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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 114, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/114\hfil Ground state solutions]
{Ground state solutions for a quasilinear Schr\"odinger equation
with singular coefficients}

\author[J. Wang, Q. Gao, L. Wang \hfil EJDE-2017/114\hfilneg]
{Jixiu Wang, Qi Gao, Li Wang}

\address{Jixiu Wang \newline
School of Mathematics and Computer Science,
Hubei University of Arts and Science, Xiangyang 441053, China}
\email{wangjixiu127@163.com}

\address{Qi Gao (corresponding author) \newline
Department of Mathematics,
School of Science,
Wuhan University of Technology,
Wuhan 430070, China}
\email{gaoq@whut.edu.cn}

\address{Li Wang \newline
College of Science,
East China Jiaotong University,
 Nanchang 330013, China}
\email{wangli.423@163.com}

\dedicatory{Communicated by Paul H. Rabinowitz}

\thanks{Submitted March 7, 2017. Published April 27, 2017.}
\subjclass[2010]{35J62, 35J60, 35J20}
\keywords{Quasilinear Schr\"odinger equations; critical exponent;
\hfill\break\indent  ground state solutions; calculus of variations}

\begin{abstract}
 In this article, we study the quasilinear Schr\"odinger equation
 with the critical exponent and singular coefficients,
 \[
 -\Delta u +V(x)u-\Delta(|u|^2)u=\lambda\frac{|u|^{q-2}u}{|x|^{\mu}}
 +\frac{|u|^{22^*(\nu)-2}u}{|x|^\nu}\quad\text{in } \mathbb{R}^N,
 \]
 where $N\geq 3$, $2<q<22^*(\mu)$, $2^*(s)=\frac{2(N-s)}{N-2}$, and
 $\lambda, \mu, \nu$ are parameters with $\lambda>0$, $\mu, \nu \in [0,2)$.
 By applying the Mountain Pass Theorem and the Concentration
 Compactness Principle, we establish the existence of the ground state solutions
 to the above problem.
\end{abstract}

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

\section{Introduction} \label{intro}

In this article we study the existence of ground state solutions of the
 quasilinear Schr\"odinger equations with singular coefficients,
\begin{equation}\label{eqS1.1}
-\Delta u +V(x)u-\Delta(|u|^2)u=\lambda\frac{|u|^{q-2}u}{|x|^{\mu}}
+\frac{|u|^{22^*(\nu)-2}u}{|x|^\nu}\quad\text{in } \mathbb{R}^N,
\end{equation}
where $N\geq 3$, $2<q<22^*(\mu)$, $2^*(s)=\frac{2(N-s)}{N-2}$, and
$\lambda, \mu, \nu$ are parameters with $\lambda>0$, $\mu, \nu \in [0,2)$.
The corresponding energy functional for \eqref{eqS1.1} is
\begin{equation}\label{energy}
\begin{aligned}
I(u)&=\frac{1}{2}\int_{\mathbb{R}^N}(1+2u^2)|\nabla
u|^2 dx+\frac{1}{2}\int_{\mathbb{R}^N}V| u |^2 dx \\
&\quad -\frac{\lambda}{q}\int_{ \mathbb{R}^N}\frac{|u|^{q}}{|x|^{\mu}} dx
 -\frac{1}{22^*(\nu)}\int_{\mathbb{R}^N}\frac{|u|^{22^*(\nu)}}{|x|^\nu} dx.
\end{aligned}
\end{equation}

Throughout this article, we assume that potential satisfies:
\begin{itemize} 
\item[(A0)] $V\in \mathcal{C}(\mathbb{R}^N,\mathbb{R})$ with $\inf V(x)=V_0>0$, and
 for each $M>0$, $\operatorname{meas}\{x\in\mathbb{R}^N:V(x)\leq M\}<+\infty$,
 where $V_0$ is a constant and meas denotes the Lebesgue measure in $\mathbb{R}^N$.
\end{itemize}
By the ground state solution of \eqref{eqS1.1}, we mean that $u\neq0$ and its
 energy is minimal among the energy of all nontrivial solutions to \eqref{eqS1.1}.

The question addressed in this paper is motivated by analogous results
for the ground state solutions of the Schr\"odinger equation
\begin{equation}\label{eqS1.2}
i\partial_t \phi=-\Delta \phi+W(x)\phi-f(|\phi|^2)\phi-k \Delta
h(|\phi|^2)h'(|\phi|^2)\phi,
\end{equation}
where $\phi:\mathbb{R}\times\mathbb{R}^N\to \mathbb{C}$, $W:\mathbb{R}^N\to\mathbb{R}$
is a given potential, $k$ is a
real constant and $f, h:\mathbb{R}^+\to \mathbb{R}$ are suitable functions.
In particular, we consider the model with $h(s)=s$,
\[
f(s)=\sqrt{\lambda\frac{|s|^{q-2}}{|x|^{\mu}}+\frac{|s|^{22^*(\nu)-2}}{|x|^\nu}},
\]
$W(x)= V(x)+\beta$. A stationary equation of the desired form is obtained by
considering standing wave solutions, $\phi(t,x)=\exp{(-i\beta t)}u(x)$.
Once we substitute the formula of standing wave solutions into \eqref{eqS1.2}
with the special choices of $h(s)$, $f(s)$ and $W(x)$ as pointed above,
 we can immediately obtain  \eqref{eqS1.1}. According to this substitution,
$u$ is a solution of \eqref{eqS1.1} if and only if $\phi$ is standing
wave solution to \eqref{eqS1.2}.

Schr\"odinger equations of this type have appeared in many physical models.
It can be used to describe different physical phenomena due to the variety
of the nonlinear term $h$. When $h(s)=s$, \eqref{eqS1.2} was used to discuss
the time evolution of the condensate wave function of super-fluid film equation
in plasma physics \cite{K,LSS}. When $h(s)= (1 + s)^{1/2}$, \eqref{eqS1.2}
models the self-channeling of a high-power ultra short laser in matter, see
\cite{BMM,CS,DHJ,Ri}.
Equation \eqref{eqS1.2} also appears in the theory of Heisenberg ferromagnets
 and magnons \cite{KIK} and in condensed
matter theory \cite{MF}. For further physical backgrounds and applications,
we refer readers to \cite{BL2,BN,LPT,PSW} and references therein.

Poppenberg-Schmitt-Wang \cite{PSW} considered the eigenvalue problem
\begin{equation}\label{eigenprb}
-\Delta u+V(x)u-(\Delta|u|^2 )u=\lambda|u|^{q-2}u,
\end{equation}
with bounded potential $V(x)$ and $q>2$, $\lambda>0$.
They showed the existence of positive ground state solutions for one
dimensional case via the constrained variational method. For the same
equation mentioned in \eqref{eigenprb}, Liu-Wang-Wang \cite{LWW} also
proved the existence of positive solution with unbounded/bounded/periodic
potential $V(x)$ when $4<q<22^*$ and $\lambda>0$ by a change of variables
and an Orlicz space, where $2^* =\frac{2N}{N-2}$ is the critical Sobolev exponent.
 In a recent work of do \'{O}-Miyagaki-Soares \cite{DMS},
the critical exponent problem is studied for the following quasilinear equation
\begin{equation}\label{critical}
-\Delta u+V(x)u-(\Delta|u|^2 )u=|u|^{22^* -2} u+|u|^{q-2}u,
\end{equation}
where $4<q<22^*$, $2^*$ is again the critical Sobolev exponent, $N\ge3$.
Applying a change of variables and the Moutain Pass Theorem, they showed
that there is a positive solution for \eqref{critical} with bounded/periodic
potential $V(x)$. Liu-Liu-Wang \cite{LLW} extended the method in \cite{DMS}
to investigate the more general Schr\"odinger equations with critical
growth
\begin{equation}\label{gseq}
\begin{aligned}
 &-\sum^N_{i,j=1} D_j (a_{ij}(u) D_i u )+\frac{1}{2}\sum^N_{i,j=1}
 D_s a_{ij}(u) D_i u D_j u +V(x)u \\
&=|u|^{22^* -2}u+|u|^{q-2}u,
\end{aligned}
\end{equation}
where $4<q<22^*$, $N\ge3$. It is easy to see that \eqref{gseq} can be
transferred into \eqref{critical} with $a_{ij}(u)=(1+2u^2 )\delta_{i,j}$.
They also obtained the existence for positive solution with bounded potential
well by the Nehari method.
Later on, via classical variation techniques, Wu-Zhou \cite{WZ} improved
the results of \cite{DMS} with unbounded potential $V(x)$, and they relax
the restriction $q>4$ to $q>2$. Moreover, Bae-Choi-Pahk \cite{BCP} studied the
existence of nodal radial solutions to the elliptic equations:
\begin{equation}\label{bae}
-\Delta u=\lambda\frac{|u|^{q-2}u}{|x|^\mu}+\frac{|u|^{2^* (\nu)-2}u}{|x|^\nu}\quad
\text{in } B_1,
\end{equation}
with $\mu, \nu> -2$, $\lambda>0$.
The singular terms addressed in our model are similar to the one in \eqref{bae}.
 Therefore, motivated by \cite{BCP} and \cite{WZ},
we notice that the existence of ground state solutions for \eqref{eqS1.1}
depends not only on the range of $q$, but also the parameter $\lambda$,
for the case when $\mu\neq0$ and $\nu\neq0$. The main theorem of our paper
is as follows:

