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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 17, pp. 1--22.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/17\hfil Existence of multiple solutions]
{Existence of multiple solutions for quasilinear elliptic
equations in $\mathbb{R}^N$}

\author[H. Yin, Z. Yang \hfil EJDE-2014/17\hfilneg]
{Honghui Yin, Zuodong Yang}  % in alphabetical order

\address{Honghui Yin \newline
Institute of Mathematics, School of Mathematical Sciences \\
Nanjing Normal University, Jiangsu Nanjing 210023, China.\newline
School of Mathematical Sciences\\
 Huaiyin Normal University,
Jiangsu Huaian 223001,  China}
\email{yinhh771109@163.com}

\address{Zuodong Yang \newline
Institute of Mathematics, School of Mathematical Sciences \\
Nanjing Normal University, Jiangsu Nanjing 210023, China.\newline
 School of Teacher Education\\
Nanjing Normal University, Jiangsu Nanjing 210097, China}
\email{zdyang\_jin@263.net}

\thanks{Submitted August 4, 2013. Published January 10, 2014.}
\subjclass[2000]{35J62, 35J50， 35J92}
\keywords{Nehari manifold; quasilinear; positive solution;
(PS)-sequence}

\begin{abstract}
 In this article, we establish the multiplicity of positive weak solution for
 the  quasilinear elliptic equation
 \begin{gather*}
 -\Delta_p u+\lambda|u|^{p-2}u=f(x) |u|^{s-2 }u+h(x)|u|^{r-2}u\quad  x\in \mathbb{R}^N,\\
 u>0\quad  x\in \mathbb{R}^N,\\
 u\in W^{1,p}(\mathbb{R}^N)
 \end{gather*}
 We show how the shape of the graph of $f$ affects the number of
 positive solutions. Our results extend the corresponding results
 in \cite{w1}.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
%\newtheorem{corollary}[theorem]{Corollary}
%\newtheorem{definition}[theorem]{Definition}
%\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

 In this article we  consider the existence of solutions for
the nonlinear quasilinear problem
\begin{equation}
\begin{gathered}
 -\Delta_p u+\lambda|u|^{p-2}u=f(x) |u|^{s-2 }u+h(x)|u|^{r-2}u\quad  x\in \mathbb{R}^N,\\
u>0\quad  x\in \mathbb{R}^N\\
u\in W^{1,p}(\mathbb{R}^N)
\end{gathered} \label{El}
\end{equation}
where $1\leq r<p<s<p^*$, $p<N$, $p^*=\frac{pN}{N-p}$ denotes the critical
Sobolev exponent, $\lambda>0$ is a parameter,
$h\in L^{\frac{p}{p-r}}(\mathbb{R}^N)\backslash\{0\}$ is nonnegative.
For the function $f$, we assume the following conditions:
\begin{itemize}
\item[(C1)] $f\in C(\mathbb{R}^N)$ and is nonnegative in $\mathbb{R}^N$;

\item[(C2)] $f^\infty=\lim_{|x|\to\infty}f(x)>0$;

\item[(C3)] There exist some points $x^1,x^2,\dots,x^k$ in $\mathbb{R}^N$
such that $f(x^i)$ are some strict maxima and satisfy
$$
f^\infty<f(x^i)=f_{\rm max}\equiv\max\{f(x)|x\in \mathbb{R}^N\}
$$
for  $i=1,2,\dots,k$.
\end{itemize}
Associated with \eqref{El}, we consider the energy functional
$$
I_\lambda(u)=\frac{1}{p}\int_{\mathbb{R}^N}|\nabla u|^{p}
+\lambda|u|^pdx-\frac{1}{s}\int_{\mathbb{R}^N}f(x)
|u|^{s}dx-\frac{1}{r}\int_{\mathbb{R}^N}h(x) |u|^{r}dx.
$$
It is well known that the functional $I_\lambda\in C^1(W^{1,p}(\mathbb{R}^N),R)$,
and that the solutions of \eqref{El} are the critical points of
the energy functional $I_\lambda$.

