2N}{2N-\mu}}(\mathbb{R}^N)$ to
$L^{\frac{2N}{\mu}}(\mathbb{R}^N)$. Then
\begin{align}\label{n1}
\mathcal{K}_\mu*|u_n|^{2_{\mu,s}^*}\rightharpoonup \mathcal{K}_\mu*|u|^{2_{\mu,s}^*}
\quad\text{in } L^{\frac{2N}{\mu}}(\mathbb{R}^N),
\end{align}
as $n\to\infty$. Note that for any measurable subset $U\subset \mathbb{R}^N$,
 we have
\begin{align*}
\int_U \left||u_n|^{2_{\mu,s}*-2}u_n u\right|^{\frac{2_s^*}{2_{\mu,s}^*}}dx
\leq \|u_n\|_{L^{2_s^*}(\mathbb{R}^N)}^{\frac{2_{\mu,s}^*-1}{2_{\mu,s}^*}}
\|u\|_{L^{2_s^*}(U)}^{\frac{1}{2_{\mu,s}^*}},
\end{align*}
which implies that $\{||u_n|^{2_{\mu,s}^*-2}u_n u|^{\frac{2_s^*}{2_{\mu,s}^*}}\}$
is equi-integrable in $L^1(\mathbb{R}^N)$. Observe that
$|u_n|^{2_{\mu,s}^*-2}u_n u\to |u|^{2_{\mu,s}^*}$ a.e. in $\mathbb{R}^N$,
 then the Vitali convergence theorem yields
\begin{align}\label{n2}
|u_n|^{2_{\mu,s}^*-2}u_n u\to |u|^{2_{\mu,s}^*}\quad\text{in }
 L^{\frac{2_s^*}{2_{\mu,s}^*}}(\mathbb{R}^N).
\end{align}
Combining \eqref{n1} with $\eqref{n2}$ and
$\frac{2_s^*}{2_{\mu,s}^*}=\frac{2N}{2N-\mu}$, we obtain the desired result
\eqref{hb}.

It follows from \eqref{eq3.9} and \eqref{eq3.10} that
\begin{align*}
&a[u]_{s,A}^2+a[w_n]_{s,A}^2+b[u]^4_{s,A}+b[w_n]_{s,A}^4
+2b[w_n]_{s,A}^2[u]_{s,A}^2 \\
&- \int_{\mathbb{R}^N}(\mathcal{K}_\mu*|u|^{2_{\mu,s}^*})|u|^{2_{\mu,s}^*}dx
-\int_{\mathbb{R}^N}
(\mathcal{K}_\mu*|w_n|^{2_{\mu,s}^*})|w_n|^{2_{\mu,s}^*}dx
=o(1).
\end{align*}
Then
\begin{align*}
a[w_n]_{s,A}^2+b[w_n]_{s,A}^4+b[w_n]_{s,A}^2[u]_{s,A}^2
-\int_{\mathbb{R}^N}
(\mathcal{K}_\mu*|w_n|^{2_{\mu,s}^*})|w_n|^{2_{\mu,s}^*}dx
=o(1).
\end{align*}
From the definition of $S_{H,L}$, we obtain
\begin{align*}
\int_{\mathbb{R}^N}(\mathcal{K}_\mu*|w_n|^{2_{\mu,s}^*})|w_n|^{2_{\mu,s}^*}dx
\leq S_{H,L}^{2_{\mu,s}^*}[w_n]_{s,A}^{22_{\mu,s}^*}.
\end{align*}
Using this and letting $n\to\infty$, we arrive at the inequality
\begin{align*}
a\eta^2+b\eta^2[u]_{s,A}^2+b\eta^4\leq
 S_{H,L}^{-2_{\mu,s}^*}\eta^{22_{\mu,s}^*},
\end{align*}
which implies
\begin{equation} \label{eq3.12}
a\eta^2+b\eta^4\leq
 S_{H,L}^{-2_{\mu,s}^*}\eta^{22_{\mu,s}^*}.
\end{equation}
When $\mu=4s$ and $S_{H,L}^{-2_{\mu,s}^*}<b$, it follows from \eqref{eq3.12}
that $\eta=0$. Thus, $u_n\to u$ in $D^s_{A}(\mathbb{R}^N,\mathbb{C})$.

