\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 $a\geq 0, b>0$, $00$ 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 $2
0$. \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 $s\in (0,1)$, $\omega>0$, $N\geq 3$, $1+\frac{\alpha}{N}
\mu=4s, a=0,b>0, V\equiv0$, 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)$, $2
S_{H,L}^{-2_{\mu,s}^*}$ 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$, $2
0$.
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 $N>1$, $00$ 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 $\frac{2}{2_{\mu,s}^*}S_{H,L}^{-4}4s$, 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 $S_{H,L}^{-2_{\mu,s}^*}4s$, 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 $a=0,V\equiv0$, $\mu=4s$, $2
S_{H,L}^{-2_{\mu,s}^*}$, 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 $2
0$
small enough such that $\mathcal{I}(t\phi_0)\leq \beta<0$ for all $0