When $p=2$ and $h(x)\equiv0$, Equation \eqref{El} becomes
\begin{equation}
\begin{gathered}
 -\Delta u+\lambda u=f(x) |u|^{s-2 }u\quad  x\in \mathbb{R}^N,\\
u>0\quad  x\in \mathbb{R}^N,\\
u\in H^1(\mathbb{R}^N).
\end{gathered} \label{e1.1}
\end{equation}
It is known that the existence of positive
solutions of \eqref{e1.1} is affected by the shape of the graph of $f(x)$.
This has been the focus of a great deal of research by several
authors \cite{b1,b2,c1,l3}.
 Specially, if $f$ is a positive constant, then
\eqref{e1.1} has a unique positive solution \cite{k1}
Adachi and Tanaka \cite{a1} showed that there exist at least four
positive solutions of the equation
\begin{equation}
\begin{gathered}
 -\Delta u+\lambda u=f(x) |u|^{s-2 }u+h(x)\quad  x\in \mathbb{R}^N,\\
u>0\quad  x\in \mathbb{R}^N,\\
u\in H^1(\mathbb{R}^N)
\end{gathered} \label{e1.2}
\end{equation}
under the assumptions $0<f(x)\leq f^\infty=\lim_{|x|\to\infty}f(x)$,
$h\in H^{-1}(R^N)\backslash\{0\}$ is nonnegative and $\|h\|_{H^{-1}}$ is
sufficiently small. Several authors have studied a generalized version of
\eqref{e1.2},
\begin{equation}
\begin{gathered}
 -\Delta u+\lambda u=f(x,u)+h(x)\quad  x\in \mathbb{R}^N,\\
u>0\quad  x\in \mathbb{R}^N,\\
u\in H^1(\mathbb{R}^N)
\end{gathered}  \label{e1.3}
\end{equation}
where $f(x,u)$ and $h(x)$ satisfy some suitable
conditions. They showed the existence of at least two positive
solutions when $\|h\|_{H^{-1}}$ is sufficiently small, see
\cite{a2,c2,j1}.

Wu \cite{w1} considered the problem \eqref{El} with $p=2$,
under some suitable assumptions on $f(x),h(x)$. The author
obtained the existence of multiple positive solution by
variational methods.
Several publications \cite{b3,b4,c3,y1} show results about the 
quasilinear elliptic equation
\begin{equation}
\begin{gathered}
 -\Delta_p u+\lambda |u|^{p-2}u=f(x,u)\quad  x\in \Omega,\\
u\in W_0^{1,p}(\Omega),\;u\neq0
\end{gathered} \label{e1.4}
\end{equation}
where $1<p<N$, $N\geq3$, $\Omega$ is an unbounded
domain in $\mathbb{R}^N$.
Because of the unboundedness of the domain, the Sobolev
compact embedding does not hold. There are many methods to
overcome the difficulty. In \cite{y1}, the authors used the
concentration-compactness principle posed by Lions and the
mountain pass lemma to solve problem \eqref{e1.4}.
In \cite{b3,b4}, the authors studied the problem in symmetric Sobolev spaces which
posses Sobolev compact embedding.

Especially, when $\lambda=1$, $ f(x,u)=q(x)u^\alpha$ and $\Omega$
is replaced by $\mathbb{R}^N$,  using a min-max procedure formulated by
Bahri and Li \cite{b2}, Citti and Uguzzoni \cite{c3} obtained the
existence of a solution $u\in W^{1,p}(\mathbb{R}^N)\cap
C_{\rm loc}^{1+\beta}(\mathbb{R}^N)$ of \eqref{e1.4} when $p\in (1,2)$, and
$\beta\in (0,1)$ is  constant. In \cite{s1}, the authors studied
the  problem
\begin{equation}
\begin{gathered}
 -\Delta_p u+\lambda a(x)u^{p-1}=f(x)u^{p^*-1}+g(x)u^q\quad  x\in \mathbb{R}^N,\\
u\in D_0^{1,p}(\mathbb{R}^N)\cap C_{loc}^{1+\beta}(\mathbb{R}^N),\quad
\lim_{|x|\to\infty}u(x)=0,
\end{gathered} \label{e1.5}
\end{equation}
which is a general case of \eqref{El}. The
authors proved that there exists a positive solution of \eqref{e1.5} for
all $\lambda$ in some interval $[0,\lambda_0)$.