\begin{theorem}\label{intro1}
Let $q$ and $\lambda$ be positive parameters, for every fixed $\mu$,
$\nu\in [0,2)$, we have the following statements:
\begin{itemize}
\item[(ii)] if $\frac{2(N+2-2\mu)}{N-2}< q<22^*(\mu)$, there exists a ground state
solution of \eqref{eqS1.1} for any $\lambda>0$;

 \item[(ii)] if $2<q
\le \frac{2(N+2-2\mu)}{N-2}$, there exists a constant $\lambda^*>0$, such that for $\lambda>\lambda^*$, \eqref {eqS1.1} has a ground state solution.
\end{itemize}
\end{theorem}

We now briefly mention the main difficulties of this problem.
As observed in \cite{LWW1},
for each fixed $\nu$, the number $22^*(\nu)$ behaves like a critical exponent
for the embedding $X\hookrightarrow L^{22^*(\nu)}(\mathbb{R}^N,|x|^{-\nu})$,
where $X=\{u\in H^1(\mathbb{R}^N):\ u^2\in H^1(\mathbb{R}^N),\sqrt{|V|}u
\in L^2(\mathbb{R}^N)\}$ is the domain of the energy functional
corresponding to \eqref{eqS1.1}. The main tool of this problem is a variant
version of the Mountain Pass Theorem \cite{SV}, which introduced a so-called
Cerami sequence. We denote it as $(C)_c$ sequence for convenience.
The action of the $(C)_c$ sequence in the modified Mountain Pass Theorem
is similar to the Palais-Smale sequence in the classical Mountain Pass
Theorem \cite{AR}. Due to lack of compactness of the embedding
 $X\hookrightarrow L^{22^*(\nu)}(\mathbb{R}^N,|x|^{-\nu})$,
it complicates the process of verifying the existence and nonvanishing
of the weak limit of a $(C)_c$ sequence. On the other hand, $X$ is not even
a vector space, which causes that the usual variation techniques cannot be
applied directly. Therefore, the choice of a suitable function space is also
important for our discussion.

The plan of this paper is as follows:
Section 2 states some preliminary results and establishes the Mountain Pass
geometry structure, Section 3 covers the compactness of $(C)_c$ sequence.
Section 4 is devoted to the proof of Theorem~\ref{intro1}.

In what follows, $C$ denotes the universal positive constant unless specified,
$L^{q}(\mathbb{R}^N)$ denotes the usual Lebesgue space with norm
$\|u\|_{q}=(\int_{\mathbb{R}^N}|u|^q dx)^{1/q}$, $1\le q<\infty$.

\section{Preliminaries}\label{sec:2}
 Let
\[
H^1(\mathbb{R}^N)=\{u\in L^2(\mathbb{R}^N):\nabla u\in L^2(\mathbb{R}^N)\}
\]
be endowed with the inner product
$$
\langle u,v\rangle_{H}=\int_{\mathbb{R}^N}\big(\nabla
u\nabla v+uv\big)dx,
$$
and the norm $\|u\|^2_{H}=\langle u,u\rangle_{H}$. We define
$$
E=\{u\in H^1(\mathbb{R}^N):\int_{\mathbb{R}^N}V(x)u^2 dx<\infty\}
$$
with the inner product
$$
\langle u,v\rangle=\int_{\mathbb{R}^N}\big[\nabla
u\nabla v+V(x)uv\big]dx,
$$
and the associated norm $\|u\|^2=\langle u,u\rangle$.
It is easy to see that both $H^1(\mathbb{R}^N)$ and $E$
are Hilbert spaces. Fortunately, thanks to \cite{BW} and \cite{ZM},
we can obtain the following lemma.

\begin{lemma}\label{lm8.1}
Let $0\leq \sigma<2$. The embedding
$E\hookrightarrow L^s(\mathbb{R}^N,|x|^{-\sigma})$ is continuous for
$2\leq s\leq 2^*(\sigma)$
and compact for $2\leq s<2^*(\sigma)$ when $V(x)$ satisfies the condition (A0).
\end{lemma}

\begin{proof}
By \cite{BW} and \cite{ZM}, the embedding $E\hookrightarrow L^p(\mathbb{R}^N)$
is continuous for $2\leq p\leq 2^*$ and
compact for $2\leq p< 2^*$ under the condition (A0).


 Let $u_n\to 0$ in $E$ as $n\to\infty$.
By the H\"older's inequality, we have
\begin{equation}\label{eqS5.10}
\begin{aligned}
\int_{\mathbb{R}^N}\frac{|u_n|^{s}}{|x|^\sigma}dx
&=\int_{\mathbb{R}^N}\frac{|u_n|^{\sigma}}{|x|^\sigma}\cdot|u_n|^{s-\sigma}dx \\
&\leq \Big(\int_{\mathbb{R}^N}\frac{u_n^{2}}{|x|^2}dx\Big)^{\sigma/2}
\Big(\int_{\mathbb{R}^N}|u_n|^{\frac{2(s-\sigma)}{2-\sigma}}dx
 \Big)^{\frac{2-\sigma}{2}}.
\end{aligned}
\end{equation}
Since $2\leq s\leq 2^*(\sigma)$ and $0\leq \sigma<2$, it follows that
$2\leq\frac{2(s-\sigma)}{2-\sigma}\leq 2^*$. Hence, applying the Hardy's
 and Sobolev inequalities to \eqref{eqS5.10}, we can obtain
$\int_{\mathbb{R}^N}\frac{|u_n|^{s}}{|x|^\sigma}dx
\leq C\|u_n\|^s$, which gives that $u_n\to 0$ in
$L^s(\mathbb{R}^N,|x|^{-\sigma})$ for $2\leq s\leq 2^*(\sigma)$ as $n\to\infty$.
Therefore, the first part of this lemma is proved.