When $\mu>4s$, it follows from \eqref{eq3.12} and the Young inequality that
\begin{align*}
a\eta^2+b\eta^4
&\leq \frac{1}{\frac{1}{2_{\mu,s}^*-1}}
(\eta^{42_{\mu,s}^*-4})^{\frac{1}{2_{\mu,s}^*-1}}
\Big[\Big(\frac{b}{2_{\mu,s}^*-1}\Big)^{2_{\mu,s}^*-1}\Big]^{\frac{1}{2_{\mu,s}^*-1}}
\\
&\quad +\frac{1}{\frac{1}{2-2_{\mu,s}^*}}
\Big(\frac{b}{2_{\mu,s}^*-1}\Big)^{-\frac{2_{\mu,s}^*-1}{2-2_{\mu,s}^*}}
S_{H,L}^{-\frac{2_{\mu,s}^*}{2-2_{\mu,s}^*}}\Big(\eta^{4-22_{\mu,s}^*}
 \Big)^{\frac{1}{2-2_{\mu,s}^*}}\\
&=b\eta^4+(2-2_{\mu,s}^*)\Big(\frac{b}{2_{\mu,s}^*-1}
 \Big)^{-\frac{2_{\mu,s}^*-1}{2-2_{\mu,s}^*}}
 S_{H,L}^{-\frac{2_{\mu,s}^*}{2-2_{\mu,s}^*}}\eta^2.
\end{align*}
Consequently,
 \begin{align*}
\Big\{a-(2-2_{\mu,s}^*)\Big(\frac{b}{2_{\mu,s}^*-1}
\Big)^{-\frac{2_{\mu,s}^*-1}{2-2_{\mu,s}^*}}
 S_{H,L}^{-\frac{2_{\mu,s}^*}{2-2_{\mu,s}^*}}\Big\}\eta^{2}\leq 0,
\end{align*}
which together with \eqref{as1} implies that $\eta=0$.
Hence $u_n\to u$ in $D^s_A(\mathbb{R}^N,\mathbb{C})$.
\end{proof}

\begin{remark}\label{re3.1} \rm
Clearly, when $a=0,V\equiv0$, $\mu=4s$, $2<p<2_s^*$ and $b>S_{H,L}^{-2_{\mu,s}^*}$,
the functional $\mathcal{I}$
also satisfies the $(PS)_c$ condition in $D^s_A(\mathbb{R}^N,\mathbb{C})$.
\end{remark}

\begin{proof}[Proof of Theorem \ref{thm1}]
We first show that \eqref{eq1} has a nontrivial global minimizer solution.
 By \eqref{eq3.1}, we know
$m:=\inf_{u\in D^s_A(\mathbb{R}^N,\mathbb{C})}\mathcal{I}(u)$
is well-defined. Now we claim that there exists $\lambda^*>0$ such that
$m<0$ for all $\lambda>\lambda^*$. Actually, we can choose
$\varphi_0\in D^s_A(\mathbb{R}^N,\mathbb{C})$ with $[\varphi_0]_{s,A}=1$ and
 $\int_{\mathbb{R}^N}h(x)|\varphi_0|^pdx>0$, then
 \begin{align*}
 \mathcal{I}(\varphi_0)
&\leq \frac{a}{2}+\frac{1}{2}\|V\|_{L^{\frac{N}{2s}}(\mathbb{R}^N)}S^{-1}
 +\frac{b}{4}-\frac{1}{22_{\mu,s}^*}
 \int_{\mathbb{R}^N}(\mathcal{K}_\mu*|\varphi_0|^{2_{\mu,s}^*})
 |\varphi_0|^{2_{\mu,s}^*}dx \\
&\quad -\frac{\lambda}{p}\int_{\mathbb{R}^N}h(x)|\varphi_0|^pdx\\
&\leq \frac{a}{2}+\frac{1}{2}\|V\|_{L^{\frac{N}{2s}}(\mathbb{R}^N)}S^{-1}
 +\frac{b}{4}-\frac{\lambda}{p}\int_{\mathbb{R}^N}h(x)|\varphi_0|^pdx<0,
 \end{align*}
for all $\lambda>\frac{p(\frac{a}{2}+\frac{1}{2}\|V\|_{L^{\frac{N}{2s}}
 (\mathbb{R}^N)}S^{-1} +\frac{b}{4})}{\int_{\mathbb{R}^N}h(x)|\varphi_0|^pdx}$.
Hence our claim holds true. Further, by Lemma \ref{lem3.1} and
\cite[Theorem 4.4]{MW}, there exists $u_1\in D^s_A(\mathbb{R}^N,\mathbb{C})$
such that $\mathcal{I}(u_1)=m$. Therefore, $u_1$ is a nontrivial
global minimizer solution of \eqref{eq1} with $\mathcal{I}(u_1)<0$.