 In this article, we consider  show the existence  of multiple positive 
solutions of \eqref{El}. Our arguments are based on a combination of the
concentration-compactness principle of Lions \cite{l1}, and
Ekeland's variational principle \cite{e1}. Our main result is the
following theorem.

\begin{theorem} \label{thm1.1} 
Assume {\rm (C1)--(C3)} hold, and $h\in
L^{\frac{p}{p-r}}(\mathbb{R}^N)\backslash{\{0\}}$ 
is nonnegative.
Then there exists $\lambda_0>0$ such that for all
$\lambda>\lambda_0$, Equation \eqref{El} has at least $k+1$ positive
solutions.
\end{theorem}

The rest of this article is organized as follows. 
In Section 2, we give some preliminaries and some properties of Nehri manifold. 
In Section 3, we  prove the main result, Theorem \ref{thm1.1}.

\section{Preliminaries}
Throughout the paper, $C,c$ will denote various positive
constants, their  values may vary from place to anther.
By the change of variables $\eta=\lambda^{-1/p}$,
$v(x)=\eta^{p/(s-p)}u(\eta x)$, Equation \eqref{El} can be
transformed into
\begin{equation}
\begin{gathered}
 -\Delta_p v+|v|^{p-2}v=f_\eta|v|^{s-2 }
v+\eta^{\frac{p(s-r)}{s-p}}h_\eta|v|^{r-2}v\quad  x\in \mathbb{R}^N,\\
v>0\quad  x\in \mathbb{R}^N,\\
v\in W^{1,p}(\mathbb{R}^N)
\end{gathered} \label{Ee}
\end{equation}
 where $f_\eta=f(\eta x),h_\eta=h(\eta x)$.

For $u\in W^{1,p}(\mathbb{R}^N)$, $c\in R$, $a\in C(\mathbb{R}^N)$
nonnegative and bounded, 
 and  $b\in L^{\frac{p}{p-r}}(\mathbb{R}^N)$ non-negative, we define
\begin{gather*}
I_{a,b}(u)=\frac{1}{p}\|u\|^p-\frac{1}{s}\int_{\mathbb{R}^N}a
|u|^{s}dx-\eta^{\frac{p(s-r)}{s-p}}\frac{1}{r}\int_{\mathbb{R}^N}b
|u|^{r}dx; \\
M_{a,b}(c)=\{u\in W^{1,p}(\mathbb{R}^N)\backslash\{0\}|\langle
I'_{a,b}(u),u\rangle=c\}; \\
\alpha_{a,b}(c)=\inf\{I_{a,b}(u)|u\in M_{a,b}(c)\},
\end{gather*}
where $\|u\|=(\int_\Omega|\nabla u|^p+|u|^pdx)^{1/p}$ is a
standard norm in $W^{1,p}(\mathbb{R}^N)$ and $I_{a,b}'$ denote the
Fr\'{e}chet derivative of $I_{a,b}$. We shall write
$M_{a,b}(0),\alpha_{a,b}(0)$ as $M_{a,b},\alpha_{a,b}$
respectively. Then, we have the following results.

\begin{lemma} \label{lem2.1} 
Suppose $a$ is a continuous bounded and
nonnegative function on $\mathbb{R}^N$, then
$\alpha_{a,0}(c)=\frac{c}{p}$ for $c>0$ and
$$
\alpha_{a,0}\leq \alpha_{a,0}(c)+\alpha_{a,0}(-c)
 -\frac{s-p}{sp}|c|\quad \text{for all }c\in\mathbb{R}.
$$
\end{lemma}