Secondly, let $\{u_n\}$ be a bounded sequence in $E$. It is clear that,
up to a subsequence, $u_n\rightharpoonup u$ in $E$ as $n\to\infty$.
Employing the H\"older's and Hardy's inequalities, we obtain
\begin{equation}\label{eqS5.11}
\begin{aligned}
\int_{\mathbb{R}^N}\frac{|u_n-u|^{s}}{|x|^\sigma}dx
&=\int_{\mathbb{R}^N}\frac{|u_n-u|^{\sigma}}{|x|^\sigma}\cdot|u_n-u|^{s-\sigma}dx \\
&\leq \Big(\int_{\mathbb{R}^N}\frac{|u_n-u|^{2}}{|x|^2}dx\Big)^{\sigma/2}
\Big(\int_{\mathbb{R}^N}|u_n-u|^{\frac{2(s-\sigma)}{2-\sigma}}dx
 \Big)^{\frac{2-\sigma}{2}}\\
&\leq \|u_n-u\|^\sigma
\Big(\int_{\mathbb{R}^N}|u_n-u|^{\frac{2(s-\sigma)}{2-\sigma}}dx
 \Big)^{\frac{2-\sigma}{2}}.
\end{aligned}
\end{equation}

Since $2\leq s< 2^*(\sigma)$ and $0\leq \sigma<2$, we have
$2\leq\frac{2(s-\sigma)}{2-\sigma}< 2^*$. It then follows that $u_n\to u$ in
$L^{\frac{2(s-\sigma)}{2-\sigma}}(\mathbb{R}^N)$. Therefore, we can obtain
 $u_n\to u$ in $L^s(\mathbb{R}^N,|x|^{-\sigma})$ for $2\leq s< 2^*(\sigma)$
from \eqref{eqS5.11}. Thus $E\hookrightarrow L^s(\mathbb{R}^N,|x|^{-\sigma})$
is compact for $2\leq s< 2^*(\sigma)$. We complete our proof.
\end{proof}

The purpose of this section is to establish the variational structure of
\eqref{eqS1.1}. And the main difficulty arises from the function space
where the energy functional \eqref{energy} is not well defined.
To overcome this difficulty, and motivated by \cite{LWW} and \cite{CJ}, we
 define a $\mathcal{C}^{\infty}$ function $f(t)$ as below:
\[
f(-t)=-f(t) \text{ on } (-\infty,0],\quad
f'(t)=\frac{1}{(1+2f^2(t))^{1/2}} \text{ on } [0,+\infty).
\]

We also need some properties on $f$.

\begin{proposition}\label{lm2.1}
The function $f(t)$ has the following properties:
\begin{itemize}
\item[(A1)] $f$ is a uniquely defined, invertible $\mathcal{C}^{\infty}$-function;

\item[(A2)] $0<f'(t)\leq 1$ for all $t\in\mathbb{R}$;

\item[(A3)] $|f(t)|\leq |t|$ for all $t\in\mathbb{R}$;

\item[(A4)] $|f(t)|\leq 2^{\frac{1}{4}}|t|^{1/2}$ for all $t\in\mathbb{R}$;

\item[(A5)] There exists a positive constant $C$ such that
\[
|f(t)|\geq
 \begin{cases}
C|t|,& |t|\leq 1,\\
C|t|^{1/2},& |t|\geq 1;
\end{cases}
\]

\item[(A6)] $\frac{f(t)}{2}\le tf'(t)\le f(t)$ for $t\ge0$;

\item[(A7)] $|f(t)f'(t)|\leq \frac{1}{\sqrt{2}}$ for all $t\in\mathbb{R}$.
\end{itemize}
\end{proposition}
We observe that a direct calculation implies that $\frac{f(t)}{t}$ is
decreasing for $t>0$ from (A6).
The proof of this proposition may be found in
\cite{CJ}, \cite{LWW,DMS,DS}.

After using the same change of variables $v=f^{-1}(u)$ as in \cite{LWW}
and \cite{CJ}, and the definition of $f$ mentioned above,
$I(u)$ can be transferred into a new functional $J(v)$:
\begin{equation}\label{newenergy}
\begin{aligned}
J(v)&=\frac{1}{2}\int_{\mathbb{R}^N}\Big[|\nabla
v|^2+V(x)f^2 (v)\Big]dx-\frac{\lambda}{q}\int_{
\mathbb{R}^N}\frac{|f(v)|^q}{|x|^\mu}dx \\
&\quad -\frac{1}{22^*(\nu)}\int_{\mathbb{R}^N}\frac{|f(v)|^{22^*(\nu)}}{|x|^\nu} dx,
\end{aligned}
\end{equation}
which is well defined on $E$. Moreover, a standard argument shows that
$J\in \mathcal{C}^1(E,\mathbb{R})$ and
 \begin{equation}\label{1stvariation}
\begin{aligned}
\langle J'(v),\varphi\rangle
&=\int_{\mathbb{R}^N}\Big[\nabla v\nabla \varphi+V(x)f(v)f'(v)\varphi\\
&\quad-\lambda\frac{|f(v)|^{q-2}}{|x|^\mu}f(v)f'(v)\varphi
 -\frac{|f(v)|^{22^*(\nu)-2}}{|x|^\nu}f(v)f'(v)\varphi\Big]dx
\end{aligned}
\end{equation}
for all $v$, $\varphi\in E$. Therefore, the nontrivial critical points
of the functional $J$ are also the nontrivial weak solutions of the
following equation
\begin{equation}\label{eulerto1st}
-\Delta v=f(v)f'(v)\Big[\lambda\frac{|f(v)|^{q-2}}{|x|^\mu}
+\frac{|f(v)|^{22^*(\nu)-2}}{|x|^\nu}-V(x)\Big] \quad \text{in } \mathbb{R}^N.
\end{equation}
According to this change of variables (see \cite{CJ,WZ}), we notice that
if $v$ is a solution of \eqref{eulerto1st}, $u=f(v)$
is also a solution of \eqref{eqS1.1}.

Now show that the functional $J$ exhibits the Mountain Pass geometry structure.
Let
$$
B(\rho)=\{v\in E:\int_{\mathbb{R}^N}\big[|\nabla
v|^2+V(x)f^2(v)\big]dx< \rho^2\}.
$$

\begin{lemma}\label{lm3.1}
For any fixed $\mu, \nu \in [0,2)$, the functional $J$ satisfies
\begin{itemize}
\item[(i)] there exist positive constants $\alpha$ and $\rho_0$,
 such that $J(v)\geq \alpha$ for
all $v\in \partial B(\rho_0)$,