 Now we prove that \eqref{eq1} has a mountain pass solution.
 Since $p\in (2,2_s^*)$, we obtain that $0$ a local minimum point of
 $\mathcal{I}$ in $D^s_A(\mathbb{R}^N,\mathbb{C})$. Define
\[
 c=\inf_{\gamma\in\Gamma}\max_{t\in[0,1]}\mathcal{I}(\gamma(t)),
\]
 where $\Gamma=\{\gamma\in C([0,1],D^s_A(\mathbb{R}^N,\mathbb{C})):
 \gamma(0)=0,\gamma(1)=u_1\}$. Then $c>0$. By Lemma \ref{lem3.1}, we know that
 $\mathcal{I}$ satisfies the conditions of the mountain-pass lemma
 (see \cite[Theorem 2.1]{AR}). Then there exists
$u_2\in D^s_A(\mathbb{R}^N,\mathbb{C})$
such that $\mathcal{I}(u_2)=c>0$ and $\mathcal{I}'(u_2)=0$.
Thus, $u_2$ is a nontrivial solution of equation \eqref{eq1}.
\end{proof}

To obtain the existence of infinitely many solutions, we introduce the
following theorem (see \cite{Chang}).

\begin{theorem}[{\cite[Theorem 5.2.23]{Chang}}] \label{thm3.1}
 Let $X$ be a Banach space, and $J\in C^1(X,\mathbb{R})$ be an even functional
satisfying the $(PS)_c$ condition. Assume $\alpha<\beta$ and either
$J(0)<\alpha$ or $J(0)>\beta$. If
further,
\begin{itemize}
\item[(1)] there are an $m$-dimensional linear subspace $E$ and a constant $\rho> 0$
such that $\sup_ {E\cap \partial B_\rho(0)}J(u)\leq \beta$,
where $\partial B_\rho(0)=\{u\in X:\|u\|=\rho\}$;

\item[(2)] there is a $j$-dimensional linear subspace $F$ such that
 $\inf_{F^\perp} J(u)>\alpha$, where $F^\perp$ is a
complementary space of $F$;

\item[(3)] $m>j$,
\end{itemize}
then $J$ has at least $m-j$ pairs of distinct critical points.
\end{theorem}