\begin{proof}
The case $p=2$ was proved by Cao-Noussair \cite[Lemma 2.2]{c1}. 
By a modification of the method given in \cite{c1}, we obtain our result. 
For the readers convenience, we give a sketch here.
For any  $c>0$, let $u\in M_{a,0}(c)$. Then
$$
\|u\|^p=\int_{\mathbb{R}^N}a|u|^sdx+c\geq c.
$$ Thus
\[
I_{a,0}(u)=\frac{1}{p}\|u\|^p-\frac{1}{s}\int_{{\bf
R^N}}a|u|^sdx 
=(\frac{1}{p}-\frac{1}{s})\|u\|^p+\frac{c}{s} \\
\geq \frac{c}{p}.
\]
To show that the equality holds, choose $v\in W^{1,p}{(\mathbb{R}^N})$ with
$\int_{{\mathbb{R}^N}}|\nabla v|^pdx=c$, for any $\sigma>0$, define
$$
u_\sigma(x)=\sigma^{\frac{N-p}{p}}v(\sigma x),\quad
w_\sigma(x)=(1+\theta)u_\sigma
$$
where $\theta>0$ being selected so that $w_\sigma\in M_{a,0}(c)$.
It is easy to see that
\begin{gather*}
\int_{{\mathbb{R}^N}}|\nabla u_\sigma|^pdx=c, \\
\int_{{\mathbb{R}^N}}|u_\sigma|^qdx
=\sigma^{\frac{(N-p)q}{p}-N}\int_{{\mathbb{R}^N}}|v|^qdx\to0
\quad\text{as }\sigma\to\infty
\end{gather*}
for $q<p^*$. Obviously, such a $\theta=\theta(\sigma)$ exists when 
$\sigma$ large enough and
$\theta\to0$ as $\sigma\to+\infty$. Therefore,
$$
I_{a,0}(w_\sigma)=\frac{1}{p}\|w_\sigma\|^p-\frac{1}{s}
\int_{{\mathbb{R}^N}}a|w_\sigma|^sdx\to\frac{c}{p}
\quad\text{as }\sigma\to+\infty.
$$
Hence
$$
\alpha_{a,0}(c)=\frac{c}{p}.
$$ 
To complete the proof of Lemma \ref{lem2.1}, let $c>0$ and $u\in M_{a,0}(-c)$. Then
$$
\|u\|^p=\int_{{\mathbb{R}^N}}a|u|^sdx-c<\int_{{\mathbb{R}^N}}a|u|^sdx.
$$
It is easy to see that there exist unique $t\in(0,1)$ such that
$v=tu\in M_{a,0}$. Then we have
\begin{align*}
I_{a,0}(v)
&=(\frac{1}{p}-\frac{1}{s})\|v\|^p \\
&=(\frac{1}{p}-\frac{1}{s})t^p\|u\|^p \\
&<(\frac{1}{p}-\frac{1}{s})\|u\|^p+\frac{c}{s}-\frac{c}{s} \\
&= I_{a,0}(u)+\frac{c}{p}+(\frac{1}{s}-\frac{1}{p})c \\
&\leq I_{a,0}(u)+\alpha_{a,0}(c)-\frac{s-p}{sp}c.
\end{align*}
The required inequality then follows by taking the infimum over
$M_{a,0}(-c)$.
\end{proof}

Define 
$$
\psi(u)=\langle
I'_{f_\eta,h_\eta}(u),u\rangle=\|u\|^p-\int_{\mathbb{R}^N}f_\eta
|u|^{s}dx-\eta^{\frac{p(s-r)}{s-p}}\int_{\mathbb{R}^N}h_\eta
|u|^{r}dx.
$$ 
Then for $u\in M_{f_\eta,h_\eta}$, we have
\begin{align*}
\langle\psi'(u),u\rangle
&= p\|u\|^p-s\int_{\mathbb{R}^N}f_\eta
|u|^{s}dx-r\eta^{\frac{p(s-r)}{s-p}}\int_{\mathbb{R}^N}h_\eta
|u|^{r}dx \\
&= (p-r)\|u\|^p-(s-r)\int_{\mathbb{R}^N}f_\eta |u|^{s}dx.
\end{align*}
Using the same methods  as \cite{t1}, we split $ M_{f_\eta,h_\eta}$
into three parts:
\begin{align*}
M^+_{f_\eta,h_\eta}
=\{u\in M_{f_\eta,h_\eta}|(p-r)\|u\|^p-(s-r)\int_{\mathbb{R}^N}f_\eta
|u|^{s}dx>0\}; \\
M^0_{f_\eta,h_\eta} 
= \{u\in M_{f_\eta,h_\eta}|(p-r)\|u\|^p-(s-r)\int_{\mathbb{R}^N}f_\eta
|u|^{s}dx=0\}; \\
M^-_{f_\eta,h_\eta}
=\{u\in M_{f_\eta,h_\eta}|(p-r)\|u\|^p-(s-r)\int_{\mathbb{R}^N}f_\eta
|u|^{s}dx<0\}.
\end{align*}
Then we have the following result.