\item[(ii)] there exists $w\in E$ such that $J(w)<0$.
\end{itemize}
\end{lemma}

\begin{proof}
(i) From (A7), we have
\begin{equation}\label{eqS0.3}
\begin{aligned}
\int_{\mathbb{R}^N}|\nabla f^2(v)|^2 dx
&=\int_{\mathbb{R}^N}|2f(v)f'(v)\nabla v|^2 dx \\
& \leq 2\int_{\mathbb{R}^N}|\nabla v|^2 dx \\
&\leq 2\int_{\mathbb{R}^N}\Big[|\nabla v|^2+V(x)f^2(v)\Big]dx.
\end{aligned}
\end{equation}
For fixed $\mu\in[0,2)$, for any $\varepsilon>0$ and $2<q<22^*(\mu)$,
there exists a constant $C(\varepsilon)>0$ such that
$\frac{|t|^q}{|x|^{\mu}}\leq \varepsilon\frac{|t|^2}{|x|^{\mu}}+
C(\varepsilon)\frac{|t|^{22^*(\mu)}}{|x|^{\mu}}$. Thus,
for any $v\in \partial B(\rho)$, by using Sobolev-Hardy inequality and
\eqref{eqS0.3},
we have
\begin{equation} \label{muineq}
\begin{aligned}
\int_{\mathbb{R}^N}\frac{|f(v)|^{22^{*}(\mu)}}{|x|^\mu} dx
&\le C\Big(\int_{\mathbb{R}^N}\big|\nabla (f^2 (v))\big|^2 dx
 \Big)^{2^*(\mu)/2} \\
&\le C\Big[\int_{\mathbb{R}^N}\left(|\nabla v|^2 +V(x)f^2 (v)\right)dx
\Big]^{2^*(\mu)/2} \\
&\le C\rho^{2^{*}(\mu)}\, .
\end{aligned}
\end{equation}
Similarly,
\begin{equation}\label{nuineq}
\int_{\mathbb{R}^N}\frac{|f(v)|^{22^{*}(\nu)}}{|x|^\nu} dx\le C\rho^{2^{*}(\nu)}.
\end{equation}
By \eqref{muineq}, it follows from (A3) and Sobolev-Hardy inequality again that
\begin{equation} \label{eqS3.2}
\begin{aligned}
\int_{\mathbb{R}^N}\frac{|f(v)|^{q}}{|x|^\mu}dx
&\leq\varepsilon\int_{
\mathbb{R}^N}\frac{|f(v)|^2}{|x|^\mu}dx+C(\varepsilon)\int_{
\mathbb{R}^N}\frac{|f(v)|^{22^*(\mu)}}{|x|^{\mu}}dx \\
&\leq C\varepsilon\int_{\mathbb{R}^N}|\nabla
v|^2 dx+C\cdot C(\varepsilon)\rho^{2^{*}(\mu)} \\
&\le C\varepsilon \rho^2 +C\cdot C(\varepsilon)\rho^{2^{*}(\mu)}.
\end{aligned}
\end{equation}
Thus, from \eqref{muineq} and \eqref{nuineq}, we obtain
\begin{align*}
J(v)
&=\frac{1}{2}\int_{\mathbb{R}^N}\Big[|\nabla v|^2+V(x)f^2(v)\Big]dx
-\frac{\lambda}{q}\int_{\mathbb{R}^N}\frac{|f(v)|^q}{|x|^\mu}dx \\
&\quad -\frac{1}{22^*(\nu)}\int_{\mathbb{R}^N}\frac{|f(v)|^{22^*(\nu)}}{|x|^\nu}dx \\
&\geq [\frac12 -\frac{\lambda}{q} C\varepsilon]\rho^2
 -\frac{\lambda}{q}C\cdot C(\varepsilon)\rho^{2^{*}(\mu)}
 -\frac{1}{22^* (\nu)}C\rho^{2^{*}(\nu)} \\
&\ge \frac14 \rho^2 -C\cdot C(\varepsilon)\rho^{2^{*}(\mu)}-C\rho^{2^{*}(\nu)},
\end{align*}
for $\varepsilon>0$ sufficiently small. Choose $\rho_0>0$ with
$\frac{1}{4}\rho^2_0 -C\cdot C(\varepsilon)\rho^{2^{*}(\mu)}_0
 -C\cdot\rho^{2^{*}(\nu)}_0=:\alpha>0$. Then we have
$J(v)\geq \alpha$ for all $v\in \partial B(\rho_0)$.

(ii) For fixed $\nu\in[0,2)$, given $\psi\in E\cap L^{22^{*}(\nu)}(\mathbb{R}^N )$
with $0<\psi\le1$, we can show that
$$
J(t\psi)\to -\infty\quad \text{as } t\to \infty.
$$
Indeed, since $0<t\psi(x)\leq t$ for $t>0$, and (A6) in Proposition~\ref{lm2.1},
it follows that
\begin{equation}\label{eqS3.3}
\frac{f(t\psi(x))}{t\psi(x)}\geq \frac{f(t)}{t} \Rightarrow
 f(t\psi(x))\geq f(t)\psi(x).
\end{equation}
Thus, for $t\ge1$, by \eqref{eqS3.3}, (A3) and (A5), we have
\begin{align*}
J(t\psi)
&\leq\frac{1}{2}\int_{\mathbb{R}^N}\Big[|\nabla
(t\psi)|^2+V(x)f^2(t\psi)\Big]dx-\frac{1}{22^*(\nu)}\int_{
\mathbb{R}^N}\frac{|f(t\psi)|^{22^*(\nu)}}{|x|^\nu}dx \\
&\leq \frac{t^2}2\int_{\mathbb{R}^N}\Big[|\nabla
\psi|^2+V(x)\psi^2\Big]dx-\frac{1}{22^*(\nu)}\int_{
\mathbb{R}^N}\frac{|f(t)\psi|^{22^*(\nu)}}{|x|^\nu}dx \\
&=\frac{t^2}{2}\int_{\mathbb{R}^N}\Big[|\nabla
\psi|^2+V(x)\psi^2\Big]dx-Ct^{2^*(\nu)}\int_{
\mathbb{R}^N}\frac{|\psi|^{22^*(\nu)}}{|x|^\nu}dx \\
&\to -\infty,\quad\text{as}\ t\to +\infty,
\end{align*}
since $2^*(\nu)>2$ for $\nu\in [0,2)$.
This implies that there exists a $t$ large and positive such that
$w=t\psi$, $J(w)<0$.
\end{proof}

It is well known that the minimization problem
$$
S=\inf\Big\{\int_{\mathbb{R}^N}|\nabla v|^2 dx :v\in
\mathcal{D}^{1,2}(\mathbb{R}^N),\int_{\mathbb{R}^N}
\frac{|v|^{2^*(\nu)}}{|x|^{\nu}}dx=1\Big\}$$
has a solution given by
$$
w_\epsilon(x)=\frac{[(N-\nu)(N-2)\epsilon]^{\frac{N-2}{4(2-\nu)}}}
{[\epsilon+|x|^{2-\nu}]^{\frac{N-2}{2(2-\nu)}}}.
$$
Let $0<R<1$, and $\varphi\in \mathcal{C}_0^{\infty}(\mathbb{R}^N,[0,1])$ be a smooth
cut-off function, such that $\varphi(x)=1$ for $|x|\le R$, $0<\varphi(x)<1$
for $R<|x|<2R$, and $\varphi(x)=0$ for $|x| \geq 2R$.
 For any $\epsilon>0$, it is known that
$-\Delta(w_\epsilon^2)=\frac{w_\epsilon^{\frac{2(N+2-2\nu)}{N-2}}}{|x|^{\nu}}$ and
$S$ can be attained by $w_\epsilon^2$. Set $u_\epsilon=\varphi
w_\epsilon$. By a similar computation to that in \cite{BCP,BrN}, we have:
\begin{gather}\label{eqS10.1}
\begin{gathered}
\int_{\mathbb{R}^N}|\nabla(u_\epsilon^2)|^{2}dx
=S^{\frac{N-\nu}{2-\nu}}+O\big(\epsilon^{\frac{N-2}{2-\nu}}\big),\\
\int_{\mathbb{R}^N} \frac{|u_\epsilon|^{22^*(\nu)}}{|x|^\nu}dx
=S^{\frac{N-\nu}{2-\nu}}+O\big(\epsilon^{\frac{N-\nu}{2-\nu}}\big),
\end{gathered} \\
\label{eqS10.3}
\int_{\mathbb{R}^N}|\nabla
u_\epsilon|^{2}dx\leq O\big(\epsilon^{\frac{N-2}{2(2-\nu)}}|\ln\epsilon|\big),\quad
\int_{\mathbb{R}^N} u_\epsilon^2 dx=O\big(\epsilon^{\frac{N-2}{2(2-\nu)}}\big), \\
\label{eqS10.5}
\int_{\mathbb{R}^N} \frac{|u_\epsilon|^q}{|x|^\mu} dx
=O\big(\epsilon^{\frac{N-\mu}{2-\nu}-\frac{N-2}{4(2-\nu)}q}\big) \quad
 \text{for } \frac{2(N-\mu)}{N-2}<q<\frac{4(N-\mu)}{N-2}.
\end{gather}

As usual, we define the Mountain Pass level $c$ of $J$ to be
\begin{equation}\label{eqS3.5}
c=\inf_{\gamma\in \Gamma}\sup_{t\in [0,1]}J(\gamma(t)),
\end{equation}
where $\Gamma=\{\gamma\in C([0,1],E):\gamma(0)=0,\gamma(1)\neq0, J(\gamma(1))<0\}$.
And it is easy to see that $c>0$ by Lemma~\ref{lm3.1}.

