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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 123, pp. 1--25.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2018/123\hfil Existence of nontrivial solutions]
{Existence of nontrivial solutions for a perturbation of Choquard
equation \\ with Hardy-Littlewood-Sobolev upper \\ critical exponent}

\author[Y. Su, H. Chen \hfil EJDE-2018/123\hfilneg]
{Yu Su, Haibo Chen}

\address{Yu Su \newline
School of Mathematics and Statistics,
Central South University, Changsha, 410083 Hunan, China. \newline
School of Mathematical Sciences,
Xinjiang Normal University,
Urumuqi, 830054 Xinjiang, China}
\email{yizai52@qq.com}

\address{Haibo Chen  (corresponding author) \newline
School of Mathematics and Statistics,
Central South University, Changsha, 410083 Hunan, China}
\email{math\_chb@163.com}


\dedicatory{Communicated by Claudianor O. Alves}

\thanks{Submitted November 8, 2017. Published June 15, 2018.}
\subjclass[2010]{35J20, 35J60}
\keywords{ Hardy-Littlewood-Sobolev upper critical exponent;
\hfill\break\indent Choquard equation}

\begin{abstract}
 In this article, we consider the  problem
 \[
 -\Delta u
 =\Big(\int_{\mathbb{R}^{N}} \frac{|u|^{2^{*}_{\mu}}}{|x-y|^{\mu}}\,\mathrm{d}y\Big)
 |u|^{2^{*}_{\mu}-2}u + f(x,u) \quad\text{in }\mathbb{R}^{N},
 \]
 where $N\geqslant3$, $\mu\in(0,N)$ and $2^{*}_{\mu}=\frac{2N-\mu}{N-2}$.
 Under suitable assumptions on  $f(x,u)$, we
 establish the existence and non-existence of nontrivial solutions via
 the variational method.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks

\section{Introduction}

In this article, we consider the problem
\begin{equation} \label{eP}
-\Delta u
=\Big(\int_{\mathbb{R}^{N}}\frac{|u|^{2^{*}_{\mu}}}{|x-y|^{\mu}}\,\mathrm{d}y\Big)
|u|^{2^{*}_{\mu}-2}u+f(x,u)\quad \text{in }\mathbb{R}^{N},
\end{equation}
where $N\geqslant3$, $\mu\in(0,N)$, $2^{*}_{\mu}=\frac{2N-\mu}{N-2}$ and
$f(x,u)$ is a sign-changing nonlinearity satisfying certain assumptions.
Equation \eqref{eP} is closely related to the nonlinear Choquard equation
as follows:
\begin{equation}\label{1}
-\Delta u+V(x)u=\left(|x|^{\mu}*|u|^{p}\right)|u|^{p-2}u \quad \text{in }
\mathbb{R}^{N},
\end{equation}
where
$\frac{2N-\mu}{N}\leqslant p\leqslant\frac{2N-\mu}{N-2}$.
For $p=2$ and $\mu=1$, the equation \eqref{1}
goes back to the description of the quantum theory of a polaron at rest by
Pekar in 1954 \cite{Pekar1954}
and the modeling of an electron trapped in its own hole in 1976
in the work of  Choquard, as a certain approximation to Hartree-Fock
theory of one-component plasma \cite{Penrose1996}.
For $p=\frac{2N-1}{N-2}$ and $\mu=1$, by using the Green function,
it is obvious that equation \eqref{1} can be regarded as a generalized version of
Schr\"odinger-Newton equation
\begin{gather*}
-\Delta u+V(x)u=|u|^{\frac{N+1}{N-2}}\phi \quad \text{in }\mathbb{R}^{N},\\
-\Delta \phi =|u|^{\frac{N+1}{N-2}} \quad \text{in }\mathbb{R}^{N}.
\end{gather*}
The existence and qualitative properties of solutions of Choquard type equations
\eqref{1} have been widely studied in the previous decades (see \cite{Moroz2016}).
Moroz and Van Schaftingen \cite{Moroz2015} considered \eqref{1}
with lower critical exponent $\frac{2N-\mu}{N}$ if the potential
$1-V(x)$ should not decay to zero at infinity faster than the inverse of
$|x|^2$. In \cite{O.Alves2016},
the authors studied  \eqref{1} with critical growth in the sense of
 Trudinger-Moser inequality and studied the existence and concentration
of the ground states.

In 2016, Gao and Yang \cite{Gao2016} firstly investigated the critical Choquard
equation
\begin{equation}\label{2}
-\Delta u = \Big(\int_{\mathbb{R}^{N}}\frac{|u|^{2^{*}_{\mu}}}{|x-y|^{\mu}}
\,\mathrm{d}y \Big)|u|^{2^{*}_{\mu}-2}u
+\lambda u\quad \text{in } \Omega,
\end{equation}
where $\Omega$ is a bounded domain of $\mathbb{R}^{N}$,
with lipschitz boundary, $N\geqslant3$, $\mu\in(0,N)$ and $\lambda>0$.
By using variational methods, they established the existence, multiplicity
 and nonexistence of nontrivial solutions to equation \eqref{2}.
In equation \eqref{2},  $\lambda u$ is a linear perturbed term.

In \cite{Mukherjee2017}, the authors studied the following critical
Choquard equation
\begin{equation}\label{3}
\begin{gathered}
-\Delta u
=\Big( \int_{\mathbb{R}^{N}} \frac{|u|^{2^{*}_{\mu}}}{|x-y|^{\mu}}
\,\mathrm{d}y\Big) |u|^{2^{*}_{\mu}-2}u +\lambda u^{-q} \text{ in }
\Omega,\\
u>0 \quad \text{in } \Omega, \quad u=0  \text{ on } \partial\Omega,
\end{gathered}
\end{equation}
where $\Omega$ is a bounded domain of $\mathbb{R}^{N}$
$(N\geqslant3)$, $\mu\in(0,N)$, $0<q<1$ and $\lambda>0$.
By using variational methods and the Nehari manifold,
they established the existence and multiplicity of nontrivial solutions to
 \eqref{3}. In equation \eqref{3}, $\lambda u^{-q}$ is a singular perturbed term.

In \cite{Gao2017JMAA}, the authors studied the critical Choquard equation
\begin{equation}\label{4}
-\Delta u=\Big(\int_{\mathbb{R}^{N}} \frac{|u|^{2^{*}_{\mu}}}{|x-y|^{\mu}}
\,\mathrm{d}y \Big) |u|^{2^{*}_{\mu}-2}u +\lambda f(u)\quad
\text{in } \Omega,
\end{equation}
where $\Omega$ is a bounded domain of $\mathbb{R}^{N}$, $N\geqslant3$
and $\mu\in(0,N)$. By using variational methods,
they established the nonexistence, existence and multiplicity of nontrivial
solutions to equation \eqref{4} with different kinds of perturbed terms.

Very recently, Alves, Gao, Squassina and Yang \cite{O.Alves2017}
studied the singularly perturbed critical Choquard equation
\[
-\varepsilon^2\Delta u + V(x)u
= \varepsilon^{\mu-3} \Big(\int_{\mathbb{R}^{N}} \frac{Q(y)G(u(y))}{|x-y|^{\mu}}
\,\mathrm{d}y \Big) Q(x)g(u) \quad \text{in } \mathbb{R}^{3},
\]
where $0<\mu<3$, $\varepsilon$ is a positive parameter, $V,Q$
are two continuous real function on $\mathbb{R}^{3}$ and $G$
is the primitive of $g$ which is of critical growth due to the
Hardy-Littlewood-Sobolev inequality.
Under suitable assumptions on $g$, they first establish the existence of
ground states for the critical Choquard equation with constant coefficient.
They also establish existence and multiplicity of semi-classical solutions
and characterize the concentration behavior by variational methods.

Inspired by \cite{Gao2016,Gao2017JMAA,Mukherjee2017}, we are interested in
the problem that how the sign-changing Hardy term or the  sign-changing
superlinear nonlocal term will effect the existence and nonexistence of
solutions for the equation \eqref{eP}.
The main difference between equation \eqref{eP} and equations \eqref{2}, \eqref{3}
and \eqref{4} are not only the working domain $\mathbb{R}^{N}$ but also
the sign-changing perturbed term $f(x,u)$.

In this paper, we study problem \eqref{eP} with two kinds of perturbation.
Our results are divided into two classes:

\subsection*{Perturbation with sign-changing Hardy term}

For problem \eqref{eP} with a sign-changing Hardy term
$f(x,u)=g(x)\frac{u}{|x|^2}$,
\begin{equation} \label{eP1}
-\Delta u
= \Big( \int_{\mathbb{R}^{N}} \frac{|u|^{2^{*}_{\mu}}}{|x-y|^{\mu}}
\,\mathrm{d}y \Big) |u|^{2^{*}_{\mu}-2}u
+ g(x) \frac{u}{|x|^2}\quad \text{ in }\mathbb{R}^{N}.
\end{equation}
We suppose that $g$ satisfies the following hypotheses:
\begin{itemize}
\item[(A7)] $g\in C(\mathbb{R}^{N})$, $g_{\rm max}$
and $g_{\rm min}$ are well-defined, where
$g_{\rm max}:=\max_{x\in\mathbb{R}^{N}}g(x)$
and $g_{\rm min}:=\min_{x\in\mathbb{R}^{N}}g(x)$;

\item[(A8)] the sets $\Omega_1:=\{ x\in \mathbb{R}^{N}|g(x)>0\}$
and $\Omega_2 :=\{ x\in \mathbb{R}^{N}|g(x)<0\}$
have finite positive Lebesgue measure;

\item[(A9)] there exist $r_{\varepsilon}$ and $r_{g}$ such that
$\Omega_1\cup \Omega_2 \subset \overline{B(0,r_{g})}
\setminus B(0,r_{\varepsilon})$, and $g(x)=0$
in $\mathbb{R}^{N}\setminus(\Omega_1\cup \Omega_2 )$,
where $0< r_{\varepsilon}<r_{g}<\infty$;


\item[(A10)] $\int_{\Omega_1}g(x)\,\mathrm{d}x
>2 \big(\frac{r_{g}}{r_{\varepsilon}}\big)^{4N}
\int_{\Omega_2 }(-g(x))\,\mathrm{d}x$;

\item[(A11)]  $g_{\rm max}\in(0,\frac{(N-2)^2}{4})$;

\item[(A12)]  $|x_1-x_2 |\geqslant2r_{\varepsilon}$ for any
$ x_1\in\Omega_1$  and $x_2 \in\Omega_2 $.
\end{itemize}

Firstly, we firstly present a nontrivial example.
Let $\widetilde{19}:=(19,0,0)$ and $-\widetilde{19}:=(-19,0,0)$,
and $\widetilde{19},-\widetilde{19}\in \mathbb{R}^{3}$.
Then
$$
g(x)=\begin{cases}
\frac{1}{10}e^{-|x-\widetilde{19}|^2}-\frac{1}{10}e^{-1}& \text{in }
 B(\widetilde{19},1),\\[3pt]
\frac{1}{10}e^{-1}-\frac{1}{10}e^{-10^{4}|x-\widetilde{19}|^2}& \text{in }
B(-\widetilde{19},0.01),\\[3pt]
0& \quad\text{otherwise}.
\end{cases}
$$
The function $g(x)$ satisfies  hypotheses (A7)--(A12).


\begin{theorem}\label{theorem1}
Let  $N\geq3$ and $\mu\in(0,N)$, then \eqref{eP1} has no weak solution when
$g(x)$ is a differential functional and
$(x\cdot \nabla g(x))$ has a fixed sign.
\end{theorem}

\begin{theorem}\label{theorem2}
Assume that {\rm (A7)--(A11)} hold. Let $N\geqslant3$ and $\mu\in(0,N)$,
then \eqref{eP1} has a ground state solution.
\end{theorem}

\subsection*{Perturbation with a sign-changing superlinear nonlocal term}

We are interested in  problem \eqref{eP}
with a sign-changing superlinear nonlocal term
\begin{equation} \label{eP2}
\begin{aligned}
-\Delta u
&=\Big(\int_{\mathbb{R}^{N}}\frac{|u|^{2^{*}_{\mu}}}{|x-y|^{\mu}}
\,\mathrm{d}y \Big) |u|^{2^{*}_{\mu}-2}u \\
&\quad  + \Big(\int_{\mathbb{R}^{N}} \frac{g(y)|u|^{p}}{|x-y|^{\mu}}
\,\mathrm{d}y\Big) g(x)|u|^{p-2}u \quad \text{in } \mathbb{R}^{N}.
\end{aligned}
\end{equation}

\begin{theorem}\label{theorem3}
Assume that {\rm (A7)--(A10), (A12)} hold. Let
$N\geqslant3$, $\mu\in(0,N)$ and $p\in(\frac{2N-\mu}{N},\frac{2N-\mu}{N-2})$,
then problem  \eqref{eP2} has a nontrivial solution.
\end{theorem}

We need to point out the main features of problem \eqref{eP} are three-fold:
(1) Because of the Hardy-Littlewood-Sobolev upper critical term,
it is difficult to establish the Pohozaev type of identity on entire space;
(2) Since the sign-changing perturbed term,
it is difficult to estimate the Mountain-Pass level $c$;
(3) The loss of compactness due to the Hardy-Littlewood-Sobolev upper critical
exponent which makes it difficult to verify the $(PS)$ condition.

We refer the readers to
\cite{Bianchi1992,Deng2007,Deng2012,Nyamoradi2014,Rodrigues2010}
for equations involving different kinds of sign-changing perturbed term,
the difference between the present paper and previous papers
not only the assumptions on perturbed term but also the method of estimate
the Mountain-Pass level $c$.

The extremal function of best constant plays a key role in estimating the
 Mountain-Pass level $c$. In previous papers,
they estimate the Mountain-Pass level $c$ by $\sigma$ small enough or
large enough (where $\sigma$ defined in \eqref{6}).
In present paper,
we estimate the Mountain-Pass level $c$ by $\sigma\in[r_{\varepsilon},r_{g}]$
(where $r_{\varepsilon}$ and $r_{g}$ defined in (A9)).

This article is organized as follows:
In Section 2, we present notation and useful preliminary lemmas.
In Section 3, we investigate the critical Choquard equation perturbed
by a sign-changing Hardy term;
In Section 4, we investigate the critical Choquard equation perturbed by
a sign-changing superlinear nonlocal term.