\begin{lemma} \label{lem2.2} 
There exists $\eta_1>0$ such that for all
$\eta\in(0,\eta_1)$, we have $M^0_{f_\eta,h_\eta}={\emptyset}$.
\end{lemma}

\begin{proof} Assume the contrary, that is
$M^0_{f_\eta,h_\eta}\neq{\emptyset}$ for all $\eta>0$. Then for 
$u\in M^0_{f_\eta,h_\eta}$, we have
\begin{gather}
\|u\|^p=\frac{s-r}{p-r}\int_{\mathbb{R}^N}f_\eta |u|^{s}dx\label{e2.1}
\\
\eta^{\frac{p(s-r)}{s-p}}\int_{\mathbb{R}^N}h_\eta
|u|^{r}dx=\|u\|^p-\int_{\mathbb{R}^N}f_\eta
|u|^{s}dx=\frac{s-p}{p-r}\int_{\mathbb{R}^N}f_\eta
|u|^{s}dx.\label{e2.2}
\end{gather}
Moreover,
\begin{align*}
\frac{s-p}{s-r}\|u\|^p
&= \|u\|^p-\int_{\mathbb{R}^N}f_\eta
|u|^{s}dx\leq\eta^{\frac{p(s-r)}{s-p}}\|h_\eta\|_{L^{\frac{p}{p-r}}}\|u\|^r \\
&= \eta^\beta\|h\|_{L^{\frac{p}{p-r}}}\|u\|^r,
\end{align*}
where $\beta=\frac{p(s-r)}{s-p}-\frac{p-r}{p}N$. Also we have
\begin{equation}
\|u\|\leq[\frac{s-r}{s-p}\eta^\beta\|h\|_{L^{\frac{p}{p-r}}}]^{\frac{1}{p-r}}.
\label{e2.3}
\end{equation}
Let $K:M_{f_\eta,h_\eta}\to R$ be given by
$$
K(u)=c(s,r)(\frac{\|u\|^{p\frac{s-1}{p-1}}}{\int_{\mathbb{R}^N}f_\eta
|u|^{s}dx})^{\frac{p-1}{s-p}}-\eta^{\frac{p(s-r)}{s-p}}
\int_{\mathbb{R}^N}h_\eta |u|^{r}dx,
$$
where
$c(s,r)=(\frac{s-r}{p-r})^{\frac{1-s}{s-p}}\frac{s-p}{p-r}$. Then
$K(u)=0$ for all $\eta>0$ and $u\in M^0_{f_\eta,h_\eta}$.  From
\eqref{e2.1} and \eqref{e2.2}, it follows that for $u\in M^0_{f_\eta,h_\eta}$,
and
\begin{equation}
K(u)=c(s,r)[\frac{(\frac{s-r}{p-r}\int_{\mathbb{R}^N}f_\eta
|u|^{s}dx)^{\frac{s-1}{p-1}}}{\int_{\mathbb{R}^N}f_\eta
|u|^{s}dx}]^{\frac{p-1}{s-p}}-\frac{s-p}{p-r}
\int_{\mathbb{R}^N}f_\eta |u|^{s}dx=0.\label{e2.4}
\end{equation}
However, by \eqref{e2.3}, the
H\"{o}lder and Sobolev inequalities and
$$
(\frac{\|u\|^{s}}{\int_{\mathbb{R}^N}f_{\rm max}
|u|^{s}dx})^{\frac{p-1}{s-p}}>(\frac{S^s}{f_{\rm max}})^{\frac{p-1}{s-p}}\quad
\text{for all } u\in M_{f_\eta,h_\eta},
$$
where $S=\inf_{u\in W^{1,p}{(\mathbb{R}^N})\backslash \{0\}}
\frac{\|u\|}{\|u\|_{L^s}}$ is the best
Sobolev constant. Also we have
\begin{align*}
K(u)&\geq  c(s,r)(\frac{\|u\|^{p\frac{s-1}{p-1}}}{\int_{\mathbb{R}^N}f_\eta
|u|^{s}dx})^{\frac{p-1}{s-p}}-\eta^\beta\|h\|_{L^{\frac{p}{p-r}}}\|u\|^r \\
&\geq \|u\|^r[c(s,r)(\frac{S^s}{f_{\rm max}})^{\frac{p-1}{s-p}}\|u\|^{1-r}
 -\eta^\beta\|h\|_{L^{\frac{p}{p-r}}}] \\
&\geq \|u\|^r[c(s,r)(\frac{S^s}{f_{\rm max}})^{\frac{p-1}{s-p}}
(\eta^\beta\frac{s-r}{s-p}\|h\|_{L^{\frac{p}{p-r}}})^{\frac{1-r}{p-r}}
-\eta^\beta\|h\|_{L^{\frac{p}{p-r}}}]
\end{align*}
for all $u\in M^0_{f_\eta,h_\eta}$, where
$\beta=\frac{p(s-r)}{s-p}-\frac{p-r}{p}N>0$ (see Lemma \ref{lemA}). Since
$\frac{1-r}{p-r}\leq0$, there exists $\eta_1>0$ such that for each
$\eta\in(0,\eta_1)$ and $u\in M^0_{f_\eta,h_\eta}$, we have
$K(u)>0$, this contradicts to \eqref{e2.4}. We can conclude that $
M^0_{f_\eta,h_\eta}={\emptyset}$ for all $\eta\in(0,\eta_1)$.
\end{proof}