Since
$$
J(f^{-1}(0))=I(0)=0,\ J(f^{-1}(t u_\epsilon))=I(tu_\epsilon)\to
-\infty\quad\text{as } t\to \infty,
$$
it follows that there exists $t_0\neq 0$ such that $J(f^{-1}(t_0 u_\epsilon))<0$.
Let
$$
\gamma_1(t)=f^{-1}(t t_0u_\epsilon),
$$
we have
$$
\gamma_1(0)=0,\gamma_1(1)=f^{-1}(t_0u_\epsilon)\neq0, J(\gamma_1 (1))
=J(f^{-1}(t_0 u_\epsilon))<0.
$$
Therefore, from the definition of the Mountain Pass level, it follows that
\begin{align*}
c&=\inf_{\gamma\in \Gamma}\sup_{t\in [0,1]}J(\gamma(t))
 \leq \sup_{t\in [0,1]}J(\gamma_1(t)) \\
&=\sup_{t\in [0,1]}J(f^{-1}(t t_0u_\epsilon))
 =\sup_{t\geq 0}J(f^{-1}(t u_\epsilon)) \\
&=\sup_{t\geq 0}I(t u_\epsilon).
\end{align*}
Taking $u_\epsilon$ as a test function, from the following lemma, we can check that
\begin{equation}\label{eqS3.7}
c< \frac{2-\nu}{4(N-\nu)}S^{\frac{N-\nu}{2-\nu}}.
\end{equation}

\begin{lemma}\label{lm3.4}
Let $\mu$, $\nu\in[0,2)$ be fixed, we have
\begin{itemize}
\item[(i)] if $\frac{2(N+2-2\mu)}{N-2}< q<22^*(\mu)$, then \eqref{eqS3.7} holds
for any $\lambda>0$;

\item[(i)] if $2<q\le \frac{2(N+2-2\mu)}{N-2}$, there exists a positive constant
$\lambda^*$, such that \eqref{eqS3.7} still holds for $\lambda>\lambda^*$.
\end{itemize}
\end{lemma}

\begin{proof}
(i) Since $I(0)=0$, $\lim_{t\to \infty}I(t u_\epsilon)=-\infty$, there exists
$t_\epsilon>0$ such that
$I(t_\epsilon u_\epsilon)=\max_{t\geq0}I(t u_\epsilon)$.
We claim that there exist positive constants $t_1$ and $t_2$ such that
$t_1\leq t_\epsilon \leq t_2$ for $\epsilon\in (0,\epsilon_0)$.
In fact, by \eqref{eqS10.1}-\eqref{eqS10.5}, there is a small $\epsilon_2 >0$
such that
\begin{equation}\label{eqS30.8}
\begin{aligned}
I(t u_\epsilon)
&\leq\frac{t^2}{2}\int_{\mathbb{R}^N}\big[|\nabla
u_\epsilon|^2+V(x)u_\epsilon^2\big]dx+\frac{t^4}{4}\int_{\mathbb{R}^N}|\nabla
u_\epsilon^2|^2 dx \\
&\quad -\frac{t^{22^*(\nu)}}{22^*(\nu)}\int_{\mathbb{R}^N}
 \frac{|u_\epsilon|^{22^*(\nu)}}{|x|^\nu} dx \\
&\leq \frac{t^2}{2}+\frac{t^4}{2}S^{\frac{N-\nu}{2-\nu}}
-\frac{t^{22^*(\nu)}}{4\cdot2^*(\nu)}S^{\frac{N-\nu}{2-\nu}}
\end{aligned}
\end{equation}
for all $\epsilon\in(0,\epsilon_2)$. Hence
\[
\frac{t_\epsilon^{22^*(\nu)}}{2\cdot2^*(\nu)}S^{\frac{N-\nu}{2-\nu}}
\leq t_\epsilon^2+t_\epsilon^4S^{\frac{N-\nu}{2-\nu}}
\]
 which implies that there exists a constant $t_2 >0$ such that
$t_\epsilon\le t_2$ for all $\epsilon\in(0,\epsilon_2)$.

Since $2^*(\mu)<\frac{2(N+2-2\mu)}{N-2}<q<22^*(\mu)$, it follows
from \eqref{eqS10.1}-\eqref{eqS10.5} that there exists
$\epsilon_1\in (0,\epsilon_2)$ such that
\begin{align*}
I(t u_\epsilon)
&\geq\frac{t^4}{4}\int_{\mathbb{R}^N}|\nabla
(u_\epsilon^2)|^2 dx-\lambda\frac{t^q}{q}
\int_{\mathbb{R}^N}\frac{|u_\epsilon|^{q}}{|x|^\mu}dx
 -\frac{t^{22^*(\nu)}}{22^*(\nu)}\int_{
\mathbb{R}^N}\frac{|u_\epsilon|^{22^*(\nu)}}{|x|^\nu}dx \\
&\geq \frac18S^{\frac{N-\nu}{2-\nu}}t^4
 -\lambda C\epsilon^{\frac{N-\mu}{2-\nu}-\frac{N-2}{4(2-\nu)}q}t^q
-\frac{1}{2^*(\nu)}S^{\frac{N-\nu}{2-\nu}}t^{22^*(\nu)},
\end{align*}
for all $\epsilon\in (0,\epsilon_1)$.
 Let
\[
\chi=\max_{0\leq t\leq1}\big[\frac18t^4-\frac{1}{2^*(\nu)}t^{22^*(\nu)}
\big]S^{\frac{N-\nu}{2-\nu}}.
\]
Then $\chi>0$. Since $\frac{N-\mu}{2-\nu}-\frac{N-2}{4(2-\nu)}q>0$,
 we can find a small $\epsilon_0<\epsilon_1$ with
$\lambda C\epsilon^{\frac{N-\mu}{2-\nu}-\frac{N-2}{4(2-\nu)}q}
\leq \frac{\chi}{2}$ for all $\epsilon\in(0,\epsilon_0)$. Thus
\[
I(t_\epsilon u_\epsilon)
\geq \max_{0\leq t\leq 1}\Big\{\frac18S^{\frac{N-\nu}{2-\nu}}t^4
 -\lambda C\epsilon^{\frac{N-\mu}{2-\nu}-\frac{N-2}{4(2-\nu)}q}t^q
-\frac{1}{2^*(\nu)}S^{\frac{N-\nu}{2-\nu}}t^{22^*(\nu)}\Big\}
\geq \frac{\chi}{2}.
\]
Combining the above inequality with \eqref{eqS30.8}, we deduce that
\[
 \frac{\chi}{2}\leq \frac{t_\epsilon^2}{2}
+\frac{t_\epsilon^4}{2}S^{\frac{N-\nu}{2-\nu}}
-\frac{t_\epsilon^{22^*(\nu)}}{4\cdot2^*(\nu)}S^{\frac{N-\nu}{2-\nu}},
\]
 which implies that there exists a $t_1>0$ such that $t_\epsilon\geq t_1$ for
all $\epsilon\in (0,\epsilon_0)$. Hence we prove our claim.