\section{Preliminaries}

$D^{1,2}(\mathbb{R}^{N})$ is the completion of $C_{0}^{\infty}(\mathbb{R}^{N})$
with respect to the norm
$$
\|u\|_{D}^2=\int_{\mathbb{R}^{N}} |\nabla u|^2\,\mathrm{d}x.
$$
It is well known that $\frac{(N-2)^2}{4}$
is the best constant in the Hardy inequality
$$
\frac{(N-2)^2}{4} \int_{\mathbb{R}^{N}} \frac{ u^2}{|x|^2} \,\mathrm{d}x
\leqslant \int_{\mathbb{R}^{N}} |\nabla u|^2\,\mathrm{d}x,\quad
\text{for any } u\in D^{1,2}(\mathbb{R}^{N}).
$$
By (A7)--(A11), we derive that
$$
\|u\|_{g}^2
=\int_{\mathbb{R}^{N}} \Big(|\nabla u|^2-g(x)\frac{ u^2}{|x|^2}\Big)
\,\mathrm{d}x,
$$
is an equivalent norm in $D^{1,2}(\mathbb{R}^{N})$,
since the following inequalities hold:
$$
\left(1-\frac{4g_{\rm max}}{(N-2)^2}\right)
\|u\|_{D}^2\leqslant\|u\|_{g}^2\leqslant
\Big(1-\frac{4g_{\rm min}}{(N-2)^2}\Big)\|u\|_{D}^2.
$$
We recall the Sobolev inequality
$$
S\Big(\int_{\mathbb{R}^{N}}|u|^{2^{*}}\,\mathrm{d}x\Big)^{2/2^*}
\leqslant \int_{\mathbb{R}^{N}} |\nabla u|^2\,\mathrm{d}x,\quad
\text{for any } u\in D^{1,2}(\mathbb{R}^{N}),
$$
where $S>0$ is the Sobolev constant (see \cite{Swanson1992}).

\begin{lemma}[Hardy-Littlewood-Sobolev inequality \cite{Lieb2001}]
Let $t,r>1$ and $\mu\in(0,N)$ with
$\frac{1}{t}+\frac{1}{r}+\frac{\mu}{N}=2$, $f\in L^{t}(\mathbb{R}^{N})$
and $h\in L^{r}(\mathbb{R}^{N})$.
There exists a sharp constant $C(N,\mu,r,t)$, independent of $f,h$
such that
\[
\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}} \frac{|f(x)||h(y)|}
{|x-y|^{\mu}} \,\mathrm{d}x\,\mathrm{d}y
\leqslant C(N,\mu,r,t) \|f\|_{t} \|h\|_{r}.
\]
If $t=r=\frac{2N}{2N-\mu}$, then
$$
C(N,\mu,r,t)=C(N,\mu)
= \pi^{\frac{\mu}{2}} \frac{\Gamma(\frac{N}{2}-\frac{\mu}{2})}
{\Gamma(N-\frac{\mu}{2})}
\Big(\frac{\Gamma(\frac{N}{2})}{\Gamma(N)}\Big)^{-1+\frac{\mu}{N}}.
$$
\end{lemma}

For $\mu\in(0,N)$, we define the best constant
\begin{equation}\label{5}
S_{H,L}:= \inf_{u\in D^{1,2}(\mathbb{R}^{N})\setminus\{0\}}
\frac{\int_{\mathbb{R}^{N}} |\nabla u|^2 \,\mathrm{d}x}
{(\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}
\frac{|u(x)|^{2^{*}_{\mu}}|u(y)|^{2^{*}_{\mu}}}{|x-y|^{\mu}}
\,\mathrm{d}x\,\mathrm{d}y)^{1/2^*_{\mu}}}\,.
\end{equation}
The authors in \cite[Lemma 1.2]{Gao2016} proved that
$S_{H,L}$ is attained in $\mathbb{R}^{N}$
by the extremal function:
\begin{equation}\label{6}
w_{\sigma}(x)
=\sigma^{\frac{2-N}{2}}w(\frac{x}{\sigma}),\quad
w(x)
=\frac{\mathfrak{C}}{(1+|x|^2)^{\frac{N-2}{2}}},
\end{equation}
where $\mathfrak{C}>0$ is a fixed constant.
By the definition of convolution, we set
\begin{gather*}
|x|^{-\mu}*(|u_n|^{2^{*}_{\mu}}):=
\int_{\mathbb{R}^{N}}\frac{|u_n(y)|^{2^{*}_{\mu}}}
{|x-y|^{\mu}}\,\mathrm{d}y\\
|x|^{-\mu}*(g|u_n|^{p})(\mathbb{R}^{N}):=
\int_{\mathbb{R}^{N}}\frac{g(y)|u_n(y)|^{p}}
{|x-y|^{\mu}}\,\mathrm{d}y,\\
|x|^{-\mu}*(g|u_n|^{p})(\Omega_i):=
\int_{\Omega_i}\frac{g(y)|u_n(y)|^{p}}
{|x-y|^{\mu}}\,\mathrm{d}y, \quad (i=1,2).
\end{gather*}
\begin{lemma}[{\cite[Lemma 2.3]{Gao2016}}] \label{lemma5}
Let $N\geqslant3$ and $0<\mu<N$. If $\{u_n\}$
is a bounded sequence in $L^{\frac{2N}{N-2}}(\mathbb{R}^{N})$
such that $u_n\to u$ almost everywhere in $\mathbb{R}^{N}$
as $n\to\infty$, then the following hold,
\begin{align*}
&\int_{\mathbb{R}^{N}}
(|x|^{-\mu}*(|u_n|^{2_{\mu}^{*}}))|u_n|^{2_{\mu}^{*}}\,\mathrm{d}x
- \int_{\mathbb{R}^{N}}
(|x|^{-\mu}*(|u_n-u|^{2_{\mu}^{*}}))|u_n-u|^{2_{\mu}^{*}}
\,\mathrm{d}x\\
&\to \int_{\mathbb{R}^{N}}
(|x|^{-\mu}*(|u|^{2_{\mu}^{*}}))|u|^{2_{\mu}^{*}}
\,\mathrm{d}x.
\end{align*}
\end{lemma}

\section{Perturbation with a sign-changing Hardy term}

In this section we study the existence and nonexistence of solutions
for the critical Choquard equation with a sign-changing Hardy term,
i.e.
\begin{equation} \label{eP1b}
-\Delta u =
\Big(\int_{\mathbb{R}^{N}}\frac{|u|^{2^{*}_{\mu}}}{|x-y|^{\mu}}
\,\mathrm{d}y\Big) |u|^{2^{*}_{\mu}-2}u
+ g(x) \frac{u}{|x|^2}, \quad \text{in } \mathbb{R}^{N}.
\end{equation}
We introduce the energy functional associated with \eqref{eP1} as
\[
I_1(u)
= \frac{1}{2} \|u\|_{D}^2
- \frac{1}{2} \int_{\mathbb{R}^{N}} g(x)\frac{|u|^2}{|x|^2}\,\mathrm{d}x
- \frac{1}{2\cdot 2^{*}_{\mu}}
\int_{\mathbb{R}^{N}} \int_{\mathbb{R}^{N}}
\frac{|u_n(x)|^{2_{\mu}^{*}}|u_n(y)|^{2_{\mu}^{*}}}{|x-y|^{\mu}}
\,\mathrm{d}x\,\mathrm{d}y.
\]

\subsection{Non-existence result}
In this subsection, if $N\geqslant3$, $\mu\in(0,N)$ and
$(x\cdot \nabla g(x))$ has a fixed sign,
we prove that problem \eqref{eP1} does not have any solution by
Pohozaev type of identity.


\begin{proof}[Proof of Theorem \ref{theorem1}]
We use the same cut-off function which was used in \cite{Bhakta2017}.
More precisely,  for $\epsilon>0$ and $\epsilon_1>0$,
we define
$\tilde{\psi}_{\epsilon,\epsilon_1}(x)
=\psi_{\epsilon}(x)\bar{\psi}_{\epsilon_1}(x)$,
where
$\psi_{\epsilon}(x)=\psi(\frac{|x|}{\epsilon})$
and
$\bar{\psi}_{\epsilon_1}(x)=\psi(\frac{|x|}{\epsilon_1})$,
$\psi$ and $\bar{\psi}$
are smooth functions in $\mathbb{R}$ with the properties
$0\leqslant \psi,\bar{\psi}\leqslant1$,
with supports of $\psi$ and $\bar{\psi}$ in
$(1,\infty)$ and $(-\infty,2)$ respectively and
$\psi(t)=1$ for $t\geqslant2$, and $\bar{\psi}(t)=1$
for $t\leqslant1$.

Let $u$ be a weak solution of problem \eqref{eP1}.
Then $u$ is smooth away from origin and hence
$( x\cdot\nabla u) \tilde{\psi}_{\epsilon,\epsilon_1}\in C^2_{c}(\mathbb{R}^{N})$
(see \cite{Bhakta2017}).
Multiplying problem \eqref{eP1} by
$( x\cdot\nabla u) \tilde{\psi}_{\epsilon,\epsilon_1}$
and integrating by parts we obtain
\begin{equation}\label{7}
\begin{aligned}
&-\int_{\mathbb{R}^{N}} \Delta u
( x\cdot\nabla u) \tilde{\psi}_{\epsilon,\epsilon_1}
\,\mathrm{d}x\\
&=\int_{\mathbb{R}^{N}}\Big(
\int_{\mathbb{R}^{N}}
\frac{|u|^{2^{*}_{\mu}}}{|x-y|^{\mu}}
\,\mathrm{d}y  \Big)
|u|^{2^{*}_{\mu}-2}u
( x\cdot\nabla u) \tilde{\psi}_{\epsilon,\epsilon_1}
\,\mathrm{d}x \\
&\quad + \int_{\mathbb{R}^{N}} g(x)
\frac{u}{|x|^2} ( x\cdot\nabla u) \tilde{\psi}_{\epsilon,\epsilon_1}
\,\mathrm{d}x,
\end{aligned}
\end{equation}
We can show that
\begin{gather}\label{8}
\lim_{\epsilon_1\to\infty}\lim_{\epsilon\to0}
-\int_{\mathbb{R}^{N}}
\Delta u ( x\cdot\nabla u) \tilde{\psi}_{\epsilon,\epsilon_1}
\,\mathrm{d}x
=-\Big(\frac{N-2}{2}\Big)
\int_{\mathbb{R}^{N}} |\nabla u|^2\,\mathrm{d}x, \\
\label{9}
\begin{aligned}
&\lim_{\epsilon_1\to\infty}
\lim_{\epsilon\to0}
\int_{\mathbb{R}^{N}}
\frac{g(x)}{|x|^2} u
( x\cdot\nabla u) \tilde{\psi}_{\epsilon,\epsilon_1}
\,\mathrm{d}x\\
&=-\Big(\frac{N-2}{2}\Big) \int_{\mathbb{R}^{N}}
\frac{g(x)}{|x|^2}u^2\,\mathrm{d}x
-\frac{1}{2}\int_{\mathbb{R}^{N}}
\frac{(x\cdot \nabla g(x))}{|x|^2}
u^2\,\mathrm{d}x.
\end{aligned}
\end{gather}
We just show the critical term,
\begin{align*}
&\int_{\mathbb{R}^{N}}
\Big(\int_{\mathbb{R}^{N}}\frac{|u(y)|^{2^{*}_{\mu}}}{|x-y|^{\mu}}
\,\mathrm{d}y\Big)
|u(x)|^{2^{*}_{\mu}-2}u(x)
( x\cdot\nabla u(x)) \tilde{\psi}_{\epsilon,\epsilon_1}(x)
\,\mathrm{d}x\\
&=-\int_{\mathbb{R}^{N}}u(x) \nabla
\Big( x \tilde{\psi}_{\epsilon,\epsilon_1}(x)
\int_{\mathbb{R}^{N}}
\frac{|u(y)|^{2^{*}_{\mu}}}{|x-y|^{\mu}}
\,\mathrm{d}y
|u(x)|^{2^{*}_{\mu}-2}u(x)\Big)
\,\mathrm{d}x\\
&=-N \int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u(y)|^{2^{*}_{\mu}}|u(x)|^{2^{*}_{\mu}}}
{|x-y|^{\mu}}
 \tilde{\psi}_{\epsilon,\epsilon_1}(x)
\,\mathrm{d}y
\,\mathrm{d}x\\
&\quad -(2^{*}_{\mu}-1)
\int_{\mathbb{R}^{N}}
( x\cdot\nabla u(x)) \tilde{\psi}_{\epsilon,\epsilon_1}(x)
\Big(
\int_{\mathbb{R}^{N}}
\frac{|u(y)|^{2^{*}_{\mu}}}{|x-y|^{\mu}}
\,\mathrm{d}y
|u(x)|^{2^{*}_{\mu}-2}\Big)\,\mathrm{d}x\\
&\quad +\mu \int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
x\cdot(x-y)\tilde{\psi}_{\epsilon,\epsilon_1}
\frac{|u(y)|^{2^{*}_{\mu}}|u(x)|^{2^{*}_{\mu}}}{|x-y|^{\mu}}
\,\mathrm{d}y
\,\mathrm{d}x\\
&\quad -\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}} x \cdot
(\psi_{\epsilon}(x)\nabla\bar{\psi}_{\epsilon_1}(x)
+\bar{\psi}_{\epsilon_1}(x)\nabla\psi_{\epsilon}(x))
\frac{|u(y)|^{2^{*}_{\mu}}|u(x)|^{2^{*}_{\mu}}}{|x-y|^{\mu}}
\,\mathrm{d}y
\,\mathrm{d}x
\end{align*}
Note that $\nabla\bar{\psi}_{\epsilon_1}(x)$
and $\nabla\psi_{\epsilon}(x)$ have supports in
$\{ \epsilon_1< |x| < 2\epsilon_1\} $
and $\{\epsilon < |x| < 2\epsilon\}$, respectively.
Since $|x\cdot(\psi_{\epsilon}(x)\nabla\bar{\psi}_{\epsilon_1}(x)
+\bar{\psi}_{\epsilon_1}(x)\nabla\psi_{\epsilon}(x))
|\leqslant C$,
applying the dominated convergence theorem, we have
\begin{align*}
&\lim_{\epsilon_1\to\infty}
\lim_{\epsilon\to0}\int_{\mathbb{R}^{N}}
\Big(\int_{\mathbb{R}^{N}}
\frac{|u(y)|^{2^{*}_{\mu}}}{|x-y|^{\mu}}
\,\mathrm{d}y
\Big)
|u(x)|^{2^{*}_{\mu}-2}u(x)
( x\cdot\nabla u(x)) \tilde{\psi}_{\epsilon,\epsilon_1}(x)
\,\mathrm{d}x\\
&=-N\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u(y)|^{2^{*}_{\mu}}|u(x)|^{2^{*}_{\mu}}}
{|x-y|^{\mu}}
\,\mathrm{d}y
\,\mathrm{d}x\\
&\quad-\lim_{\epsilon_1\to\infty}
\lim_{\epsilon\to0}
(2^{*}_{\mu}-1)
\int_{\mathbb{R}^{N}}
( x\cdot\nabla u(x)) \tilde{\psi}_{\epsilon,\epsilon_1}(x)
\Big(\int_{\mathbb{R}^{N}}
\frac{|u(y)|^{2^{*}_{\mu}}}{|x-y|^{\mu}}
\,\mathrm{d}y|u(x)|^{2^{*}_{\mu}-2}
\Big)\,\mathrm{d}x\\
&\quad +\mu \int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
x\cdot(x-y)
\frac{|u(y)|^{2^{*}_{\mu}}|u(x)|^{2^{*}_{\mu}}}{|x-y|^{\mu}}
\,\mathrm{d}y
\,\mathrm{d}x,
\end{align*}
which implies
\begin{align*}
&\lim_{\epsilon_1\to\infty}\lim_{\epsilon\to0}
2^{*}_{\mu}\int_{\mathbb{R}^{N}}
\Big(\int_{\mathbb{R}^{N}}
\frac{|u(y)|^{2^{*}_{\mu}}}{|x-y|^{\mu}}\,\mathrm{d}y
\Big) |u(x)|^{2^{*}_{\mu}-2}u(x)
( x\cdot\nabla u(x)) \tilde{\psi}_{\epsilon,\epsilon_1}(x)
\,\mathrm{d}x\\
&=-N\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u(y)|^{2^{*}_{\mu}}|u(x)|^{2^{*}_{\mu}}}
{|x-y|^{\mu}}
\,\mathrm{d}y
\,\mathrm{d}x \\
&\quad +\mu \int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
x\cdot(x-y)
\frac{|u(y)|^{2^{*}_{\mu}}|u(x)|^{2^{*}_{\mu}}}{|x-y|^{\mu}}
\,\mathrm{d}y
\,\mathrm{d}x,
\end{align*}
Similarly,
\begin{align*}
&\lim_{\epsilon_1\to\infty}
\lim_{\epsilon\to0}2^{*}_{\mu}
\int_{\mathbb{R}^{N}}
\Big(\int_{\mathbb{R}^{N}}
\frac{|u(x)|^{2^{*}_{\mu}}}{|x-y|^{\mu}}
\,\mathrm{d}x
\Big)|u(y)|^{2^{*}_{\mu}-2}u(y)
( y\cdot\nabla u(y)) \tilde{\psi}_{\epsilon,\epsilon_1}(y)
\,\mathrm{d}y\\
&=-N\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u(y)|^{2^{*}_{\mu}}|u(x)|^{2^{*}_{\mu}}}
{|x-y|^{\mu}}
\,\mathrm{d}y
\,\mathrm{d}x \\
&\quad +\mu \int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
x\cdot(x-y)
\frac{|u(y)|^{2^{*}_{\mu}}|u(x)|^{2^{*}_{\mu}}}{|x-y|^{\mu}}
\,\mathrm{d}y
\,\mathrm{d}x.
\end{align*}
Hence, we know that
\begin{equation}\label{10}
\begin{aligned}
&\lim_{\epsilon_1\to\infty}
\lim_{\epsilon\to0}
2^{*}_{\mu}
\int_{\mathbb{R}^{N}}
\Big(
\int_{\mathbb{R}^{N}}
\frac{|u(y)|^{2^{*}_{\mu}}}{|x-y|^{\mu}}
\,\mathrm{d}y
\Big)
|u(x)|^{2^{*}_{\mu}-2}u(x)
( x\cdot\nabla u(x)) \tilde{\psi}_{\epsilon,\epsilon_1}(x)
\,\mathrm{d}x\\
&=\frac{\mu-2N}{2\cdot2^{*}_{\mu}}
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u(y)|^{2^{*}_{\mu}}|u(x)|^{2^{*}_{\mu}}}
{|x-y|^{\mu}}
\,\mathrm{d}y
\,\mathrm{d}x\\
&=-\Big(\frac{N-2}{2}\Big)
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u(y)|^{2^{*}_{\mu}}|u(x)|^{2^{*}_{\mu}}}
{|x-y|^{\mu}}
\,\mathrm{d}y
\,\mathrm{d}x.
\end{aligned}
\end{equation}
Therefore, putting \eqref{8}--\eqref{10} into \eqref{7},
we obtain
\begin{align*}
&-
\Big(\frac{N-2}{2}\Big)
\Big(\int_{\mathbb{R}^{N}}
|\nabla u|^2
\,\mathrm{d}x
-\int_{\mathbb{R}^{N}}
\frac{g(x)}{|x|^2}u^2
\,\mathrm{d}x\Big)\\
&=-\frac{1}{2}
\int_{\mathbb{R}^{N}}
\frac{(x\cdot \nabla g(x))}{|x|^2}u^2
\,\mathrm{d}x
-\Big(\frac{N-2}{2}\Big)
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u(y)|^{2^{*}_{\mu}}|u(x)|^{2^{*}_{\mu}}}
{|x-y|^{\mu}}
\,\mathrm{d}y
\,\mathrm{d}x.
\end{align*}
Also from \eqref{eP1}, we have
\[
\int_{\mathbb{R}^{N}}
|\nabla u|^2
\,\mathrm{d}x
-\int_{\mathbb{R}^{N}}
\frac{g(x)}{|x|^2}u^2
\,\mathrm{d}x
=\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u(y)|^{2^{*}_{\mu}}|u(x)|^{2^{*}_{\mu}}}
{|x-y|^{\mu}}
\,\mathrm{d}y
\,\mathrm{d}x.
\]
Then we obtain
\[
\int_{\mathbb{R}^{N}}
\frac{(x\cdot \nabla g(x))}{|x|^2}
u^2
\,\mathrm{d}x
=0,
\]
which is not possible if $(x\cdot \nabla g(x))$ has a fixed sign and
$u\not\equiv0$.
\end{proof}