By Lemma \ref{lem2.2} for $\eta\in(0,\eta_1)$ we write
$M_{f_\eta,h_\eta}=M^+_{f_\eta,h_\eta}\cup M^-_{f_\eta,h_\eta}$ and
define
$$
\alpha^+_{f_\eta,h_\eta}=\inf_{u\in M^+_{f_\eta,h_\eta}}I_{f_\eta,h_\eta},\quad
\alpha^-_{f_\eta,h_\eta}=\inf_{u\in M^-_{f_\eta,h_\eta}}I_{f_\eta,h_\eta}.
$$ 
The following Lemma shows
that the minimizers on $M_{f_\eta,h_\eta}$ are ``usually'' critical
points for $I_{f_\eta,h_\eta}$.

\begin{lemma} \label{lem2.3} 
For $\eta\in(0,\eta_1)$, if $u_0$ is a local
minimizer for $I_{f_\eta,h_\eta}$ on $M_{f_\eta,h_\eta}$, then
$I'_{f_\eta,h_\eta}(u_0)=0$ in $W^{-1}(\mathbb{R}^N)$, where
$W^{-1}(\mathbb{R}^N)$ is the dual space of $W^{1,p}(\mathbb{R}^N)$.
\end{lemma}

 \begin{proof}  If $u_0$ is a local minimizer for $I_{f_\eta,h_\eta}$ on
 $M_{f_\eta,h_\eta}$, then $u_0$ is a solution of the optimization
 problem
$$
\text{minimize $I_{f_\eta,h_\eta}(u)$ subject  to $\psi(u)=0$}.
$$
Hence, by the theory of Lagrange multipliers, there exists
$\theta\in \mathbb{R}$ such that
$$
I'_{f_\eta,h_\eta}(u_0)=\theta\psi'(u_0)\quad\text{in }W^{-1}(\mathbb{R}^N).
$$ 
This implies
$$
\langle I'_{f_\eta,h_\eta}(u_0),u_0\rangle=\theta\langle
\psi'(u_0),u_0\rangle.
$$ 
Since $u_0\in M_{f_\eta,h_\eta}$ and by
Lemma \ref{lem2.2}, $M^0_{f_\eta,h_\eta}=\emptyset$ when $\eta\in(0,\eta_1)$, we
have
$$
\langle I'_{f_\eta,h_\eta}(u_0),u_0\rangle=0\;\text{and}\;\langle
\psi'(u_0),u_0\rangle\neq0.
$$ 
So we obtain $\theta=0$. This completes the proof.
\end{proof}

 For each $u\in W^{1,p}(\mathbb{R}^N)\backslash\{0\}$, we
 define
$$
t_{\rm max}=(\frac{p-r}{s-r}\frac{\|u\|^{p}}{\int_{\mathbb{R}^N}f_\eta
|u|^{s}dx})^{\frac{1}{s-p}}>0.
$$ 
Then we have the following Lemma.

\begin{lemma} \label{lem2.4} 
There exists $\eta_2>0$ such that for each $u\in
W^{1,p}(\mathbb{R}^N)\backslash\{0\}$ and $\eta\in(0,\eta_2)$, we
have

(i) there is a unique $t^-=t^-(u)>t_{\rm max}>0$ such that $t^-u\in
M^-_{f_\eta,h_\eta}$ and $I_{f_\eta,h_\eta}(t^-u)=\max_{t\geq
t_{\max}}I_{f_\eta,h_\eta}(tu)$;