For $\epsilon\in (0,\epsilon_0)$, from \eqref{eqS10.1}-\eqref{eqS10.5}, we have
\begin{align*}
I(t_\epsilon u_\epsilon)
&\leq\frac{t_\epsilon^4}{4}\int_{\mathbb{R}^N}|\nabla
(u_\epsilon^2)|^2 dx-\frac{t_\epsilon^{22^*(\nu)}}{22^*(\nu)}\int_{
\mathbb{R}^N}\frac{|u_\epsilon|^{22^*(\nu)}}{|x|^\nu}dx \\
&\quad-\frac{\lambda}{q}t_1^q\int_{\mathbb{R}^N}\frac{|u_\epsilon|^q}{|x|^\mu}dx
+\frac{t_2^2}{2}\int_{\mathbb{R}^N}\big[|\nabla u_\epsilon|^2
 +V(x)u_\epsilon^2\big]dx \\
&\leq\Big(\frac{t_\epsilon^4}{4}-\frac{t_\epsilon^{22^*(\nu)}}{22^*(\nu)}\Big)
S^{\frac{N-\nu}{2-\nu}}+O\left(\epsilon^{\frac{N-2}{2(2-\nu)}}|\ln\epsilon|\right)-
C\lambda\epsilon^{\frac{N-\mu}{2-\nu}-\frac{N-2}{4(2-\nu)}q} \\
&\leq \frac{2-\nu}{4(N-\nu)}S^{\frac{N-\nu}{2-\nu}}
+O\left(\epsilon^{\frac{N-2}{2(2-\nu)}}|\ln\epsilon|\right)
-C\lambda\epsilon^{\frac{N-\mu}{2-\nu}-\frac{N-2}{4(2-\nu)}q} \\
&< \frac{2-\nu}{4(N-\nu)}S^{\frac{N-\nu}{2-\nu}},
\end{align*}
for $\epsilon>0 $ small enough and
$\frac{N-\mu}{2-\nu}-\frac{N-2}{4(2-\nu)}q<\frac{N-2}{2(2-\nu)}$.
Therefore we can find a small $\bar{\epsilon}>0$ such that
$$
\sup_{t\geq 0}J(f^{-1}(tu_{\bar{\epsilon}}))
=\sup_{t\geq 0}I(tu_{\bar{\epsilon}})
=I(t_{\bar{\epsilon}}u_{\bar{\epsilon}})
<\frac{2-\nu}{4(N-\nu)}S^{\frac{N-\nu}{2-\nu}}.
$$
Moreover, from \eqref{eqS30.8}, we conclude that
$J(f^{-1}(tu_{\bar{\epsilon}}))=I(tu_{\bar{\epsilon}})\to -\infty$ as $t\to +\infty$,
which shows that there exists a $\bar{t}>0$ such that
$J(f^{-1}(\bar{t}u_{\bar{\epsilon}}))<0$. By taking
$\bar{\gamma}(t)=f^{-1}(t\bar{t}u_{\bar{\epsilon}})$, we have
$\bar{\gamma}\in \Gamma$ and $c\leq \max_{t\in[0,1]}J(\bar{\gamma}(t))
<\frac{2-\nu}{4(N-\nu)}S^{\frac{N-\nu}{2-\nu}}$
for any $\lambda>0$.

(ii) In the proof of this part, we first rewrite $I$ to be $I_\lambda$.
Let $u_0\in \mathcal{C}_0^{\infty}(\mathbb{R}^N)$ with $u_0\neq0$ and
define $t_\lambda>0$ such that
$I_\lambda(t_\lambda u_0)=\sup_{t\geq0}I_\lambda(t u_0)$.
We claim that $t_\lambda\to 0$ as $\lambda\to +\infty$.

We will prove our claim by contradiction. Suppose that the claim is not true.
Then there exists a constant $t_0>0$ and a sequence $\{\lambda_n\}$ such that
$t_{\lambda_n}\geq t_0$ as $\lambda_n\to +\infty$ for all $n$.
Without loss of generality, we may assume that $\lambda_n\geq 1$ for all $n$.
Let $t_n=t_{\lambda_n}$ and $I_1=I_{\lambda}|_{\lambda=1}$, then
$0\leq I_{\lambda_n}(t_nu_0)\leq I_1(t_nu_0)$ for all $n$.
Then it follows that
\begin{align*}
I_1 (t_n u_0 )
&= \frac{t^4_n }{4}\int_{\mathbb{R}^N}|\nabla u_0 |^2 dx
 +\frac{t^2_n }{2}\int_{\mathbb{R}^N}\left[|\nabla u_0 |^2 +V(x)u^2_0 \right]dx \\
&\quad-\frac{t^q_n }{q}\int_{\mathbb{R}^N}\frac{|u_0 |^q}{|x|^{\mu}}dx
 -\frac{t^{22^* (\nu)}_n }{22^* (\nu)}
 \int_{\mathbb{R}^N}\frac{|u_0 |^{22^* (\nu)}}{|x|^{\nu}}dx \\
&\le \frac{t^4_n }{4}\int_{\mathbb{R}^N}|\nabla u_0 |^2 dx+\frac{t^2_n }{2}
 \int_{\mathbb{R}^N}\left[|\nabla u_0 |^2 +V(x)u^2_0 \right]dx \\
&\quad-\frac{t^{22^* (\nu)}_n }{22^* (\nu)}\int_{\mathbb{R}^N}
 \frac{|u_0 |^{22^* (\nu)}}{|x|^{\nu}}dx
\to -\infty,\quad \text{as } t_n \to\infty,
\end{align*}
which gives us a contradiction since $22^* (\nu)>4>2$.
Thus $t_n$ is bounded from above. Moreover, we also have
\begin{equation}\label{eqS3.8}
\begin{aligned}
& I_{\lambda_n}(t_nu_0)\\
& \leq\frac{t_n^4}{4}\int_{\mathbb{R}^N}|\nabla
u_0^2|^2 dx+\frac{t_n^2}{2}\int_{\mathbb{R}^N}\big[|\nabla
u_0|^2+V(x)u_0^2\big]dx-\lambda_n\frac{t_n^{q}}{q}\int_{
\mathbb{R}^N}\frac{|u_0|^{q}}{|x|^\mu}dx \\
&\leq C -\lambda_n\frac{t_0^q}{q}\int_{
\mathbb{R}^N}\frac{|u_0|^q}{|x|^\mu}dx\to -\infty
\end{aligned}
\end{equation}
as $n\to \infty$, which contradicts $I_{\lambda_n}(t_nu_0)\geq 0$.
Hence our claim holds. Since $t_\lambda\to 0$ as
$\lambda\to +\infty$ and
$I_{\lambda}(t_\lambda u_0)\leq \frac{t_\lambda^4}{4}\int_{\mathbb{R}^N}|\nabla
u_0^2|^2 dx+\frac{t_\lambda^2}{2}\int_{\mathbb{R}^N}\big[|\nabla
u_0|^2+V(x)u_0^2\big]dx$, we can obtain that $I_{\lambda}(t_\lambda u_0)\to 0$
as $\lambda\to +\infty$. Therefore, there exists
$\lambda^*>0$ such that $\sup_{t\geq 0}I_\lambda(tu_0)
< \frac{2-\nu}{4(N-\nu)}S^{\frac{N-\nu}{2-\nu}}$ for any $\lambda>\lambda^*$.
This implies that
$c< \frac{2-\nu}{4(N-\nu)}S^{\frac{N-\nu}{2-\nu}}$ for all $\lambda>\lambda^*$.
The proof is complete.
\end{proof}



\section{Compactness of the $(C)_c$ sequence}\label{sec:3}

Since the compactness of $(C)_c$ sequence plays an important role in our process,
we will pay much more attention on the $(C)_c$ sequence of $J$ in this section.
Recall that  $\{v_n\}$ is a $(C)_c$ sequence of $J$ if $J(v_n)\to c$
and $(1+\|v_n\|)J'(v_n)\to 0$ as $n\to \infty$.

\begin{lemma}\label{lm4.1}
Any $(C)_c$ sequence $\{v_n\}\subset E$ of $J$ is bounded in $H^1(\mathbb{R}^N)$.
\end{lemma}

\begin{proof}
Let $\{v_n\}\subset E$ be a $(C)_c$ sequence of $J$ at level $c$,
that is,
\[
 J(v_n)\to c\quad \text{and}\quad (1+\|v_n\|)J'(v_n)\to 0\quad
 \text{as } n\to \infty.
 \]
 Choosing $\varphi_n=\frac{f(v_n)}{f'(v_n)}$, it is easy to see that
$\varphi_n \in E$ since $v_n \in E$ and the definition of $f'$.
 Then, by (A3) and (A4), we obtain
$\|\varphi_n\|\leq 5\|v_n\|$ and $\langle J'(v_n),\varphi_n\rangle\to 0$
as $n\to \infty$.