\subsection{Existence of a ground state solution}

In this subsection, we study the existence of ground state solution for problem
\eqref{eP1} on$\mathbb{R}^{N}$.
The following Lemma plays an important role in estimating the Mountain-Pass levels.

\begin{lemma}\label{lemma6}
Assume that {\rm (A7)--(A10)} hold. Then for all
$\sigma\in[r_{\varepsilon},r_{g}]$,
we have
$$
\int_{\mathbb{R}^{N}}
g(x)
\frac{|w_{\sigma}(x)|^2}{|x|^2}
\,\mathrm{d}x>0.
$$
\end{lemma}

\begin{proof}
By using \eqref{6}, we have
\begin{equation}\label{11}
w_{\sigma}(x)
=\frac{\mathfrak{C}\sigma^{\frac{2-N}{2}}}{(1
+|\frac{x}{\sigma}|^2)^{\frac{N-2}{2}}}.
\end{equation}
According to (A8), (A9) and \eqref{11}, we obtain
\begin{align*}
\int_{\mathbb{R}^{N}} g(x) \frac{|w_{\sigma}(x)|^2} {|x|^2}\,\mathrm{d}x
=& \int_{\mathbb{R}^{N}} g(x)
\frac{\mathfrak{C}^2\sigma^{2-N}}
{(1+|\frac{x}{\sigma}|^2)^{N-2}|x|^2}\,\mathrm{d}x\\
=& \int_{\Omega_1\cup\Omega_2 }
g(x)\frac{\mathfrak{C}^2\sigma^{N-2}}{(\sigma^2 +|x|^2)^{N-2}
|x|^2}\,\mathrm{d}x.
\end{align*}
Since
$\Omega_1\cup\Omega_2 \subset \overline{B(0,r_{g})}
\setminus B(0,r_{\varepsilon})$,
we have $|x|\in[r_{\varepsilon},r_{g}]$
in $\Omega_1\cup\Omega_2 $. From the fact that
$\int_{\Omega_1} g(x)\,\mathrm{d}x>0$
and $\int_{\Omega_2 }g(x)\,\mathrm{d}x<0$, we obtain
\[
\int_{\mathbb{R}^{N}} g(x) \frac{|w_{\sigma}(x)|^2}{|x|^2}\,\mathrm{d}x
\geqslant \frac{\mathfrak{C}^2\sigma^{N-)}}{(\sigma^2
+r_{g}^2)^{N-2}r_{g}^2}\int_{\Omega_1}g(x)\,\mathrm{d}x
+\frac{\mathfrak{C}^2\sigma^{N-2}}{(\sigma^2+r_{\varepsilon}^2)^{N-2}
r_{\varepsilon}^2}\int_{\Omega_2 } g(x)\,\mathrm{d}x.
\]
Keeping in mind that $\int_{\Omega_2 }g(x)\,\mathrm{d}x<0$ and
$\sigma\in[r_{\varepsilon},r_{g}]$,
we know that
\begin{equation}\label{12}
\begin{aligned}
\int_{\mathbb{R}^{N}}g(x)\frac{|w_{\sigma}(x)|^2}{|x|^2}\,\mathrm{d}x
\geqslant& \frac{\mathfrak{C}^2\sigma^{N-2}}{(2r_{g}^2)^{N-2}r_{g}^2}
\int_{\Omega_1}g(x)\,\mathrm{d}x+\frac{\mathfrak{C}^2\sigma^{N-2}}
{(2r_{\varepsilon}^2)^{N-2}r_{\varepsilon}^2}\int_{\Omega_2 }g(x)
\,\mathrm{d}x\\
=&\frac{\mathfrak{C}^2\sigma^{N-2}}{2^{N-2}r_{g}^{2N-2}}\int_{\Omega_1}g(x)
\,\mathrm{d}x
+\frac{\mathfrak{C}^2\sigma^{N-2}}
{2^{N-2}
r_{\varepsilon}^{2N-2}}
\int_{\Omega_2 }g(x)
\,\mathrm{d}x.
\end{aligned}
\end{equation}
By (A10),
we have
\begin{equation}\label{13}
\begin{aligned}
\int_{\Omega_1}g(x)\,\mathrm{d}x
>
2\big(\frac{r_{g}}{r_{\varepsilon}}\big)^{4N}
\int_{\Omega_2 }(-g(x))\,\mathrm{d}x
>
\big(\frac{r_{g}}{r_{\varepsilon}}\big)^{2N-2}
\int_{\Omega_2 }(-g(x))\,\mathrm{d}x.
\end{aligned}
\end{equation}
Inserting \eqref{13} into \eqref{12},
we deduce that
$\int_{\mathbb{R}^{N}}g(x)\frac{|w_{\sigma}(x)|^2}{|x|^2}\,\mathrm{d}x>0.
$
\end{proof}

We show that the functional $I_1$ satisfies the Mountain-Pass geometry,
and estimate the Mountain-Pass levels.

\begin{lemma}\label{lemma7}
Assume that the hypotheses of Theorem \ref{theorem2} hold,
there exists a $(PS)_{c}$ sequence of $I_1$ at a level $c$, where
$0<c<c^{*}=\frac{N+2-\mu}{4N-2\mu}S_{H,L}^{\frac{2N-\mu}{N+2-\mu}}$.
\end{lemma}

\begin{proof}
\textbf{Step 1.} We prove that $I_1$
satisfies all the conditions in Mountain-pass theorem.
\begin{itemize}
\item[(i)] $I_1(0)=0$;

\item[(ii)] For any $u\in D^{1,2}(\mathbb{R}^{N})\setminus\{0\}$,
we have
\begin{align*}
I_1(u)
\geqslant& \frac{1}{2}
\|u\|_{g}^2
-\frac{1}{2\cdot2^{*}_{\mu}S_{H,L}^{2^{*}_{\mu}}}
\|u\|_{D}^{2\cdot 2^{*}_{\mu}}\\
\geqslant& \frac{1}{2}
\Big(1-\frac{4g_{\rm max}}{(N-2)^2}\Big)
\|u\|_{D}^2
-\frac{1}{2\cdot2^{*}_{\mu}S_{H,L}^{2^{*}_{\mu}}}
\|u\|_{D}^{2\cdot 2^{*}_{\mu}}.
\end{align*}
Because of $2<2\cdot 2^{*}_{\mu}$,
there exists a sufficiently small positive number
$\rho$ such that
$$
\vartheta:=\inf_{\|u\|_{D}=\rho}I_1(u)>0=I_1(0).
$$

\item[(iii)] Given
$u\in D^{1,2}(\mathbb{R}^{N})\setminus\{0\}$
such that
$\lim_{t\to\infty}
I_1(tu)=-\infty$.
We could choose $t_{u}>0$
corresponding to $u$
such that
$I_1(tu)<0$
for all
$t>t_{u}$
and
$\|t_{u} u\|_{D}>\rho$.
Set
$$
c=\inf_{\Upsilon\in\Gamma_{u}}
\max_{t\in [0,1]} I_1(\Upsilon(t)),
$$
where
$\Gamma_{u}=\{
\Upsilon\in C([0,1],D^{1,2}(\mathbb{R}^{N})):
\Upsilon(0)=0,
\Upsilon(1)=t_{u}u\}$.
\end{itemize}
\smallskip

\noindent \textbf{Step 2.} Here we show
$0<c<c^{*}$.
Using Lemma \ref{lemma6},
there exists
$\sigma\in[r_{\varepsilon},r_{g}]$
such that
$\int_{\mathbb{R}^{N}}
g(x)\frac{|w_{\sigma}(x)|^2}
{|x|^2}\,\mathrm{d}x>0$.
For all
$t\geqslant0$,
we obtain
\begin{align*}
0<c\leqslant&
\sup_{t\geqslant0}
I_1(tw_{\sigma})\\
\leqslant&
\frac{N+2-\mu}{4N-2\mu}
\Big(
\frac{
\int_{\mathbb{R}^{N}}
|\nabla w_{\sigma}(x)|^2
\,\mathrm{d}x
-
\int_{\mathbb{R}^{N}}
g(x)
\frac{|w_{\sigma}(x)|^2}{|x|^2}
\,\mathrm{d}x}
{(\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}
\frac{|w_{\sigma}(x)|^{2^{*}_{\mu}}|w_{\sigma}(y)|^{2^{*}_{\mu}}}{|x-y|^{\mu}}
\,\mathrm{d}x\,\mathrm{d}y)^{1/2^*_{\mu}}}
\Big)^{\frac{2N-\mu}{N+2-\mu}}\\
<&
\frac{N+2-\mu}{4N-2\mu}
\Big(
\frac{
\int_{\mathbb{R}^{N}}
|\nabla w_{\sigma}(x)|^2
\,\mathrm{d}x}
{
(\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}
\frac{|w_{\sigma}(x)|^{2^{*}_{\mu}}|w_{\sigma}(y)|^{2^{*}_{\mu}}}{|x-y|^{\mu}}
\,\mathrm{d}x\,\mathrm{d}y)^{1/2^*_{\mu}}
}
\Big)^{\frac{2N-\mu}{N+2-\mu}}\\
=&
\frac{N+2-\mu}{4N-2\mu}S_{H,L}^{\frac{2N-\mu}{N+2-\mu}}.
\end{align*}
which means
$0<c<\frac{N+2-\mu}{4N-2\mu}S_{H,L}^{\frac{2N-\mu}{N+2-\mu}}$.
\end{proof}

\begin{lemma}\label{lemma8}
Assume that the hypotheses of Theorem \ref{theorem2} hold.
If $\{u_n\}$ is a $(PS)_{c}$ sequence of
$I_1$, then
$\{u_n\}$
is bounded in
$D^{1,2}(\mathbb{R}^{N})$.
\end{lemma}

\begin{proof}
The $(PS)_{c}$ sequence $\{u_n\}$ defined in Lemma \ref{lemma7}.
From the definition of $(PS)_{c}$ sequence, we have
\begin{align*}
c^{*}+\|u_n\|_{D}
\geqslant
c^{*}+o(1)\|u_n\|_{D}
\geqslant&
I(u_n)
-\frac{1}{2}
\langle I'(u_n),u_n\rangle\\
=&
\frac{2^{*}_{\mu}-1}{2\cdot2^{*}_{\mu}}
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u_n(x)|^{2_{\mu}^{*}}|u_n(y)|^{2_{\mu}^{*}}}
{|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y.
\end{align*}
Applying above inequality and (A11),
we know
\begin{align*}
c^{*}
\geqslant
I(u_n)
\geqslant&
\frac{1}{2}
\Big(1-\frac{4g_{\rm max}}{(N-2)^2}\Big)
\|u_n\|^2_{D}
-
\frac{1}{2^{*}_{\mu}-1}
(c^{*}+\|u_n\|_{D}).
\end{align*}
Set
\begin{align*}
f_1(t)
=
\frac{1}{2}
\Big(1-\frac{4g_{\rm max}}{(N-2)^2}\Big)
t^2
-\frac{1}{2^{*}_{\mu}-1}t
-\frac{2^{*}_{\mu}c^{*}}{2^{*}_{\mu}-1}.
\end{align*}
We have two solutions of $f_1(\cdot)$ as follows:
\begin{align*}
t'=&
\frac{\frac{1}{2^{*}_{\mu}-1}+\sqrt{(\frac{1}{2^{*}_{\mu}-1})^2
+\frac{2\cdot2^{*}_{\mu}\cdot c^{*}}{2^{*}_{\mu}-1}
(1-\frac{4g_{\rm max}}{(N-2)^2})}}
{1-\frac{4g_{\rm max}}{(N-2)^2}}>0,\\
t^{''}=&
\frac{
\frac{1}{2^{*}_{\mu}-1}
-\sqrt{(\frac{1}{2^{*}_{\mu}-1})^2
+\frac{2\cdot2^{*}_{\mu}\cdot c^{*}}{2^{*}_{\mu}-1}
(1-\frac{4g_{\rm max}}{(N-2)^2})}}
{1-\frac{4g_{\rm max}}{(N-2)^2}}<0.
\end{align*}
Therefore,
$0\leqslant\|u_n\|_{D}\leqslant t'$,
this implies that
$\{u_n\}$
is bounded in
$D^{1,2}(\mathbb{R}^{N})$.
\end{proof}