(ii) if $\int_{\mathbb{R}^N}h_\eta |u|^{r}dx>0$, then there is a
unique $0<t^+=t^+(u)<t_{\rm max}<t^-=t^-(u)$ such that $t^+u \in
M^+_{f_\eta,h_\eta}$, $t^-u \in M^-_{f_\eta,h_\eta}$ and
$$
I_{f_\eta,h_\eta}(t^+u)=\min_{t^-\geq t\geq0}I_{f_\eta,h_\eta}(tu),\quad
I_{f_\eta,h_\eta}(t^-u)=\max_{t\geq t_{\rm max}}I_{f_\eta,h_\eta}(tu).
$$
\end{lemma}

\begin{proof} (i) Since $h(x)$ is nonnegative, then 
$\int_{\mathbb{R}^N}h_\eta |u|^{r}dx\geq0$.
Let
$$
m(t)=t^{p-r}\|u\|^p-t^{s-r}\int_{\mathbb{R}^N}f_\eta |u|^{s}dx,
$$ 
clearly, $m(t)$ is increasing in $(0,t_{\rm max})$ and is
decreasing in $(t_{\rm max},+\infty)$, also, we
have
$$
m(0)=0,\;\lim_{t\to+\infty}m(t)=-\infty,
$$ 
i.e.
$m(t)$ is concave and achieve its maximum at $t_{\rm max}$. Moreover,
\begin{align*}
m(t_{\rm max})
&= (\frac{p-r}{s-r}\frac{\|u\|^{p}}{\int_{\mathbb{R}^N}f_\eta
|u|^{s}dx})^{\frac{p-r}{s-p}}\|u\|^p-(\frac{p-r}{s-r}
 \frac{\|u\|^{p}}{\int_{\mathbb{R}^N}f_\eta
|u|^{s}dx})^{\frac{s-r}{s-p}}\int_{\mathbb{R}^N}f_\eta |u|^{s}dx \\
&= \frac{\|u\|^{p\frac{s-r}{s-p}}}{(\int_{\mathbb{R}^N}f_\eta
|u|^{s}dx)^{\frac{p-r}{s-p}}}[(\frac{p-r}{s-r})^{\frac{p-r}{s-p}}
 -(\frac{p-r}{s-r})^{\frac{s-r}{s-p}}] \\
&= \frac{s-p}{s-r}(\frac{p-r}{s-r})^{\frac{p-r}{s-p}}
 (\frac{\|u\|^{s}}{\int_{\mathbb{R}^N}f_\eta
|u|^{s}dx})^{\frac{p-r}{s-p}}\|u\|^r \\
&\geq \frac{s-p}{s-r}(\frac{p-r}{s-r})^{\frac{p-r}{s-p}}
 (\frac{S^{s}}{f_{\rm max}})^{\frac{p-r}{s-p}}\|u\|^r 
= C\|u\|^r.
\end{align*}
Since $C>0$,
$$
0\leq\eta^{\frac{p(s-r)}{s-p}}\int_{\mathbb{R}^N}h_\eta
|u|^{r}dx\leq\eta^\beta\|h\|_{L^{\frac{p}{p-r}}}\|u\|^r
$$ 
and
$\beta>0$, there exists $\eta_2>0$, such that for any
$\eta\in(0,\eta_2)$, we
have
$$
m(t_{\rm max})>\eta^{\frac{p(s-r)}{s-p}}\int_{{R}^N}h_\eta |u|^{r}dx.
$$ 

Case (a): $\int_{{R}^N}h_\eta |u|^{r}dx=0$.
Then there is unique $t^->t_{\max}$ such that $m(t^-)=0$ and
$m'(t^-)<0$.
Now
\[
\langle\psi'(t^-u),t^-u\rangle
= (p-r)\|t^-u\|^p-(s-r)\int_{\mathbb{R}^N}f_\eta |t^-u|^{s}dx 
= (t^-)^{r+1}m'(t^-)<0
\]
and
\begin{align*}
\langle
I'_{f_\eta,h_\eta}(t^-u),t^-u\rangle
&= \|t^-u\|^p-\int_{\mathbb{R}^N}f_\eta |t^-u|^{s}dx-\eta^{\frac{p(s-r)}{s-p}}\int_{\mathbb{R}^N}h_\eta |t^-u|^{r}dx \\
&= (t^-)^r[(t^-)^{p-r}\|u\|^p-(t^-)^{s-r}\int_{\mathbb{R}^N}f_\eta
|u|^{s}dx] \\
&= (t^-)^rm(t^-)=0 .
\end{align*}
Thus, $t^-u\in M^-_{f_\eta,h_\eta}$. Moreover, we have
$$
\frac{d}{dt}I_{f_\eta,h_\eta}(tu)=0,\quad
\frac{d^2}{dt^2}I_{f_\eta,h_\eta}(tu)<0,\quad \text{for }t=t^-.
$$
Then we have $I_{f_\eta,h_\eta}(t^-u)=\max_{t\geq
t_{\max}}I_{f_\eta,h_\eta}(tu)$.