By defining two real functions
\[
\psi(t,x)=\lambda\frac{|t|^{q-2}t}{|x|^{\mu}}+\frac{|t|^{22^*(\nu)-2}t}{|x|^\nu},
\quad
\Psi(t,x)=\int_0^t\psi(s,x)ds,
\]
 choosing $\sigma=\max\{\mu, \nu\}$, we find that there exists a constant
$\tau \in (4,22^*(\nu))$ such that
$$
\lim_{t\to 0}\frac{|x|^\sigma[t\psi(t,x)-\tau\Psi(t,x)]}{t^2}=0
$$
and
$$
\lim_{|t|\to +\infty}\frac{|x|^\sigma[t\psi(t,x)-\tau\Psi(t,x)]}{t^\tau}=+\infty
\quad \hbox{uniformly for } x\in\mathbb{R}^N.
$$
Therefore, there exists $r>0$ such that
\begin{equation}\label{eqS4.1}
t\psi(t,x)-\tau\Psi(t,x)\ge0,\quad\text{for any } |t|>r, \; x\in\mathbb{R}^N.
\end{equation}
Moreover, for any $\varepsilon>0$, there exists a positive constant
$C(\varepsilon)$ such that
\begin{equation}\label{eqS4.2}
|t\psi(t,x)-\tau\Psi(t,x)|\leq \varepsilon\frac{|t|^2}{|x|^\sigma}
+C(\varepsilon)\frac{|t|^{22^*(\nu)}}{|x|^\sigma},\quad \forall t\in\mathbb{R}\,, \;
 x\in\mathbb{R}^N.
\end{equation}
With $\varphi_n$ defined before, we can deduce from \eqref{eqS4.1} that
\begin{equation}\label{eqS4.3}
\begin{aligned}
c+o(1)&=  J(v_n)-\frac{1}{\tau}\langle J'(v_n),\varphi_n\rangle \\
&=\frac12\int_{\mathbb{R}^N}|\nabla v_n|^2 dx
 -\frac1\tau\int_{\mathbb{R}^N}\Big(1+\frac{2f^2(v_n)}{1+2f^2(v_n)}\Big)|\nabla
v_n|^2 dx \\
&\quad+\big(\frac12-\frac1\tau\big)\int_{\mathbb{R}^N}V(x)f^2(v_n)dx \\
&\quad +\int_{\mathbb{R}^N}
 \big[\frac1\tau\psi(f(v_n),x)f(v_n)-\Psi(f(v_n),x)\big]dx \\
&\geq \big(\frac12-\frac2\tau\big)\int_{\mathbb{R}^N}|\nabla v_n|^2 dx
 +\big(\frac12-\frac1\tau\big)\int_{\mathbb{R}^N}V(x)f^2(v_n)dx \\
&\quad+\int_{B}\big[\frac1\tau\psi(f(v_n),x)f(v_n)-\Psi(f(v_n),x)\big]dx,
\end{aligned}
\end{equation}
where $B=\{x\in\mathbb{R}^{N}:|f(v_n)|\le r\}$.
By \eqref{eqS4.2}, there is a constant $M>V_0$ such that
\begin{equation}\label{eqS4.4}
\big|\frac1\tau t\psi(t,x)-\Psi(t,x)\big|
\leq \big(\frac14-\frac1{2\tau}\big)M \frac{t^2}{|x|^\sigma},\quad \text{for any }
 |t|\leq r,\; x\in\mathbb{R}^N,
\end{equation}
where $V_0$ is given in (A0).

Let $A=\{x\in\mathbb{R}^N:V(x)\leq M\}$. By \eqref{eqS4.4} and (A0), we obtain
\begin{align*}
&\big(\frac14-\frac1{2\tau}\big) \int_{\mathbb{R}^N}V(x)f^2(v_n)dx
 +\int_{B}\big[\frac1\tau\psi(f(v_n),x)f(v_n)-\Psi(f(v_n),x)\big]dx \\
&\geq \big(\frac14-\frac1{2\tau}\big)\int_{B\cap\left\{x\in\mathbb{R}^N : |x|>1\right\}}V(x)f^2(v_n)dx-\big(\frac14-\frac1{2\tau}\big)M\int_B \frac{|f(v_n)|^2}{|x|^\sigma}dx \\
&=\big(\frac14-\frac1{2\tau}\big)
\int_{B\cap \{x\in\mathbb{R}^N: |x|>1\}}V(x)f^2(v_n)dx \\
&\quad-\big(\frac14-\frac1{2\tau}\big)M
\int_{B\cap\{x\in\mathbb{R}^N: |x|>1\}} \frac{|f(v_n)|^2}{|x|^\sigma}dx \\
&\quad-\big(\frac14-\frac1{2\tau}\big)M
 \int_{B\cap\{x\in\mathbb{R}^N: |x|\le1\}} \frac{|f(v_n)|^2}{|x|^\sigma}dx \\
&\ge\big(\frac14-\frac1{2\tau}\big)
 \int_{B\cap\{x\in\mathbb{R}^N: |x|>1\}}\big(V(x)-M\big)f^2(v_n)dx \\
&\quad-\big(\frac14-\frac1{2\tau}\big)
 M\int_{B\cap\{x\in\mathbb{R}^N: |x|\le1\}} \frac{|f(v_n)|^2}{|x|^\sigma}dx \\
&\ge\big(\frac14-\frac1{2\tau}\big)
 \int_{A\cap B\cap\{x\in\mathbb{R}^N: |x|>1\}}\big(V_0 -M\big)r^2 dx\\
&\quad-\big(\frac14-\frac1{2\tau}\big)M
 \int_{B\cap\{x\in\mathbb{R}^N: |x|\le1\}} \frac{r^2}{|x|^\sigma}dx \\
&\ge\big(\frac14-\frac1{2\tau}\big)\big(V_0 -M\big)r^2
 \operatorname{meas}(A\cap B\cap\left\{x\in\mathbb{R}^N: |x|>1\right\}) \\
&\quad-\big(\frac14-\frac1{2\tau}\big)Mr^2
 \int_{\{x\in\mathbb{R}^N: |x|\le1\}} \frac{1}{|x|^\sigma}dx \\
&\geq \big(\frac14-\frac1{2\tau}\big)\big(V_0 -M\big)r^2
 \operatorname{meas}(A)-\big(\frac14-\frac1{2\tau}\big)r^2 M
 \int^1_0 \rho^{N-\sigma-1}d\rho \\
&\geq \big(\frac14-\frac1{2\tau}\big)\big(V_0 -M\big)r^2
 \operatorname{meas}(A)-\big(\frac14-\frac1{2\tau}\big)r^2M\,,
\end{align*}
which implies
\begin{equation}\label{eqS4.3b}
\begin{aligned}
 &\big(\frac12-\frac2\tau\big)\int_{\mathbb{R}^N}|\nabla v_n|^2 dx
 +\big(\frac14-\frac1{2\tau}\big)\int_{\mathbb{R}^N}V(x)f^2(v_n)dx \\
 &\leq \big(\frac14-\frac1{2\tau}\big)r^2
\left[\big(M-V_0\big)\operatorname{meas}(A)+M\right]+c+o(1).
\end{aligned}
\end{equation}
Therefore,
\[
 \int_{\mathbb{R}^N}\Big[|\nabla v_n|^2+V(x)f^2(v_n)\Big]dx\leq C\,,
\]
since $\operatorname{meas}(A)$ is finite according to the assumption (A0).