To check that functional $I_1$ satisfies the $(PS)_{c}$ condition,
we give the following Lemma.

\begin{lemma}\label{lemma9}
Assume that the hypotheses  of Theorem \ref{theorem2} hold.
If $\{u_n\}$ is a bounded sequence in
$D^{1,2}(\mathbb{R}^{N})$, up to a subsequence,
$u_n\rightharpoonup u$ in $D^{1,2}(\mathbb{R}^{N})$
and $u_n\to u \text{a.e. in }~~\mathbb{R}^{N}$
as $n\to\infty$, then
\begin{align*}
\int_{\mathbb{R}^{N}}g(x)
\frac{|u_n|^2}{|x|^2} \,\mathrm{d}x
\to
\int_{\mathbb{R}^{N}}
g(x)
\frac{|u|^2}{|x|^2}
\,\mathrm{d}x.
\end{align*}
In addition,
for any $\varphi\in D^{1,2}(\mathbb{R}^{N})$,
\begin{align*}
\int_{\mathbb{R}^{N}} g(x) \frac{u_n\varphi}{|x|^2}
\,\mathrm{d}x
\to \int_{\mathbb{R}^{N}} g(x)
\frac{u\varphi}{|x|^2}
\,\mathrm{d}x.
\end{align*}
as $n\to\infty$.
\end{lemma}

\begin{proof}
\textbf{Step 1.}
Define $v_n:=u_n-u$.
According to (A8), (A9) and
Br\'{e}zis-Lieb lemma in
\cite{Brezis1983},
we have
\begin{equation}\label{14}
\int_{\Omega_1}
g(x)
\frac{|u_n|^2}{|x|^2}
\,\mathrm{d}x
=\int_{\Omega_1}
g(x)
\frac{|v_n|^2}{|x|^2}
\,\mathrm{d}x
+\int_{\Omega_1}
g(x)
\frac{|u|^2}{|x|^2}
\,\mathrm{d}x
+o(1),\quad \text{as } n\to\infty,
\end{equation}
and
\begin{equation}\label{15}
\int_{\Omega_2 }
g(x)
\frac{|u_n|^2}{|x|^2}
\,\mathrm{d}x
=\int_{\Omega_2 }
g(x)\frac{|v_n|^2}{|x|^2}
\,\mathrm{d}x
+\int_{\Omega_2 }
g(x)\frac{|u|^2}{|x|^2}
\,\mathrm{d}x
+o(1),\quad\text{as } n\to\infty.
\end{equation}
Combining \eqref{14} and \eqref{15},
we obtain
\begin{equation}\label{16}
\int_{\Omega_1\cup\Omega_2 }
g(x) \frac{|u_n|^2}{|x|^2}
\,\mathrm{d}x
=\int_{\Omega_1\cup\Omega_2 }
g(x) \frac{|v_n|^2}{|x|^2}
\,\mathrm{d}x
+\int_{\Omega_1\cup\Omega_2 }
g(x)\frac{|u|^2}{|x|^2}
\,\mathrm{d}x
+o(1),
\end{equation}
as $n\to\infty$.
\smallskip

\noindent\textbf{Step 2.}
Furthermore, we estimate the term involving $v_n$ in
\eqref{16}. Since
$v_n\rightharpoonup0$
in $D^{1,2}(\mathbb{R}^{N})$,
we have $v_n\to0$
in
$L^2(B(0,r_{g})\setminus B(0,r_{\varepsilon}))$.
According to (A8) and (A9),
we obtain
\begin{equation}\label{17}
\begin{aligned}
\lim_{n\to\infty}
\int_{\Omega_1\cup\Omega_2 }
g(x)
\frac{|v_n|^2}{|x|^2}
\,\mathrm{d}x
=& \lim_{n\to\infty}
\int_{B(0,r_{g})\setminus B(0,r_{\varepsilon})}
g(x)
\frac{|v_n|^2}{|x|^2}
\,\mathrm{d}x\\
\leqslant&
\frac{g_{\rm max}}{r_{\varepsilon}^2}
\lim_{n\to\infty}
\int_{B(0,r_{g})\setminus B(0,r_{\varepsilon})}
|v_n|^2
\,\mathrm{d}x=0.
\end{aligned}
\end{equation}
Keeping in mind that
$g_{\rm min}<0$,
similar to
\eqref{17},
we obtain
\begin{equation}\label{18}
\begin{aligned}
\lim_{n\to\infty}
\int_{\Omega_1\cup\Omega_2 }
g(x)
\frac{|v_n|^2}{|x|^2}
\,\mathrm{d}x
=&
\lim_{n\to\infty}
\int_{B(0,r_{g})\setminus B(0,r_{\varepsilon})}
g(x)
\frac{|v_n|^2}{|x|^2}
\,\mathrm{d}x\\
\geqslant&
\frac{g_{\rm min}}{r_{\varepsilon}^2}
\lim_{n\to\infty}
\int_{B(0,r_{g})\setminus B(0,r_{\varepsilon})}
|v_n|^2
\,\mathrm{d}x\to0.
\end{aligned}
\end{equation}
Combining \eqref{17} and \eqref{18}, we obtain
\begin{align*}
0\leqslant \lim_{n\to\infty}
\int_{\Omega_1\cup\Omega_2 }
g(x) \frac{|v_n|^2}{|x|^2}
\,\mathrm{d}x
\leqslant 0,
\end{align*}
then
\begin{equation}\label{19}
\lim_{n\to\infty}
\int_{\Omega_1\cup\Omega_2 }
g(x)
\frac{|v_n|^2}{|x|^2}
\,\mathrm{d}x
=0.
\end{equation}
\smallskip

\noindent
\textbf{Step 3.}
Putting  \eqref{19} into \eqref{16}, we know
\begin{equation}\label{20}
\int_{\Omega_1\cup\Omega_2 }
g(x) \frac{|u_n|^2}{|x|^2}
\,\mathrm{d}x
= \int_{\Omega_1\cup\Omega_2 }
g(x) \frac{|u|^2}{|x|^2}
\,\mathrm{d}x
+ o(1)\quad \text{as } n\to\infty.
\end{equation}
Since
$g\equiv0$  in $\mathbb{R}^{N}\setminus(\Omega_1\cup\Omega_2 )$,
by \eqref{20}, we have
\[
\int_{\mathbb{R}^{N}}
g(x) \frac{|u_n|^2}{|x|^2}
\,\mathrm{d}x
= \int_{\mathbb{R}^{N}}
g(x) \frac{|u|^2}{|x|^2} \,\mathrm{d}x
+ o(1),\quad \text{as } n\to\infty.
\]
\smallskip

\noindent \textbf{Step 4.}
In addition, the boundedness of
$u_n$
in
$D^{1,2}(\mathbb{R}^{N})$
yields that
$u_n$
are bounded in
$L^2(\Omega_1,|x|^{-2})$
and
$L^2(\Omega_2 ,|x|^{-2})$,
respectively.
Therefore,
up to a subsequence,
we have the following weak convergence
\begin{gather*}
g(x)u_n \rightharpoonup
g(x)u \quad \text{in }L^2(\Omega_1,|x|^{-2}),\\
g(x)u_n
\rightharpoonup
g(x)u \quad \text{in } L^2(\Omega_2 ,|x|^{-2}).
\end{gather*}
Then
\[
g(x)u_n
\rightharpoonup g(x)u \quad \text{in }
L^2(\Omega_1\cup\Omega_2 ,|x|^{-2}).\\
\]
Since $g\equiv0$ in
$\mathbb{R}^{N}\setminus(\Omega_1\cup\Omega_2 )$,
we know that
\[
g(x)u_n \rightharpoonup
g(x)u \quad \text{in }
L^2(\mathbb{R}^{N},|x|^{-2}).
\]
For any
$\varphi\in D^{1,2}(\mathbb{R}^{N})$,
we have
\[
\int_{\mathbb{R}^{N}}
g(x)\frac{u_n\varphi}{|x|^{-2}}
\,\mathrm{d}x
\to
\int_{\mathbb{R}^{N}}
g(x)\frac{u \varphi}{|x|^2}
\,\mathrm{d}x.
\]
\end{proof}

Now we check functional $I_1$ satisfies the $(PS)_{c}$ condition.

\begin{lemma}\label{lemma10}
Assume that the hypotheses of Theorem
\ref{theorem2} hold.
If $\{u_n\}$
is a
$(PS)_{c}$
sequence of
$I_1$
with
$0<c<\frac{N+2-\mu}{4N-2\mu}S_{H,L}^{\frac{2N-\mu}{N+2-\mu}}$,
then
$\{u_n\}$
has a convergent subsequence.
\end{lemma}

\begin{proof}
\textbf{Step 1.}
Since $D^{1,2}(\mathbb{R}^{N})$
is a reflexive space.
And
$\{u_n\}$
is a bounded sequence in
$D^{1,2}(\mathbb{R}^{N})$,
up to a subsequence,
we can assume that
\begin{gather*}
u_n\rightharpoonup u \quad\text{in } D^{1,2}(\mathbb{R}^{N}), \quad
u_n\to u \quad\text{a.e. in }\mathbb{R}^{N},\\
u_n\to u \quad\text{in } L^{r}_{loc}(\mathbb{R}^{N})\quad\text{for all }
 r\in[1,2^{*}).
\end{gather*}
Then
\[
|u_n|^{2^{*}_{\mu}}
\rightharpoonup
|u|^{2^{*}_{\mu}}\quad \text{in }
L^{\frac{2N}{2N-\mu}}(\mathbb{R}^{N}),
\quad\text{as }n\to\infty.
\]
By the Hardy-Littlewood-Sobolev inequality,
the Riesz potential defines a linear continuous map from
$L^{\frac{2N}{2N-\mu}}(\mathbb{R}^{N})$
to $L^{\frac{2N}{\mu}}(\mathbb{R}^{N})$,
we know
$$
|x|^{-\mu}*|u_n|^{2^{*}_{\mu}}
\rightharpoonup
|x|^{-\mu}*|u|^{2^{*}_{\mu}} \quad
\text{in } L^{\frac{2N}{\mu}}(\mathbb{R}^{N}),
\quad\text{as }n\to\infty.
$$
Combining with the fact that
$$
|u_n|^{2^{*}_{\mu}-2} u_n
\rightharpoonup
|u|^{2^{*}_{\mu}-2}u
\quad \text{in }
L^{\frac{2N}{N+2-\mu}}(\mathbb{R}^{N}),
\quad\text{as n } \to\infty,
$$
we obtain
$$
(|x|^{-\mu}*|u_n|^{2^{*}_{\mu}})
|u_n|^{2^{*}_{\mu}-2}u_n
\rightharpoonup
(|x|^{-\mu}*|u|^{2^{*}_{\mu}})
|u|^{2^{*}_{\mu}-2}u
\quad \text{in } L^{\frac{2N}{N+2}}(\mathbb{R}^{N})\quad \text{as } n\to\infty.
$$
For any $\varphi\in D^{1,2}(\mathbb{R}^{N})$,
we obtain
\begin{align*}
0\leftarrow\langle I'_1(u_n), \varphi\rangle
&=\int_{\mathbb{R}^{N}}
\nabla u_n
\nabla \varphi
\,\mathrm{d}x
-\int_{\mathbb{R}^{N}}
g(x)\frac{u_n\varphi}{|x|^2}
\,\mathrm{d}x\\
&\quad - \int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u_n(x)|^{2^{*}_{\mu}}|u_n(y)|^{2^{*}_{\mu}-2}u_n(y)\varphi(y)}
{|x-y|^{\mu}}
\,\mathrm{d}y
\,\mathrm{d}x.
\end{align*}
Passing to the limit as $n\to\infty$,
by using Lemma \ref{lemma9}, we obtain
\begin{align*}
0&=\int_{\mathbb{R}^{N}}
\nabla u \nabla \varphi \,\mathrm{d}x
-\int_{\mathbb{R}^{N}} g(x)\frac{u\varphi}{|x|^2} \,\mathrm{d}x \\
&\quad -\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u(x)|^{2^{*}_{\mu}}|u(y)|^{2^{*}_{\mu}-2}u(y)\varphi(y)}
{|x-y|^{\mu}}
\,\mathrm{d}y
\,\mathrm{d}x,
\end{align*}
for any
$\varphi\in D^{1,2}(\mathbb{R}^{N})$,
which means that
$u$
is a weak solution of problem
\eqref{eP1}.
Taking
$\varphi=u\in D^{1,2}(\mathbb{R}^{N})$
as a test function in \eqref{eP1},
we have
\[
\int_{\mathbb{R}^{N}}
|\nabla u|^2
\,\mathrm{d}x
-
\int_{\mathbb{R}^{N}}
g(x)\frac{|u|^2}{|x|^2}
\,\mathrm{d}x
=
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u(x)|^{2^{*}_{\mu}}|u(y)|^{2^{*}_{\mu}}u(y)}
{|x-y|^{\mu}}
\,\mathrm{d}y
\,\mathrm{d}x,
\]
which implies that
$\langle I'_1(u), u\rangle=0$.
\smallskip