\begin{proof}[Proof of Theorem \ref{thm2}]
Clearly, $\mathcal{I}$ is an even functional. By Remark \ref{re3.1},
$\mathcal{I}$ satisfies the $(PS)_c$ condition. Choose
$E=D^s_A(\mathbb{R}^N,\mathbb{C})$ and $F=\emptyset$, then
$F^\bot=D^s_A(\mathbb{R}^N,\mathbb{C})$.
We can choose $\phi_0\in D^s_A(\mathbb{R}^N,\mathbb{C})$ such that
$[\phi_0]_{s,A}=1$ and $\int_{\mathbb{R}^N}h(x)|\phi_0|^pdx>0$. Then
\begin{align*}
\mathcal{I}(t\phi_0)
&=\frac{b}{4}t^4[\phi_0]_{s,A}^4-t^{22_{\mu,s}^*}\frac{1}{22_{\mu,s}^*}
 \int_{\mathbb{R}^N}(\mathcal{K}*|\phi_0|^{2_{\mu,s}^*})|\phi_0|^{2_{\mu,s}^*}dx\\
&\quad -t^p\frac{\lambda}{p}\int_{\mathbb{R}^N}h(x)|\phi_0|^pdx\\
&\leq \frac{b}{4}t^4[\phi_0]_{s,A}^4
-\frac{\lambda}{p}t^p\int_{\mathbb{R}^N}h(x)|\phi_0|^pdx\\
&=\Big[\frac{b}{4}t^{4-p}[\phi_0]_{s,A}^4
-\frac{\lambda}{p}\int_{\mathbb{R}^N}h(x)|\phi_0|^pdx\Big]t^p,
\end{align*}
for all $t>0$. It follows from $2<p<4$ that there exist $\beta<0$ and $\rho>0$
small enough such that $\mathcal{I}(t\phi_0)\leq \beta<0$ for all $0<t<\rho$.
Thus, we obtain
\[
\sup_{E\cap\partial B_\rho(0)}\mathcal{I}(u)\leq \beta<0,
\]
where $\partial B_\rho(0):=\{u\in D^s_A(\mathbb{R}^N,\mathbb{C}):[u]_{s,A}=\rho\}$.
By \eqref{eq3.1}, we easily deduce
\begin{align*}
\inf_{u\in F^\bot}\mathcal{I}(u)>-\infty.
\end{align*}
Therefore, $\mathcal{I}$ satisfies the conditions of Theorem \ref{thm3.1}.
Hence $\mathcal{I}$ has infinitely many pairs distinct critical points in
$D^s_A(\mathbb{R}^N,\mathbb{C})$, that is, equation \eqref{eq1} has infinitely
many pairs distinct solutions.
 Let $u\in D^s_A(\mathbb{R}^N,\mathbb{C})\setminus\{0\}$ be a solution of
 \eqref{eq1}. Then
\[
 b[u]_{s,A}^4=\int_{\mathbb{R}^N}(\mathcal{K}_\mu*|u|^{2_s^*})|u|^{2_s^*}dx
 +\lambda\int_{\mathbb{R}^N}h(x)|u|^pdx.
\]
 It follows that
\[
 b[u]_{s,A}^4\leq S_{H,L}^{-2_{\mu,s}^*}[u]_{s,A}^{22_{\mu,s}^*}
 +\lambda\|h\|_{L^{\frac{2_s^*}{2_s^*-p}}(\mathbb{R}^N)}S^{-\frac{p}{2}}[u]_{s,A}^p.
\]
 Since $b>S^{-2_{\mu,s}^*}$ and $2_{\mu,s}^*=2$ by $\mu=4s$,
we have
\[
(b-S_{H,L}^{-2_{\mu,s}^*})[u]_{s,A}^4\leq \lambda\|h\|_{L^{\frac{2_s^*}{2_s^*-p}}(\mathbb{R}^N)}S^{-\frac{p}{2}}[u]_{s,A}^p.
\]
Further,
\[
\Big[(b-S_{H,L}^{-2_{\mu,s}^*})[u]_{s,A}^{4-p}-
\lambda\|h\|_{L^{\frac{2_s^*}{2_s^*-p}}(\mathbb{R}^N)}S^{-\frac{p}{2}}\Big]
[u]_{s,A}^p\leq 0,
\]
which implies 
\[
[u]_{s,A}\leq \bigg[\frac{\lambda\|h\|_{L^{\frac{2_s^*}{2_s^*-p}}(\mathbb{R}^N)}
S^{-\frac{p}{2}}}{b-S_{H,L}^{-2_{\mu,s}^*}}\bigg]^{\frac{1}{4-p}}.
\]
This completes the proof.
\end{proof}

\subsection*{Acknowledgements}
 M. Xiang was supported by the National Natural Science Foundation of China
 (No. 11601515).


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\end{document}