Moreover, by (A5) and using Sobolev inequality again, we have
\begin{align*}
\int_{\mathbb{R}^N}|v_n|^2 dx
&=\int_{\{|v_n|\leq 1\}}|v_n|^2 dx+\int_{\{|v_n|> 1\}}|v_n|^2 dx\\
&\leq C\int_{\mathbb{R}^N}V(x)f^2(v_n)dx
+\int_{\mathbb{R}^N}|v_n|^{2^*}dx \\
&\leq C\int_{\mathbb{R}^N}V(x)f^2(v_n)dx
+C\Big[\int_{\mathbb{R}^N}|\nabla v_n|^2 dx\Big]^{2^*/2} <+\infty,
\end{align*}
where $2^* =2N/(N-2)$ is the critical Sobolev exponent.
Hence $\{v_n\}$ is bounded in $H^1(\mathbb{R}^N)$, which completes our proof.
\end{proof}

\begin{lemma}\label{lm4.2}
Let $\{v_n\}\subset E$ be a (C)$_c$ sequence of $J$. If
$c<\frac{2-\nu}{4(N-\nu)}S^{\frac{N-\nu}{2-\nu}}$, there exist positive constants
$R$ and $\xi$,  and a sequence $\{y_n\}\subset\mathbb{R}^N$, such that
 $$
\limsup_{n\to \infty}\int_{B_R(y_n)}|v_n|^2dx\geq \xi.
$$
\end{lemma}

\begin{proof}
Suppose that the conclusion is not true. It then follows from
\cite[Lemma 1.21]{W} that $v_n\to 0$ in $L^s(\mathbb{R}^N)$ for all $2<s<2^*$.
By  H\"older's inequality, Hardy's inequality and Lemma~\ref{lm4.1}, we have
\[
\int_{\mathbb{R}^N}\frac{|v_n|^{q}}{|x|^\mu}dx
\leq C\Big(\int_{\mathbb{R}^N}|v_n|^{\frac{2(q-\mu)}{2-\mu}}dx
\Big)^{\frac{2-\mu}{2}}.
\]
Since $2<q<2^*(\mu)$ and $0\leq \mu<2$, then $2<\frac{2(q-\mu)}{2-\mu}<2^*$,
we can obtain $v_n\to 0$ in $L^{\frac{2(q-\mu)}{2-\mu}}(\mathbb{R}^N)$, hence
$$
v_n\to 0 \quad \text{in } L^{q}(\mathbb{R}^N,|x|^{-\mu}), \text{ for }
 2<q<2^*(\mu).
$$
Then by (A4), Lemma \ref{lm4.1} and the interpolation, we deduce that
\begin{equation}\label{eqS4.6}
f(v_n)\to 0\quad \text{in } L^{q}(\mathbb{R}^N,|x|^{-\mu}), \text{ for }
 2<q<22^*(\mu).
\end{equation}
By passing to a subsequence of $\{v_n\}$ and Lemma~\ref{lm4.1} we may assume that
\begin{gather*}
\int_{\mathbb{R}^N}\Big(1+\frac{2f^2(v_n)}{1+2f^2(v_n)}\Big)|\nabla
v_n|^2 dx+\int_{\mathbb{R}^N}V(x)f^2(v_n)dx\to b, \\
\int_{\mathbb{R}^N}\frac{|f(v_n)|^{22^*(\nu)}}{|x|^\nu}dx
\to d.
\end{gather*}
On the other hand,
\begin{align*}
&S\Big(\int_{\mathbb{R}^N}\frac{|f(v_n)|^{22^*(\nu)}}{|x|^\nu}dx
\Big)^{\frac{2}{2^*(\nu)}}\\
&\leq\int_{\mathbb{R}^N}|\nabla f^2(v_n)|^2 dx
=\int_{\mathbb{R}^N}\frac{4f^2(v_n)}{1+2f^2(v_n)}|\nabla v_n|^2 dx \\
&\leq \int_{\mathbb{R}^N}\Big(1+\frac{2f^2(v_n)}{1+2f^2(v_n)}\Big)|\nabla
v_n|^2 dx+\int_{\mathbb{R}^N}V(x)f^2(v_n)dx.
\end{align*}
By passing to a subsequence of $\{v_n\}$ to the both sides of the above inequality,
we obtain
$Sd^{\frac{2}{2^*(\nu)}}\leq b$. On the other hand, it can be deduced
from \eqref{eqS4.6} that $0=\lim_{n\to \infty}\langle J'(v_n),w_n\rangle=b-d$,
where $w_n=\frac{f(v_n)}{f'(v_n)}$. Therefore, $b=d\geq S^{\frac{N-\nu}{2-\nu}}$.
And, by \eqref{eqS4.6} again, we obtain
\begin{align*}
c&=\lim_{n\to \infty}J(v_n) \\
&= \lim_{n\to \infty}\Big[\frac12\int_{\mathbb{R}^N}\Big(|\nabla
v_n|^2+V(x)f^2(v_n)\Big)dx
 -\frac1{22^*(\nu)}\int_{\mathbb{R}^N}\frac{|f(v_n)|^{22^*(\nu)}}{|x|^\nu}dx\Big] \\
&\ge \lim_{n\to \infty}\Big[\frac14\int_{\mathbb{R}^N}
 \Big(1+\frac{2f^2(v_n)}{1+2f^2(v_n)}\Big)|\nabla
v_n|^2 dx+\frac14\int_{\mathbb{R}^N}V(x)f^2(v_n) dx \\
&\quad -\frac1{22^*(\nu)}\int_{\mathbb{R}^N}\frac{|f(v_n)|^{22^*(\nu)}}{|x|^\nu}dx\Big] \\
&=\Big(\frac14-\frac1{22^*(\nu)}\Big)d \\
&\geq \frac{2-\nu}{4(N-\nu)}S^{\frac{N-\nu}{2-\nu}},
\end{align*}
which gives us a contradiction since
 $c< \frac{2-\nu}{4(N-\nu)}S^{\frac{N-\nu}{2-\nu}}$. The proof is complete.
\end{proof}

\section{Proof of the main result}\label{sec:4}


\begin{proof}[Proof of Theorem \ref{intro1}]
Let $c$ be the Mountain Pass level given in \eqref{eqS3.5}.
From Lemma~\ref{lm3.1}, Lemma~\ref{lm3.4} and the modified Mountain Pass
Theorem \cite{SV}, $J$ has a $(C)_c$ sequence $\{v_n\}\subset E$.
By  Lemma \ref{lm4.1}, we may assume that $v_n\rightharpoonup v$ in
$H^1(\mathbb{R}^N)$
and $f(v_n)\rightharpoonup f(v)$ in $E$, under the assumption (A0),
for any $0\leq \sigma<2$, by Lemma~\ref{lm8.1} the embedding
$E\hookrightarrow L^r(\mathbb{R}^N,|x|^{-\sigma})$ is continuous for
$2\leq r\leq 2^*(\sigma)$, it is also compact for $2\leq r<2^*(\sigma)$,
which implies
%\label{eqS4.7}
\begin{gather*}
f(v_n)\to f(v)\quad \text{in } L^s(\mathbb{R}^N,|x|^{-\mu}),
 \text{ for } 2\leq s<22^*(\mu), \\
f(v_n)\rightharpoonup f(v)\quad \text{in } L^{22^*(\nu)}(\mathbb{R}^N,|x|^{-\nu}),
\end{gather*}
with $\mu, \nu \in [0,2 )$.
Hence, we have $\langle J'(v_n),\varphi\rangle\to \langle J'(v),\varphi\rangle=0$
for any $\varphi\in \mathcal{C}_0^{\infty}(\mathbb{R}^N)$, that is, $v$
is a weak solution of \eqref{eulerto1st}.
We conclude from Lemma \ref{lm3.4} that for any $\mu, \nu \in [0,2 )$,
$c< \frac{2-\nu}{4(N-\nu)}S^{\frac{N-\nu}{2-\nu}}$ holds when either of the
following statement holds: (1) $\frac{2(N+2-2\mu)}{N-2}< q<22^*(\mu)$ and each
$\lambda>0$;
(2) $2<q\leq \frac{2(N+2-2\mu)}{N-2}$ and each $\lambda>\lambda^*$ for some
positive constant $\lambda^*$.
Moreover, by Lemma~\ref{lm4.2}, there exists a constant
$\xi>0$ such that
$$
\int_{\mathbb{R}^N}|v|^2 dx=\lim_{n\to \infty}\int_{\mathbb{R}^N}|v_n|^2 dx\geq \xi>0,
$$
which implies that $v$ is a nontrivial solution of problem \eqref{eulerto1st}.
Hence $u=f(v)$ is a nontrivial solution of
problem \eqref{eqS1.1}.

Finally, letting $e=\inf\{J(v): v\in E,v\neq 0, J'(v)=0\}$,
it is easy to see that $e$ is attained by the lower semi-continuity.
The proof is complete.
\end{proof}

\subsection*{Acknowledgments}
This research was supported by the National Natural Science Foundation
of China (No. 11501186, 11326145, 11501231, 11561024).

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\end{document}