\noindent\textbf{Step 2.}
From $\langle I'_1(u), u\rangle=0$, we obtain
\begin{align*}
I_1(u)
=&
I_1(u)
-\frac{1}{2}
\langle I'_1(u), u\rangle\\
=&\Big(\frac{1}{2}
-\frac{1}{2\cdot2^{*}_{\mu}}\Big)
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u(x)|^{2_{\mu}^{*}}|u(y)|^{2_{\mu}^{*}}}
{|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y
\geqslant0.
\end{align*}
Define $v_n:=u_n-u$, then we know
$v_n\rightharpoonup0$ in $D^{1,2}(\mathbb{R}^{N})$.
According to the Br\'{e}zis-Lieb lemma,
Lemma \ref{lemma5} and Lemma \ref{lemma9},
we have
\begin{equation}\label{21}
\begin{aligned}
c\leftarrow
I_1(u_n)
=& \frac{1}{2}
\|u_n\|_{D}^2
-\frac{1}{2}
\int_{\mathbb{R}^{N}}g(x)
\frac{|u_n|^2}{|x|^2}
\,\mathrm{d}x\\
&-\frac{1}{2\cdot2^{*}_{\mu}}
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u_n(x)|^{2_{\mu}^{*}}|u_n(y)|^{2_{\mu}^{*}}}
{|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y\\
=& I_1(u)+ \frac{1}{2}
\|v_n\|_{D}^2
-\frac{1}{2\cdot2^{*}_{\mu}}
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|v_n(x)|^{2_{\mu}^{*}}|v_n(y)|^{2_{\mu}^{*}}}
{|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y\\
\geqslant&
\frac{1}{2}
\|v_n\|_{D}^2
-\frac{1}{2\cdot2^{*}_{\mu}}
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|v_n(x)|^{2_{\mu}^{*}}|v_n(y)|^{2_{\mu}^{*}}}
{|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y,
\end{aligned}
\end{equation}
since $I_1(u)\geqslant0$.
Similarly, since $\langle I'_1(u), u\rangle=0$,
we obtain
\begin{equation}\label{22}
\begin{aligned}
o(1)
=&\langle I'_1(u_n), u_n\rangle\\
=&
\|u_n\|_{D}^2
-
\int_{\mathbb{R}^{N}}
g(x)
\frac{|u_n|^2}{|x|^2}
\,\mathrm{d}x
-
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u_n(x)|^{2_{\mu}^{*}}|u_n(y)|^{2_{\mu}^{*}}}
{|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y\\
=&
\langle I'_1(u), u\rangle+
\|v_n\|_{D}^2
-
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|v_n(x)|^{2_{\mu}^{*}}|v_n(y)|^{2_{\mu}^{*}}}
{|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y\\
=&
\|v_n\|_{D}^2
-
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|v_n(x)|^{2_{\mu}^{*}}|v_n(y)|^{2_{\mu}^{*}}}
{|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y.
\end{aligned}
\end{equation}
From this equality, there exists a nonnegative constant
$b$
such that
\[
\int_{\mathbb{R}^{N}}
|\nabla v_n|^2
\,\mathrm{d}x
\to
b
\quad\text{and}\quad
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|v_n(x)|^{2_{\mu}^{*}}|v_n(y)|^{2_{\mu}^{*}}}
{|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y
\to
b,
\]
as
$n\to\infty$.
From  \eqref{21} and
\eqref{22},
we obtain
\begin{equation}\label{23}
c\geqslant \frac{N+2-\mu}{4N-2\mu}b.
\end{equation}
By the definition of the best constant
$S_{H,L}$
in \eqref{5}, we have
\[
S_{H,L}
\Big(
\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}
\frac{|v_n(x)|^{2^{*}_{\mu}}|v_n(y)|^{2^{*}_{\mu}}}{|x-y|^{\mu}}
\,\mathrm{d}x\,\mathrm{d}y
\Big)^{1/2_\mu^*}
\leqslant
\int_{\mathbb{R}^{N}}
|\nabla v_n|^2
\,\mathrm{d}x,
\]
which gives $S_{H,L}b^{1/2_\mu^*}\leqslant b$.
Thus we have that either $b=0$ or
$b\geqslant S_{H,L}^{\frac{2N-\mu}{N+2-\mu}}$.

If $b\geqslant S_{H,L}^{\frac{2N-\mu}{N+2-\mu}}$,
then  from \eqref{23}, we obtain
\[
 \frac{N+2-\mu}{4N-2\mu}S_{H,L}^{\frac{2N-\mu}{N+2-\mu}}
\leqslant \frac{N+2-\mu}{4N-2\mu}b\leqslant c.
\]
This is in contradiction to
$c<\frac{N+2-\mu}{4N-2\mu}S_{H,L}^{\frac{2N-\mu}{N+2-\mu}}$.
Thus $b=0$, and
$\|u_n-u\|_{D}\to0,$
as $n\to\infty$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{theorem2}]
\textbf{Step 1.}
Applying Lemma \ref{lemma7},
we obtain that $I_1$
possesses a mountain pass geometry.
Then from the Maintain Pass Theorem,
there is a sequence
$\{u_n\}\subset D^{1,2}(\mathbb{R}^{N})$
satisfying
$I_1(u_n)\to c$
and
$I'_1(u_n)\to0$,
where
$0<\vartheta\leqslant c<\frac{N+2-\mu}{4N-2\mu}S_{H,L}^{\frac{2N-\mu}{N+2-\mu}}$.
Moreover,
according to Lemma \ref{lemma8} and Lemma \ref{lemma10},
$\{u_n\}$
satisfying
$(PS)_{c}$
condition.
We have a nontrivial solution
$u_{0}$
to problem \eqref{eP1}.
In following text, we show the existence of ground state solution to problem
\eqref{eP1}.
\smallskip

\noindent\textbf{Step 2.}
Define
\begin{gather*}
K_1=\{u\in D^{1,2}(\mathbb{R}^{N})|\langle I'_1(u),u\rangle=0,~u\not=0 \},\\
E_1=\{I_1(u)|u\in K_1\}.
\end{gather*}
In Step 1, we have
$u_{0}\not=0$
and
$\langle I'_1(u_{0}),u_{0}\rangle=0$.
Hence,
we know
$K_1\not=\emptyset$.

Now, we claim that
any limit point of a sequence in $K_1$ is different from zero.
For any $u\in K_1$,
according to
$\langle I'_1(u),u\rangle=0$
and \eqref{5},
it follows that
\begin{align*}
0=\langle I'_1(u),u\rangle
=&\|u\|^2_{g}
-\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}
\frac{|u(x)|^{2^{*}_{\mu}}|u(y)|^{2^{*}_{\mu}}}{|x-y|^{\mu}}
\,\mathrm{d}x\,\mathrm{d}y\\
\geqslant&
\Big(1-\frac{4g_{\rm max}}{(N-2)^2}\Big)
\|u\|^2_{D}
-\frac{1}{S_{H,L}^{2^{*}_{\mu}}}
\|u\|_{D}^{2\cdot 2^{*}_{\mu}}.
\end{align*}
From the above expression,
we obtain
\begin{align*}
\Big(1-\frac{4g_{\rm max}}{(N-2)^2}\Big)
\|u\|^2_{D}
\leqslant&
\frac{1}{S_{H,L}^{2^{*}_{\mu}}}
\|u\|_{D}^{ 2\cdot2^{*}_{\mu}},
\end{align*}
which gives
\begin{align*}
0<
\Big(
\Big(1-\frac{4g_{\rm max}}{(N-2)^2}\Big)
S_{H,L}^{2^{*}_{\mu}}
\Big)
^{\frac{1}{2\cdot2^{*}_{\mu}-2}}
\leqslant
\|u\|_{D},\quad \text{for any }u\in K_1.
\end{align*}
Hence, any limit point of a sequence in $K_1$ is different from zero.
Now, we claim that $E_1$
has an infimum.
In fact, for any $u\in K_1$, we have
\[
0=
\langle I'_1(u),u\rangle
=\|u\|^2_{g}
-\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}
\frac{|u(x)|^{2^{*}_{\mu}}|u(y)|^{2^{*}_{\mu}}}{|x-y|^{\mu}}
\,\mathrm{d}x\,\mathrm{d}y.
\]
Then
\begin{align*}
I_1(u)
=&\frac{1}{2}
\|u\|^2_{g}
-\frac{1}{2\cdot2^{*}_{\mu}}
\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}
\frac{|u(x)|^{2^{*}_{\mu}}|u(y)|^{2^{*}_{\mu}}}{|x-y|^{\mu}}
\,\mathrm{d}x\,\mathrm{d}y\\
=&\Big(
\frac{1}{2}
-\frac{1}{2\cdot2^{*}_{\mu}}
\Big)
\|u\|^2_{g}\\
\geqslant&
\Big(
\frac{1}{2}
-\frac{1}{2\cdot2^{*}_{\mu}}
\Big)
\Big(1-\frac{4g_{\rm max}}{(N-2)^2}\Big)
\|u\|^2_{D}\\
\geqslant&
\Big(\frac{1}{2}
-\frac{1}{2\cdot2^{*}_{\mu}}
\Big)
\Big(
\Big(1-\frac{4g_{\rm max}}{(N-2)^2}\Big)
S_{H,L}
\Big)
^{\frac{2\cdot2^{*}_{\mu}}{2\cdot2^{*}_{\mu}-2}}
>0.
\end{align*}
Therefore, we obtain
$$
0<\Big(
\frac{1}{2}
-\frac{1}{2\cdot2^{*}_{\mu}}
\Big)
\Big(
\Big(1-\frac{4g_{\rm max}}{(N-2)^2}\Big)
S_{H,L}
\Big)
^{\frac{2\cdot2^{*}_{\mu}}{2\cdot2^{*}_{\mu}-2}}\leqslant\overline{E}_1
:=\inf\{I_1(u)|u\in K_1\}.
$$
\smallskip

\noindent\textbf{Step 3.}
(i) For each
$u\in D^{1,2}(\mathbb{R}^{N})$
with
$u\not\equiv0$,
and
$t\in (0,\infty)$,
we set
\begin{gather*}
f_2 (t)
=I_1(tu)
=\frac{t^2}{2}
\|u\|^2_{g}
-\frac{t^{2\cdot2^{*}_{\mu}}}{2\cdot2^{*}_{\mu}}
\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}
\frac{|u(x)|^{2^{*}_{\mu}}|u(y)|^{2^{*}_{\mu}}}{|x-y|^{\mu}}
\,\mathrm{d}x\,\mathrm{d}y, \\
f_2 '(t)
=t\|u\|^2_{g}
-t^{2\cdot2^{*}_{\mu}-1}
\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}
\frac{|u(x)|^{2^{*}_{\mu}}|u(y)|^{2^{*}_{\mu}}}{|x-y|^{\mu}}
\,\mathrm{d}x\,\mathrm{d}y.
\end{gather*}
This implies  that
$f_2 '(\cdot)=0$
if and only if
$\|u\|^2_{g}
=t^{2\cdot2^{*}_{\mu}-2}
\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}
\frac{|u(x)|^{2^{*}_{\mu}}|u(y)|^{2^{*}_{\mu}}}{|x-y|^{\mu}}
\,\mathrm{d}x\,\mathrm{d}y$.
Set
\[
f_3(t)
=t^{2\cdot2^{*}_{\mu}-2}
\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}
\frac{|u(x)|^{2^{*}_{\mu}}|u(y)|^{2^{*}_{\mu}}}{|x-y|^{\mu}}
\,\mathrm{d}x\,\mathrm{d}y.
\]
We know that
$\lim_{t\to0}f_3(t)\to0$,
$\lim_{t\to\infty}f_3(t)\to\infty$
and $f_3(\cdot)$ is strictly increasing on
$(0,\infty)$.
This shows that $f_2 (\cdot)$
admits a unique critical point
$t_{u}$ on $(0,\infty)$
such that $f_2 (\cdot)$
takes the maximum at $t_{u}$. This is showing that
$t_{u}u\in K_1$.

To prove the uniqueness of $t_{u}$,
let us assume that $0<\bar{t}<\bar{\bar{t}}$
satisfy $f_2 '(\bar{t})=f_2 '(\bar{\bar{t}})=0$.
We obtain
\begin{align*}
\|u\|^2_{g}
=&\bar{t}^{2\cdot2^{*}_{\mu}-2}
\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}
\frac{|u(x)|^{2^{*}_{\mu}}|u(y)|^{2^{*}_{\mu}}}{|x-y|^{\mu}}
\,\mathrm{d}x\,\mathrm{d}y\\
=&\bar{\bar{t}}^{2\cdot2^{*}_{\mu}-2}
\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}
\frac{|u(x)|^{2^{*}_{\mu}}|u(y)|^{2^{*}_{\mu}}}{|x-y|^{\mu}}
\,\mathrm{d}x\,\mathrm{d}y.
\end{align*}
Since
$\bar{t}^{2\cdot2^{*}_{\mu}-2}
<\bar{\bar{t}}^{2\cdot2^{*}_{\mu}-2}$,
the above equality leads to the contradiction:
$u=0$.

Hence, for each $u\in D^{1,2}(\mathbb{R}^{N})$
with $u\not\equiv0$,
there exists a unique
$t_{u} > 0$ such that
$t_{u}u\in K_1$.

\noindent (ii)
Set $\Phi(u)=\langle I'_1(u),u\rangle$,
for any
$u\in K_1$,
then
\begin{align*}
\langle\Phi'(u),u\rangle
=&
\langle\Phi'(u),u\rangle
-q\Phi(u)\\
\leqslant&
(2-q)\|u\|^2_{g}
-(2\cdot 2^{*}_{\mu}-q)
\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}
\frac{|u(x)|^{2^{*}_{\mu}}|u(y)|^{2^{*}_{\mu}}}{|x-y|^{\mu}}
\,\mathrm{d}x\,\mathrm{d}y\\
\leqslant&
(2-q)\|u\|^2_{g}<0,
\end{align*}
where
$2<q<2\cdot 2^{*}_{\mu}$.
Thus,
for any
$u\in K_1$,
we obtain
$\Phi'(u)\not=0$.

\noindent (iii)
If $u\in K_1$
and
$I_1(u)=\overline{E}_1$
then since
$\overline{E}_1$
is the minimum of $I_1$ on $K_1$,
Lagrange multiplier theorem implies that there exists
$\lambda\in \mathbb{R}$ such
that
$I'_1(u)=\lambda \Phi'(u)$.
Thus
$$
\langle \lambda \Phi'(u),u\rangle =\langle I'_1(u),u\rangle=  \Phi(u)=0.
$$
According to the previous result,
$\lambda=0$, and so,
$I'_1(u)=0$,
Then $u$ is ground state
solution for
problem
\eqref{eP1}.
\smallskip

\noindent\textbf{Step 4.}
By the Ekeland variational principle,
there exists
$\{\bar{u}_n\}\subset K_1$
and
$\lambda_n\in \mathbb{R}$
such that
$$
I_1(\bar{u}_n)\to \overline{E}_1\quad\text{and}\quad
I'_1(\bar{u}_n)-\lambda_n\Phi'(\bar{u}_n)\to 0 \quad
\text{in }
(D^{1,2}(\mathbb{R}^{N}))^{-1},
$$
we can show that
$\{\bar{u}_n\}$ is bounded in
$D^{1,2}(\mathbb{R}^{N})$.
Hence, taking into account that
$$
|\langle I'_1(\bar{u}_n),\bar{u}_n\rangle
-\langle\lambda_n\Phi'(\bar{u}_n),\bar{u}_n\rangle|
\leqslant
\| I'_1(\bar{u}_n)-\lambda_n\Phi'(\bar{u}_n)\|_{D^{-1}}\|\bar{u}_n\|_{D}\to0,
$$
we have
$$
\langle I'_1(\bar{u}_n),\bar{u}_n\rangle-\lambda_n\langle
\Phi'(\bar{u}_n),\bar{u}_n\rangle\to0,
$$
Using  that
$\langle I'_1(\bar{u}_n),\bar{u}_n\rangle=0$
and
$\langle \Phi'(\bar{u}_n),\bar{u}_n\rangle\not=0$,
we conclude that
$\lambda_n\to0$.
Consequently,
$I'_1(\bar{u}_n)\to0$
in
$(D^{1,2}(\mathbb{R}^{N}))^{-1}$.
Hence
$\{\bar{u}_n\}$
is a
$(PS)_{\overline{E}_1}$
sequence of
$I_1$.

By Lemma \ref{lemma10},
we obtain that
$\{\bar{u}_n\}$
has a strongly convergent subsequence
(still denoted by
$\{\bar{u}_n\}$).
Hence,
there exists
$\bar{u}_{0}\in D^{1,2}(\mathbb{R}^{N})$
such that
$\bar{u}_n\to\bar{u}_{0}$
in
$D^{1,2}(\mathbb{R}^{N})$.
By using Step 2,
we know
$\bar{u}_{0}\not=0$.
By weak lower semicontinuity of
$\|\cdot\|_{g}$,
we have
\begin{align*}
\overline{E}_1
\leqslant
I_1(\bar{u}_{0})
=&
\Big(
\frac{1}{2}
-\frac{1}{2\cdot2^{*}_{\mu}}
\Big)
\|\bar{u}_{0}\|^2_{g}
\\
\leqslant&
\liminf_{n\to\infty}
\Big(
\frac{1}{2}
-
\frac{1}{2\cdot2^{*}_{\mu}}
\Big)
\|\bar{u}_n\|^2_{g}
\\
=&
\liminf_{n\to\infty}
I_1(\bar{u}_n)
=
\lim_{n\to\infty}
I_1(\bar{u}_n)
=\overline{E}_1,
\end{align*}
which implies that
$I_1(\bar{u}_{0})=\overline{E}_1$.
Therefore, $\bar{u}_{0}$
is a ground state solution of problem
\eqref{eP1}.
\end{proof}

\section{Perturbation with a sign-changing superlinear nonlocal term}

In this section,
we  study the existence of nontrivial solutions for the critical Choquard
equation with a sign-changing superlinear nonlocal term,
i.e.
\begin{equation} \label{eP2b}
-\Delta u
=\Big(
\int_{\mathbb{R}^{N}}
\frac{|u|^{2^{*}_{\mu}}}{|x-y|^{\mu}}
\,\mathrm{d}y \Big)
|u|^{2^{*}_{\mu}-2}u
+\Big(
\int_{\mathbb{R}^{N}}
\frac{g(y)|u|^{p}}
{|x-y|^{\mu}}
\,\mathrm{d}y
 \Big)
g(x)|u|^{p-2}u.
\end{equation}
We introduce the energy functional associated with \eqref{eP2} as
\begin{align*}
I_2 (u)
=& \frac{1}{2}
\|u\|_{D}^2
-\frac{1}{2p}
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{g(x)g(y)|u_n(x)|^{p}|u_n(y)|^{p}}
{|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y\\
&-\frac{1}{2\cdot 2^{*}_{\mu}}
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|u_n(x)|^{2_{\mu}^{*}}|u_n(y)|^{2_{\mu}^{*}}}
{|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y.
\end{align*}
The following Lemma plays an important role in estimating the Mountain-Pass levels.

\begin{lemma}\label{lemma11}
Assume that {\rm (A7)--(A10), (A12)} hold. Let
$N\geqslant3$,
$\mu\in(0,N)$
and
$p\in(\frac{2N-\mu}{N},\frac{2N-\mu}{N-2})$.
Then for all
$\sigma\in[r_{\varepsilon},r_{g}]$,
we have
$$
\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}
\frac{g(x)g(y)|w_{\sigma}(x)|^{p}|w_{\sigma}(y)|^{p}}
{|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y>0.
$$
\end{lemma}


\begin{proof}
According to (A8), (A9) and \eqref{11}, we obtain
\begin{equation}\label{24}
\begin{aligned}
&\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{g(x)g(y)|w_{\sigma}(x)|^{p}|w_{\sigma}(y)|^{p}}
{|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y\\
&=
\int_{\Omega_1\cup\Omega_2 }
\int_{\Omega_1\cup\Omega_2 }
\frac{\mathfrak{C}^{2p}\sigma^{p(N-2)}g(x)g(y)}{
(\sigma^2+|x|^2)^{\frac{p}{2}(N-2)}
|x-y|^{\mu}
(\sigma^2+|y|^2)^{\frac{p}{2}(N-2)}}
\,\mathrm{d}x
\,\mathrm{d}y\\
&=
\Big(
\int_{\Omega_1}
\int_{\Omega_1}
\frac{\mathfrak{C}^{2p}\sigma^{p(N-2)}g(x)g(y)}{
(\sigma^2+|x|^2)^{\frac{p}{2}(N-2)}
|x-y|^{\mu}
(\sigma^2+|y|^2)^{\frac{p}{2}(N-2)}}
\,\mathrm{d}x
\,\mathrm{d}y\\
&\quad +
\int_{\Omega_2 }
\int_{\Omega_1}
\frac{\mathfrak{C}^{2p}\sigma^{p(N-2)}g(x)g(y)}{
(\sigma^2+|x|^2)^{\frac{p}{2}(N-2)}
|x-y|^{\mu}
(\sigma^2+|y|^2)^{\frac{p}{2}(N-2)}}
\,\mathrm{d}x
\,\mathrm{d}y\\
&\quad +\int_{\Omega_1}
\int_{\Omega_2 }
\frac{\mathfrak{C}^{2p}\sigma^{p(N-2)}g(x)g(y)}{
(\sigma^2+|x|^2)^{\frac{p}{2}(N-2)}
|x-y|^{\mu}
(\sigma^2+|y|^2)^{\frac{p}{2}(N-2)}}
\,\mathrm{d}x
\,\mathrm{d}y\\
&\quad +
\int_{\Omega_2 }
\int_{\Omega_2 }
\frac{\mathfrak{C}^{2p}\sigma^{p(N-2)}g(x)g(y)}{
(\sigma^2+|x|^2)^{\frac{p}{2}(N-2)}
|x-y|^{\mu}
(\sigma^2+|y|^2)^{\frac{p}{2}(N-2)}}
\,\mathrm{d}x
\,\mathrm{d}y
\Big)\\
&= \mathfrak{C}^{2p}\sigma^{p(N-2)}
\Big(A_1
+A_2
+A_3
+A_4
\Big).
\end{aligned}
\end{equation}
Since $\Omega_1\cup\Omega_2 \subset \overline{B(0,r_{g})}
\setminus B(0,r_{\varepsilon})$,
we have
$|x|,|y|\in[r_{\varepsilon},r_{g}]$
in $\Omega_1$.
By using
$g(x),g(y)>0$
on
$\Omega_1$,
$\sigma\in[r_{\varepsilon},r_{g}]$
and Fubini's theorem,
we obtain
\begin{equation}\label{25}
\begin{aligned}
A_1
\geqslant&
\int_{\Omega_1}
\int_{\Omega_1}
\frac{g(x)g(y)}{
(\sigma^2+r_{g}^2)^{p(N-2)}
|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y\\
\geqslant&
\int_{\Omega_1}
\int_{\Omega_1}
\frac{g(x)g(y)}{
(\sigma^2+r_{g}^2)^{p(N-2)}
(|x|+|y|)^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y\\
\geqslant&
\frac{1}
{(\sigma^2+r_{g}^2)^{p(N-2)}
|2r_{g}|^{\mu}}
\int_{\Omega_1}
\int_{\Omega_1}
g(x)g(y)
\,\mathrm{d}x
\,\mathrm{d}y\\
\geqslant&
\frac{1}
{2^{p(N-2)+\mu}\cdot r_{g}^{2p(N-2)+\mu}}
\int_{\Omega_1}
g(x)
\,\mathrm{d}x
\int_{\Omega_1}
g(y)
\,\mathrm{d}y.
\end{aligned}
\end{equation}
Similar to \eqref{25}, we have
\begin{equation}\label{26}
A_4
\geqslant
\frac{1}
{2^{p(N-2)+\mu}\cdot r_{g}^{2p(N-2)+\mu}}
\int_{\Omega_2 }
g(x)
\,\mathrm{d}x
\int_{\Omega_2 }
g(y)
\,\mathrm{d}y\geqslant0.
\end{equation}
Keeping in mind that
$g(x)>0$
on
$\Omega_1$
and
$g(y)<0$
on
$\Omega_2 $.
Since
$x\in \Omega_1$,
$y\in \Omega_2 $,
$\Omega_1\cap\Omega_2 =\emptyset$
and (A12), we have
$|x-y|\geqslant2r_{\varepsilon}$.
Then
\begin{equation}\label{27}
\begin{aligned}
A_2
\geqslant&
\int_{\Omega_2 }
\int_{\Omega_1}
\frac{g(x)g(y)}{
(\sigma^2+r_{\varepsilon}^2)^{p(N-2)}
|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y\\
\geqslant&
\int_{\Omega_2 }
\int_{\Omega_1}
\frac{g(x)g(y)}{
(\sigma^2+r_{\varepsilon}^2)^{p(N-2)}
|2r_{\varepsilon}|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y\\
\geqslant&
\frac{1}
{2^{p(N-2)+\mu}\cdot r_{\varepsilon}^{2p(N-2)+\mu}}
\int_{\Omega_1}
g(x)
\,\mathrm{d}x
\int_{\Omega_2 }
g(y)
\,\mathrm{d}y.
\end{aligned}
\end{equation}
Similar to \eqref{27},
we have
\begin{equation}\label{28}
\begin{aligned}
A_3
\geqslant&
\frac{1}
{2^{p(N-2)+\mu}\cdot r_{\varepsilon}^{2p(N-2)+\mu}}
\int_{\Omega_2 }
g(x)
\,\mathrm{d}x
\int_{\Omega_1}
g(y)
\,\mathrm{d}y\\
=&
\frac{1}
{2^{p(N-2)+\mu}\cdot r_{\varepsilon}^{2p(N-2)+\mu}}
\int_{\Omega_1}
g(x)
\,\mathrm{d}x
\int_{\Omega_2 }
g(y)
\,\mathrm{d}y.
\end{aligned}
\end{equation}
Combining
\eqref{25},
\eqref{27}
and
\eqref{28},
we deduce that
\begin{equation}\label{29}
\begin{aligned}
A_1 +A_2 +A_3
&\geqslant \frac{1}{2^{p(N-2)+\mu}}
\int_{\Omega_1}
g(x)
\,\mathrm{d}x
\Big(\frac{1}
{r_{g}^{2p(N-2)+\mu}}
\int_{\Omega_1}
g(y)
\,\mathrm{d}y \\
&\quad +\frac{2}
{r_{\varepsilon}^{2p(N-2)+\mu}}
\int_{\Omega_2 }
g(y)
\,\mathrm{d}y\Big).
\end{aligned}
\end{equation}
By (A10), we have
\begin{equation}\label{30}
\begin{aligned}
\int_{\Omega_1}g(y)\,\mathrm{d}y
>& 2 \big(\frac{r_{g}}{r_{\varepsilon}}\big)^{4N}
\int_{\Omega_2 }(-g(y))\,\mathrm{d}y\\
>& 2\big(\frac{r_{g}}{r_{\varepsilon}}\big)^{4N-\mu}
\int_{\Omega_2 }(-g(y))\,\mathrm{d}y\\
>&2\big(\frac{r_{g}}{r_{\varepsilon}}\big)^{2p(N-2)+\mu}
\int_{\Omega_2 }(-g(y))\,\mathrm{d}y.
\end{aligned}
\end{equation}
Inserting \eqref{30} into
\eqref{29},
we deduce that
\begin{equation}\label{31}
\begin{aligned}
A_1+A_2 +A_3>0.
\end{aligned}
\end{equation}
Inserting \eqref{26} and \eqref{31} into
\eqref{24}, we obtain
\begin{align*}
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{g(x)g(y)|w_{\sigma}(x)|^{p}|w_{\sigma}(y)|^{p}}
{|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y
\geqslant
A_1+A_2 +A_3+A_4
>0.
\end{align*}
\end{proof}

We show that the functional $I_2 $
satisfies the Mountain-Pass geometry,
and estimate the Mountain-Pass levels.

\begin{lemma}\label{lemma12}
Assume that the hypotheses of Theorem \ref{theorem3}
hold. Then there exists a
$(PS)_{c}$
sequence of
$I_2 $
at a level
$c$,
where
$0<c<c^{*}=
\frac{N+2-\mu}{4N-2\mu}S_{H,L}^{\frac{2N-\mu}{N+2-\mu}}$.
\end{lemma}

\begin{proof}
\textbf{Step 1.}
For any $u\in D^{1,2}(\mathbb{R}^{N})\setminus\{0\}$,
we have
\begin{align*}
I_2 (u)
\geqslant&
\frac{1}{2}
\|u\|_{D}^2
-\frac{C_1}{2p}
\|u\|_{D}^{2p}
-\frac{1}{2\cdot2^{*}_{\mu}S_{H,L}^{2^{*}_{\mu}}}
\|u\|_{D}^{2\cdot 2^{*}_{\mu}}.
\end{align*}
We just prove that
$I_2 $ satisfies the above condition in Mountain-pass theorem,
the others similar to Lemma \ref{lemma7}.
\smallskip

\noindent\textbf{Step 2.}
Using Lemma \ref{lemma11},
there exists $\sigma\in[r_{\varepsilon},r_{g}]$
such that
$$
\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}
\frac{g(x)g(y)|w_{\sigma}(x)|^{p}|w_{\sigma}(y)|^{p}}
{|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y>0.
$$
Let
\begin{equation}\label{32}
\begin{gathered}
B_1
:=\int_{\mathbb{R}^{N}}
|\nabla w_{\sigma}|^2
\,\mathrm{d}x>0,
\\
B_2
:=\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{g(x)g(y)|w_{\sigma}(x)|^{p}|w_{\sigma}(y)|^{p}}
{|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y>0,\\
B_3
:=\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{|w_{\sigma}(x)|^{2^{*}_{\mu}}|w_{\sigma}(y)|^{2^{*}_{\mu}}}
{|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y>0.
\end{gathered}
\end{equation}
Set
\[
h_1(t)
=I_2 (tw_{\sigma})
=\frac{t^2}{2}
B_1
-\frac{t^{2p}}{2p}
B_2
-\frac{t^{2\cdot2^{*}_{\mu}}}{2\cdot2^{*}_{\mu}}
B_3.
\]
We know that
$h_1(0)=0$, $\lim_{t\to\infty}h_1(t)=-\infty$.
On the interval
$[0,\infty)$,
we can see that
$h'_1(t)=0$
if and only if
\begin{equation}\label{33}
h'_1(t)
=tB_1
-t^{2p-1}B_2
-t^{2\cdot2^{*}_{\mu}-1}
B_3
=0.
\end{equation}
From \eqref{33},
we have
\begin{equation}\label{34}
B_1
=t^{2p-2}
B_2
+t^{2\cdot2^{*}_{\mu}-2}
B_3.
\end{equation}
Define
\begin{equation}\label{35}
k(t)
=t^{2p-2}
B_2
+t^{2\cdot2^{*}_{\mu}-2}
B_3.
\end{equation}
By Lemma \ref{lemma11} and \eqref{32},
we know that
$B_2 >0$.
Observe
$k(\cdot)$
is strictly increasing on
$[0,\infty)$,
$k(t)=0$
if and only if
$t=0$,
and
$\lim_{t\to\infty}k(t)=\infty$.
From the fact that
$h'_1(t)=t(B_1-k(t))=0$,
we have two solutions
$t_1$
and $t_2 $
such that $t_1=0$
and $t_2 $
satisfies
\begin{equation}
B_1
=t_2 ^{2p-2}
B_2
+t_2 ^{2\cdot2^{*}_{\mu}-2}
B_3.\label{36}
\end{equation}
By $B_1,B_2 ,B_3>0$
and \eqref{36},
we have $t_2 >0$
and
\begin{equation}\label{37}
\begin{aligned}
h_1(t_2 )
=&\frac{
t_2 ^2
}{2}
B_1
-\frac{t_2 ^{2p}}{2p}
B_2
-\frac{t_2 ^{2\cdot2^{*}_{\mu}}}{2\cdot2^{*}_{\mu}}
B_3\\
=&
\frac{
t_2 ^2
}{2}
\Big(
t_2 ^{2p-2}
B_2
+
t_2 ^{2\cdot2^{*}_{\mu}-2}
B_3
\Big)
-\frac{t_2 ^{2p}}{2p}
B_2
-\frac{t_2 ^{2\cdot2^{*}_{\mu}}}{2\cdot2^{*}_{\mu}}
B_3\\
=&
\Big(
\frac{1}{2}
-
\frac{1}{2p}
\Big)
t_2 ^{2p}
B_2
+\Big(
\frac{1}{2}
-
\frac{1}{2\cdot2^{*}_{\mu}}
\Big)
t_2 ^{2\cdot2^{*}_{\mu}}
B_3>0.
\end{aligned}
\end{equation}
So $h_1$ does not
achieve its maximum
at $t_1=0$.


Next, we prove that
$h_1$
achieves its maximum at
$t_2 $.
Applying
\eqref{36},
we know
\begin{equation}\label{38}
\begin{aligned}
h'_1(t)
=&
t(B_1
-k(t))\\
=&
t[
(t_2 ^{2p-2}-t^{2p-2})
B_2
+
(t_2 ^{2\cdot2^{*}_{\mu}-2}-t^{2\cdot2^{*}_{\mu}-2})
B_3]
>0, \quad\text{for } t\in(0,t_2 ),
\end{aligned}
\end{equation}
and
\begin{equation}\label{39}
\begin{aligned}
h'_1(t)
=t[
(t_2 ^{2p-2}-t^{2p-2})
B_2
+(t_2 ^{2\cdot2^{*}_{\mu}-2}-t^{2\cdot2^{*}_{\mu}-2})
B_3]
<0, \quad\text{for } t\in(t_2 ,\infty).
\end{aligned}
\end{equation}
Let
$t_3=\big(2^{*}_{\mu}\cdot\frac{B_1}{B_3}\big)^{\frac{1}{2\cdot2^{*}_{\mu}-2}}$.
Since $t_3>0$, we have
\begin{equation}\label{40}
\begin{aligned}
h_1(t_3)
=\frac{t_3^2}{2}
B_1
-\frac{t_3^{2p}}{2p}
B_2
-\frac{t_3^{2\cdot2^{*}_{\mu}}}{2\cdot2^{*}_{\mu}}
B_3
=-\frac{t_3^{2p}}{2p}
B_2 <0.
\end{aligned}
\end{equation}
Now,
we claim that
$t_2 <t_3$.

Suppose on the contrary that $t_2 =t_3$,
we obtain
$$
h_1(t_3)<0<h_1(t_2 ),
$$
which contradicts with
$t_2 =t_3$.

Suppose on the contrary that
$t_2 >t_3$.
Applying
$h_1(t_1)=0$,
\eqref{40},
\eqref{38}
and
$t_3\in (t_1,t_2 )$,
we obtain
$$0=h_1(t_1)<h_1(t_3)<0,$$
which is a contradiction.
Hence,
$t_2 <t_3$.

According to Extreme value theorem,
we know that $h_1$
achieves its maximum on compact set
$[0,t_3]$.
Applying
$h_1'(t_2 )=0$,
\eqref{38}
and
\eqref{39},
we obtain that
$h_1(t_2 )$
is the maximum of
$h_1$
on
$[0,t_3]$.

By using
\eqref{39}
and
$h_1(t_3)<0$,
we obtain
$h_1(t)<0$
for
$t\in(t_3,\infty)$.
Hence,
we deduce that
$h_1(t_2 )$
is the maximum of
$h_1$
on
$[0,\infty)$.
\smallskip

\noindent\textbf{Step 3.}
Set
$h_2 (t)=\frac{t^2}{2}
B_1
-\frac{t^{2\cdot2^{*}_{\mu}}}{2\cdot2^{*}_{\mu}}
B_3$.
Similar to Step 2,
we obtain that
the maximum of
$h_2 $
attained at
$t_4=\big(\frac{B_1}{B_3}\big)^{\frac{1}{2\cdot2^{*}_{\mu}-2}}>0$.

Next, we prove that
$t_4>t_2 $.
By
\eqref{33},
we have
\[
h'_1(t_4)
=
t_4
(B_1
-t_4^{2\cdot2^{*}_{\mu}-1}
B_3)
-t_4^{2p-1}
B_2
=-t_4^{2p-1}
B_2<0.
\]
Similar to the proof of $t_3>t_2 $
in Step 2,
we know that
$t_4>t_2 $.

Furthermore, we show that
$\max_{t\geqslant0}
h_1(t)
<\max_{t\geqslant0}
h_2 (t)$.
We have
\begin{equation}\label{41}
\begin{aligned}
\max_{t\geqslant0}
h_1(t)
=
h_1(t_2 )
=&
\Big(
\frac{1}{2}
-\frac{1}{2p}
\Big)
t_2 ^{2p}
B_2
+\Big(
\frac{1}{2}
-\frac{1}{2\cdot2^{*}_{\mu}}
\Big)
t_2 ^{2\cdot2^{*}_{\mu}}
B_3\\
<&t_2 ^2
\Big(
\frac{1}{2}
-
\frac{1}{2\cdot2^{*}_{\mu}}
\Big)
\Big(
t_2 ^{2p-2}
B_2
+
t_2 ^{2\cdot2^{*}_{\mu}-2}
B_3
\Big)\\
=&t_2 ^2
\Big(
\frac{1}{2}
-
\frac{1}{2\cdot2^{*}_{\mu}}
\Big)
B_1,
\end{aligned}
\end{equation}
and
\begin{equation}\label{42}
\begin{aligned}
\max_{t\geqslant0}
h_2 (t)
=
h_2 (t_4)
=&t_4^2
\Big(
\frac{1}{2}
-\frac{1}{2\cdot2^{*}_{\mu}}
\Big)
B_1.
\end{aligned}
\end{equation}
According to
\eqref{41},
\eqref{42}
and
$t_2 <t_4$,
we know
\[
\begin{aligned}
\max_{t\geqslant0}
h_1(t)
<&
t_2 ^2
\Big(
\frac{1}{2}
-
\frac{1}{2\cdot2^{*}_{\mu}}
\Big)
B_1\\
<&
t_4^2
\Big(
\frac{1}{2}
-
\frac{1}{2\cdot2^{*}_{\mu}}
\Big)
B_1=\max_{t\geqslant0}
h_2 (t)
=
h_2 (t_4).
\end{aligned}
\]
\smallskip
%%  page 25

\noindent\textbf{Step 4.}
From the above argument,
we have
\[
0<c\leqslant
\sup_{t\geqslant0}
I_2 (tw_{\sigma})
=\max_{t\geqslant0}
h_1(t)
<h_2 (t_4)
=\frac{N+2-\mu}{4N-2\mu}S_{H,L}^{\frac{2N-\mu}{N+2-\mu}},
\]
which means that
$0<c<\frac{N+2-\mu}{4N-2\mu}S_{H,L}^{\frac{2N-\mu}{N+2-\mu}}$.
\end{proof}

\begin{lemma}\label{lemma13}
Assume that the hypotheses of Theorem \ref{theorem3} hold.
If $\{u_n\}$
is a
$(PS)_{c}$
sequence of
$I_2 $,
then
$\{u_n\}$
is bounded in
$D^{1,2}(\mathbb{R}^{N})$.
\end{lemma}

\begin{proof}
Similar to the proof of Lemma \ref{lemma8},
we have Lemma \ref{lemma13}.
We omit it.
\end{proof}

To check that the functional $I_2 $
satisfies
$(PS)_{c}$ condition,
we give the following Lemmas.

\begin{lemma}\label{lemma14}
Assume that the hypotheses of Theorem
\ref{theorem3}
hold.
If
$\{u_n\}\subset D^{1,2}(\mathbb{R}^{N})$
is a sequence converging weakly to
$u\in D^{1,2}(\mathbb{R}^{N})$
as
$n\to\infty$,
then
\[
\begin{aligned}
\int_{\Omega_1\cup\Omega_2 }
\int_{\Omega_1\cup\Omega_2 }
\frac{g(x)g(y)|u_n(x)-u(x)|^{p}|u_n(y)-u(y)|^{p}}
{|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y
=0.
\end{aligned}
\]
\end{lemma}

\begin{proof}
Set $v_n:=u_n-u$,
then we know $v_n\rightharpoonup0$
in $D^{1,2}(\mathbb{R}^{N})$.
According to (A8), (A9) and
Hardy-Littlewood-Sobolev inequality,
we have
\begin{equation}\label{43}
\begin{aligned}
&\lim_{n\to\infty}
\int_{\Omega_1\cup\Omega_2 }
\int_{\Omega_1\cup\Omega_2 }
\frac{g(x)g(y)|v_n(x)|^{p}|v_n(y)|^{p}}
{|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y\\
&\leqslant
g^2_{\rm max}
\lim_{n\to\infty}
\int_{\Omega_1\cup\Omega_2 }
\int_{\Omega_1\cup\Omega_2 }
\frac{|v_n(x)|^{p}|v_n(y)|^{p}}
{|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y\\
&\leqslant
g^2_{\rm max}
C
\|v_n\|^{2p}_{L^{\frac{2Np}{2N-\mu}}(\Omega_1\cup\Omega_2 )}\to0
\quad \text{(since $\frac{2Np}{2N-\mu}\in(2,2^{*})$)}.
\end{aligned}
\end{equation}
On the other hand,
\begin{equation}\label{44}
\begin{aligned}
&\lim_{n\to\infty}
\int_{\Omega_1\cup\Omega_2 }
\int_{\Omega_1\cup\Omega_2 }
\frac{g(x)g(y)|v_n(x)|^{p}|v_n(y)|^{p}}
{|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y\\
&=
\lim_{n\to\infty}
\Big(
\int_{\Omega_1}
\int_{\Omega_1}
\frac{g(x)g(y)|v_n(x)|^{p}|v_n(y)|^{p}}
{|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y
 \\
&\quad +
\int_{\Omega_1}
\int_{\Omega_2 }
\frac{g(x)g(y)|v_n(x)|^{p}|v_n(y)|^{p}}
{|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y\\
&\quad +
\int_{\Omega_2 }
\int_{\Omega_1}
\frac{g(x)g(y)|v_n(x)|^{p}|v_n(y)|^{p}}
{|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y\\
&\quad +
\int_{\Omega_2 }
\int_{\Omega_2 }
\frac{g(x)g(y)|v_n(x)|^{p}|v_n(y)|^{p}}
{|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y
\Big)\\
&=J_1+J_2 +J_3+J_4.
\end{aligned}
\end{equation}
Applying  (A8) and (A9),
we have
\begin{equation}\label{45}
J_1\geqslant0\quad\text{and}\quad
J_4\geqslant0.
\end{equation}
Keeping in mind that $g(x)>0$
on $\Omega_1$
and $g(y)<0$
on $\Omega_2 $.
According to
$g_{\rm min}<0$,
$p\in(\frac{2N-\mu}{N},\frac{2N-\mu}{N-2})\subset[1,2^{*})$
and
Fubini's theorem,
we obtain
\begin{equation}\label{46}
\begin{aligned}
J_2
\geqslant&
g_{\rm max} g_{\rm min}
\lim_{n\to\infty}
\int_{\Omega_1}
\int_{\Omega_2 }
\frac{|v_n(x)|^{p}|v_n(y)|^{p}}
{|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y\\
\geqslant&
\frac{g_{\rm max} g_{\rm min}}{(2r_{\varepsilon})^{\mu}}
\lim_{n\to\infty}
\int_{\Omega_1}
\int_{\Omega_2 }
|v_n(x)|^{p}|v_n(y)|^{p}
\,\mathrm{d}x
\,\mathrm{d}y\\
=&
\frac{g_{\rm max} g_{\rm min}}{(2r_{\varepsilon})^{\mu}}
\Big(
\lim_{n\to\infty}
\int_{\Omega_1}
|v_n(x)|^{p}
\,\mathrm{d}x
\Big)
\Big(
\lim_{n\to\infty}
\int_{\Omega_2 }
|v_n(y)|^{p}
\,\mathrm{d}y
\Big)\to0.
\end{aligned}
\end{equation}
Similar to \eqref{46}, we have
\begin{equation}\label{47}
J_3
\geqslant
0.
\end{equation}
Putting
\eqref{45}--\eqref{47} into \eqref{44},
we obtain
\begin{equation}\label{48}
\begin{aligned}
\lim_{n\to\infty}
\int_{\Omega_1\cup\Omega_2 }
\int_{\Omega_1\cup\Omega_2 }
\frac{g(x)g(y)|v_n(x)|^{p}|v_n(y)|^{p}}
{|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y
\geqslant0.
\end{aligned}
\end{equation}
Combining
\eqref{43}
and
\eqref{48},
we have
\[
0\leqslant
\lim_{n\to\infty}
\int_{\Omega_1\cup\Omega_2 }
\int_{\Omega_1\cup\Omega_2 }
\frac{g(x)g(y)|v_n(x)|^{p}|v_n(y)|^{p}}
{|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y
\leqslant0.
\]
Thus,
\[
\begin{aligned}
\lim_{n\to\infty}
\int_{\Omega_1\cup\Omega_2 }
\int_{\Omega_1\cup\Omega_2 }
\frac{g(x)g(y)|v_n(x)|^{p}|v_n(y)|^{p}}
{|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y
=0.
\end{aligned}
\]
\end{proof}

\begin{lemma}\label{lemma15}
Assume that the hypotheses of Theorem
\ref{theorem3}
hold.
If
$\{u_n\}$
is a bounded sequence in
$L^{\frac{2Np}{2N-\mu}}(\Omega_1\cup\Omega_2 )$
such that
$u_n\to u$ a.e. on $\Omega_1\cup\Omega_2 $
as
$n\to\infty$,
then
\begin{align*}
&\int_{\Omega_1\cup\Omega_2 }
\int_{\Omega_1\cup\Omega_2 }
\frac{g(x)g(y)|u_n(x)|^{p}|u_n(y)|^{p}}
{|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y\\
&-
\int_{\Omega_1\cup\Omega_2 }
\int_{\Omega_1\cup\Omega_2 }
\frac{g(x)g(y)|u(y)|^{p}|u(x)|^{p}}
{|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y\\
&=
\int_{\Omega_1\cup\Omega_2 }
\int_{\Omega_1\cup\Omega_2 }
\frac{g(x)g(y)|u_n(x)-u(x)|^{p}|u_n(y)-u(y)|^{p}}
{|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y
+
o(1),
\end{align*}
\end{lemma}

\begin{proof} \textbf{Step 1.}
Since
$\{u_n\}$
is a bounded sequence in
$L^{\frac{2Np}{2N-\mu}}(\Omega_1\cup\Omega_2 )$
and
$u_n\to u ~~\mathrm{a.e.~~on}~~\Omega_1\cup\Omega_2 $
as
$n\to\infty$.
Set
$v_n:=u_n-u$.
By using
Hardy-Littlewood-Sobolev inequality
and
Br\'{e}zis-Lieb lemma in
\cite{Brezis1983},
we have
\begin{equation}\label{49}
\int_{\Omega_i}
g(x)|u_n|^{p}
\,\mathrm{d}x\\
=
\int_{\Omega_i}
g(x)|v_n|^{p}
\,\mathrm{d}x
+\int_{\Omega_i}
g(x)|u|^{p}
\,\mathrm{d}x
+o(1),
\end{equation}
for $=1,2$, as $n\to\infty$,
and
\begin{equation}\label{50}
|x|^{-\mu}*(g|u_n|^{p})(\Omega_i)\\
=|x|^{-\mu}*(g|v_n|^{p})(\Omega_i)
+|x|^{-\mu}*[g|u|^{p}](\Omega_i)
+o(1),
\end{equation}
for $i=1,2$, as $n\to\infty$.

By using \cite[Lemma 2.3]{Gao2017JMAA},
we know that
\begin{equation}\label{51}
\begin{aligned}
&\int_{\Omega_i}
\big(
|x|^{-\mu}*(g|u_n|^{p})(\Omega_i)
\big)
g|u_n|^{p}
\,\mathrm{d}x\\
&= \int_{\Omega_i}
\big(
|x|^{-\mu}*(g|v_n|^{p})(\Omega_i)
\big)
g|v_n|^{p}
\,\mathrm{d}x\\
&\quad +
\int_{\Omega_i}
\big(
|x|^{-\mu}*(g|u|^{p})(\Omega_i)
\big)
g|u|^{p}
\,\mathrm{d}x
+o(1),\quad
(i=1,2), \text{ as }n\to\infty.
\end{aligned}
\end{equation}
\smallskip

\noindent\textbf{Step 2.}
Since $y\in \Omega_1$,
$x\in \Omega_2 $
and (A12),
we have
$2r_{\varepsilon}\leqslant|x-y|\leqslant2r_{g}$.
According to the properties of convolution
and Fubini's theorem,
we obtain
\begin{equation}\label{52}
\begin{aligned}
&\int_{\Omega_2 }
\left(
|x|^{-\mu}*(g|v_n|^{p})(\Omega_1)
\right)
g(|u_n|^{p}-|v_n|^{p})
\,\mathrm{d}x\\
&=
\int_{\Omega_2 }
\int_{\Omega_1}
\frac{g(y)|u_n(y)|^{p}}
{|x-y|^{\mu}}
g(x)(|u_n(x)|^{p}-|v_n(x)|^{p})
\,\mathrm{d}y
\,\mathrm{d}x\\
&=
\int_{\Omega_1}
\int_{\Omega_2 }
\frac{g(x)|u_n(x)|^{p}}
{|y-x|^{\mu}}
g(y)|u_n(y)|^{p}
\,\mathrm{d}x
\,\mathrm{d}y\\
&\quad -
\int_{\Omega_1}
\int_{\Omega_2 }
\frac{g(x)|v_n(x)|^{p}}
{|y-x|^{\mu}}
g(y)|u_n(y)|^{p}
\,\mathrm{d}x
\,\mathrm{d}y
\quad\text{(Fubini's theorem)}\\
&=
\int_{\Omega_1}
\left(
|y|^{-\mu}*g(|u_n|^{p})(\Omega_2 )
\right)(g|u_n|^{p})
\,\mathrm{d}y\\
&-\int_{\Omega_1}
\left(
|y|^{-\mu}*g(|v_n|^{p})(\Omega_2 )
\right)(g|u_n|^{p})
\,\mathrm{d}y \quad\text{(convolution)} \\
&=
\int_{\Omega_1}
\left(
|y|^{-\mu}*g(|u_n|^{p})(\Omega_2 )
-
|y|^{-\mu}*g(|v_n|^{p})(\Omega_2 )
\right)(g|u_n|^{p})
\,\mathrm{d}y\\
&=
\int_{\Omega_1}
\left(
|y|^{-\mu}*g(|u_n|^{p}-|v_n|^{p})(\Omega_2 )
\right)(g|u_n|^{p})
\,\mathrm{d}y \quad\text{(distributivity)}.
\end{aligned}
\end{equation}
Since $\{u_n\}$ is a bounded sequence in
$L^{\frac{2Np}{2N-\mu}}(\Omega_1)$
and
$u_n\to u$ a.e. on $\Omega_1$
as $n\to\infty$,
we have
\begin{equation}\label{53}
g(y)
|v_n(y)|^{p}\rightharpoonup0
~\text{in }
L^{\frac{2Np}{(2N-\mu)(p-1)}}(\Omega_1),\quad\text{as }n\to\infty.
\end{equation}
Similar to
\eqref{53},
we obtain
\begin{equation}\label{54}
g(x)
|v_n(x)|^{p}\rightharpoonup0 \quad\text{in }
L^{\frac{2Np}{(2N-\mu)(p-1)}}(\Omega_2 ), \quad\text{as } n\to\infty.
\end{equation}
\smallskip

\noindent\textbf{Step 3.}
Taking the limit $n\to\infty$,
by
\eqref{49},
\eqref{50}
and
\eqref{52}--\eqref{54},
we deduce that
\begin{equation}\label{55}
\begin{aligned}
&\int_{\Omega_2 }
\left(
|x|^{-\mu}*(g|u_n|^{p})(\Omega_1)
\right)
g|u_n|^{p}
\,\mathrm{d}x
-\int_{\Omega_2 }
\left(
|x|^{-\mu}*(g|v_n|^{p})(\Omega_1)
\right)
g|v_n|^{p}
\,\mathrm{d}x\\
&= \int_{\Omega_2 }
\left(
|x|^{-\mu}*(g(|u_n|^{p}-|v_n|^{p}))(\Omega_1)
\right)
g(|u_n|^{p}-|v_n|^{p})
\,\mathrm{d}x
\\
&\quad +
\int_{\Omega_2 }
\left(
|x|^{-\mu}*(g(|u_n|^{p}-|v_n|^{p}))(\Omega_1)
\right)
g|v_n|^{p}
\,\mathrm{d}x\\
&\quad +
\int_{\Omega_1}
\left(
|y|^{-\mu}*g(|u_n|^{p}-|v_n|^{p})(\Omega_2 )
\right)
g|v_n|^{p}
\,\mathrm{d}y \quad\text{(by \eqref{52})}\\
&=
\int_{\Omega_2 }
\left(
|x|^{-\mu}*(g |u|^{p}))(\Omega_1)
\right)
g|u|^{p}
\,\mathrm{d}x
+o(1).
\end{aligned}
\end{equation}
Similar to
\eqref{55},
we obtain
\begin{equation}\label{56}
\begin{aligned}
&\int_{\Omega_1}
\left(
|x|^{-\mu}*(g|u_n|^{p})(\Omega_2 )
\right)
g|u_n|^{p}
\,\mathrm{d}x\\
&=
\int_{\Omega_1}
\left(
|x|^{-\mu}*(g|v_n|^{p})(\Omega_2 )
\right)
g|v_n|^{p}
\,\mathrm{d}x\\
&\quad +\int_{\Omega_1}
\left(
|x|^{-\mu}*(g|u|^{p})(\Omega_2 )
\right)
g|u|^{p}
\,\mathrm{d}x
+o(1), \quad\text{as }n\to\infty.
\end{aligned}
\end{equation}
Combining
\eqref{51},
\eqref{55}
and
\eqref{56},
we obtain
\begin{align*}
&\int_{\Omega_1\cup\Omega_2 }
\int_{\Omega_1\cup\Omega_2 }
\frac{g(x)g(y)|u_n(x)|^{p}|u_n(y)|^{p}}
{|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y\\
&=
\int_{\Omega_1\cup\Omega_2 }
\int_{\Omega_1\cup\Omega_2 }
\frac{g(x)g(y)|v_n(x)|^{p}|v_n(y)|^{p}}
{|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y\\
&\quad +
\int_{\Omega_1\cup\Omega_2 }
\int_{\Omega_1\cup\Omega_2 }
\frac{g(x)g(y)|u(y)|^{p}|u(x)|^{p}}
{|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y
+o(1)\quad \text{as } n\to\infty.
\end{align*}
\end{proof}

\begin{lemma}\label{lemma16}
Assume that the hypotheses of Theorem
\ref{theorem3}
hold.
If
$\{u_n\}$
is a bounded sequence in
$D^{1,2}(\mathbb{R}^{N})$,
up to a subsequence,
$u_n\rightharpoonup u$
in
$D^{1,2}(\mathbb{R}^{N})$
and
$u_n\to u$ a.e. in $\mathbb{R}^{N}$
as $n\to\infty$,
then
\[
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{g(x)g(y)|u_n(x)|^{p}|u_n(y)|^{p}}
{|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y
\to
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{g(x)g(y)|u(x)|^{p}|u(y)|^{p}}
{|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y.
\]
In addition, for any
$\varphi\in D^{1,2}(\mathbb{R}^{N})$,
\begin{align*}
&\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{g(x)g(y)|u_n(y)|^{p}|u_n(x)|^{p-2}u_n(x)\varphi(x)}
{|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y\\
&\to \int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{g(x)g(y)|u(y)|^{p}|u(x)|^{p-2}u(x)\varphi(x)}
{|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y.
\end{align*}
as
$n\to\infty$.
\end{lemma}

\begin{proof}
\textbf{Step 1.}
By Lemmas \ref{lemma14}
and  \ref{lemma15},
we have
\begin{equation}\label{57}
\begin{aligned}
&\int_{\Omega_1\cup\Omega_2 }
\int_{\Omega_1\cup\Omega_2 }
\frac{g(x)g(y)|u_n(x)|^{p}|u_n(y)|^{p}}
{|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y\\
&=
\int_{\Omega_1\cup\Omega_2 }
\int_{\Omega_1\cup\Omega_2 }
\frac{g(x)g(y)|u(y)|^{p}|u(x)|^{p}}
{|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y
+o(1), \quad\text{as }n\to\infty.
\end{aligned}
\end{equation}
Since
$g\equiv0$
in
$\mathbb{R}^{N}\setminus(\Omega_1\cup\Omega_2 )$,
by
\eqref{57},
we have
\begin{align*}
&\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{g(x)g(y)|u_n(x)|^{p}|u_n(y)|^{p}}
{|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y \\
&\to
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{g(x)g(y)|u(x)|^{p}|u(y)|^{p}}
{|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y\quad \text{as }n\to\infty.
\end{align*}
\smallskip

\noindent \textbf{Step 2.}
By the Hardy-Littlewood-Sobolev inequality,
\begin{equation}\label{58}
\int_{\Omega_i}
\frac{g(y)|u_n|^{p}}
{|x-y|^{\mu}}
\,\mathrm{d}y
\rightharpoonup
\int_{\Omega_i}
\frac{g(y)|u|^{p}}
{|x-y|^{\mu}}
\,\mathrm{d}y \quad
\text{in }
L^{\frac{2N}{\mu}}(\Omega_i),
(i=1,2),
\text{ as }n\to\infty.
\end{equation}
From \eqref{58},
we obtain
\begin{equation}\label{59}
\int_{\Omega_1\cup\Omega_2 }
\frac{g(y)|u_n|^{p}}
{|x-y|^{\mu}}
\,\mathrm{d}y
\rightharpoonup
\int_{\Omega_1\cup\Omega_2 }
\frac{g(y)|u|^{p}}
{|x-y|^{\mu}}
\,\mathrm{d}y,
\quad \text{in }
L^{\frac{2N}{\mu}}(\Omega_1\cup\Omega_2 ) \quad\text{as }n\to\infty.
\end{equation}
Since
$g\equiv0$
in
$\mathbb{R}^{N}\setminus(\Omega_1\cup\Omega_2 )$,
by
\eqref{59},
we have
\begin{equation}\label{60}
\int_{\mathbb{R}^{N}}
\frac{g(y)|u_n|^{p}}
{|x-y|^{\mu}}
\,\mathrm{d}y
\rightharpoonup
\int_{\mathbb{R}^{N}}
\frac{g(y)|u|^{p}}
{|x-y|^{\mu}}
\,\mathrm{d}y\quad \text{in }
L^{\frac{2N}{\mu}}(\mathbb{R}^{N})
\text{ as } n\to\infty.
\end{equation}
Similar to
\eqref{60},
we obtain
\begin{equation}\label{61}
g(x)|u_n|^{p-2}u_n
\rightharpoonup
g(x)|u|^{p-2}u\quad
\text{in }
L^{\frac{2Np}{(2N-\mu)(p-1)}}(\mathbb{R}^{N})
\text{ as }n\to\infty.
\end{equation}
Combining \eqref{60} and \eqref{61},
$\mathrm{as~n\to\infty}$,
we find that
\[
\Big(
\int_{\mathbb{R}^{N}}
\frac{g(y)|u_n|^{p}}
{|x-y|^{\mu}}
\,\mathrm{d}y
\Big)
g(x)|u_n|^{p-2}u_n
\rightharpoonup
\Big(
\int_{\mathbb{R}^{N}}
\frac{g(y)|u|^{p}}
{|x-y|^{\mu}}
\,\mathrm{d}y
\Big)
g(x)|u|^{p-2}u
\]
in $L^{\frac{2Np}{2Np-2N+\mu}}(\mathbb{R}^{N})$.
Thus,
for any
$\varphi\in D^{1,2}(\mathbb{R}^{N})$,
\begin{align*}
&\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{g(x)g(y)|u_n(y)|^{p}|u_n(x)|^{p-2}u_n(x)\varphi(x)}
{|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y\\
&\to
\int_{\mathbb{R}^{N}}
\int_{\mathbb{R}^{N}}
\frac{g(x)g(y)|u(y)|^{p}|u(x)|^{p-2}u(x)\varphi(x)}
{|x-y|^{\mu}}
\,\mathrm{d}x
\,\mathrm{d}y.
\end{align*}
\end{proof}

\begin{lemma}\label{lemma17}
Assume that the hypotheses of Theorem \ref{theorem3} hold.
If $\{u_n\}$
is a
$(PS)_{c}$
sequence of
$I_2 $
with
$0<c<\frac{N+2-\mu}{4N-2\mu}S_{H,L}^{\frac{2N-\mu}{N+2-\mu}}$,
then
$\{u_n\}$
has a convergent subsequence.
\end{lemma}

The proof of the above lemma is similar to that of Lemma \ref{lemma10}.
We omit it.

\begin{proof}[Proof of Theorem \ref{theorem3}]
Applying Lemma \ref{lemma12},
we obtain that $I_2 $
possesses a mountain pass geometry.
Then from the Maintain Pass Theorem,
there is a sequence
$\{u_n\}\subset D^{1,2}(\mathbb{R}^{N})$
satisfying
$I_2 (u_n)\to c$
and
$I'_2 (u_n)\to0$,
where
\[
0<\vartheta\leqslant c<\frac{N+2-\mu}{4N-2\mu}S_{H,L}^{\frac{2N-\mu}{N+2-\mu}}.
\]
Moreover, according to Lemma \ref{lemma13} and Lemma \ref{lemma17},
$\{u_n\}$
satisfying
$(PS)_{c}$
condition.
Hence,
we have a nontrivial solution
$\tilde{u}_{0}$
to problem \eqref{eP2}.
\end{proof}

\subsection*{Acknowledgments}
This research was supported by the National Natural Science Foundation
of China 11671403.

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\end{document}