Case (b): $\int_{\mathbb{R}^N}h_\eta |u|^{r}dx>0$.
There are unique $t^+$ and $t^-$ such that $0<t^+<t_{\rm max}<t^-$
such that
$$
m(t^+)=\eta^{\frac{p(s-r)}{s-p}}\int_{\mathbb{R}^N}h_\eta
|u|^{r}dx=m(t^-)
$$
and $m'(t^+)>0>m'(t^-)$. Similar to the argument
in Case a, we have $t^\pm u\in M^\pm_{f_\eta,h_\eta}$, and
$I_{f_\eta,h_\eta}(t^-u)\geq I_{f_\eta,h_\eta}(tu)\geq
I_{f_\eta,h_\eta}(t^+u)$ for each $t\in[t^+,t^-]$, and
$I_{f_\eta,h_\eta}(tu)\geq I_{f_\eta,h_\eta}(t^+u)$ for each
$t\in[0,t^+]$.

(ii) By case (b) it follows  part (i)
\end{proof}

To establish the existence of a local minimum for
$I_{f_\eta,h_\eta}$ on $M_{f_\eta,h_\eta}$, we need the following
results.

\begin{lemma} \label{lem2.5} 
(i) For each $u\in M^+_{f_\eta,h_\eta}$, we have
$\int_{\mathbb{R}^N}h_\eta |u|^{r}dx>0$ and $I_{f_\eta,h_\eta}(u)<0$. In
particular $\alpha_{f_\eta,h_\eta}\leq\alpha^+_{f_\eta,h_\eta}<0$.

(ii) $I_{f_\eta,h_\eta}$ is coercive and bounded below on
$M_{f_\eta,h_\eta}$ for all $\eta\in
(0,(\frac{s-p}{s-r})^{\frac{1}{\beta}})$. Moreover,
$\alpha_{f_\eta,h_\eta}\to0$ as $\eta\to0$.
\end{lemma}

\begin{proof} (i) For each $u\in M^+_{f_\eta,h_\eta}$, we have
\begin{gather*}
(p-r)\|u\|^p-(s-r)\int_{\mathbb{R}^N}f_\eta |u|^{s}dx>0, \\
\|u\|^p=\int_{\mathbb{R}^N}f_\eta
|u|^{s}dx+\eta^{\frac{p(s-r)}{s-p}}\int_{\mathbb{R}^N}h_\eta
|u|^{r}dx.
\end{gather*}
By (C1), we have
$$
\eta^{\frac{p(s-r)}{s-p}}\int_{\mathbb{R}^N}h_\eta |u|^{r}dx
=\|u\|^p-\int_{\mathbb{R}^N}f_\eta
|u|^{s}dx>\frac{s-p}{p-r}\int_{\mathbb{R}^N}f_\eta |u|^{s}dx\geq0
$$
and
\begin{align*}
I_{f_\eta,h_\eta}(u)
&= (\frac{1}{p}-\frac{1}{s})\int_{\mathbb{R}^N}f_\eta
|u|^{s}dx+(\frac{1}{p}-\frac{1}{r})\eta^{\frac{p(s-r)}{s-p}}\int_{\mathbb{R}^N}h_\eta
|u|^{r}dx \\
&< (\frac{1}{p}-\frac{1}{s})\int_{\mathbb{R}^N}f_\eta
|u|^{s}dx+(\frac{1}{p}-\frac{1}{r})\frac{s-p}{p-r}\int_{\mathbb{R}^N}f_\eta
|u|^{s}dx \\
&= (s-p)(\frac{1}{ps}-\frac{1}{pr})\int_{\mathbb{R}^N}f_\eta
|u|^{s}dx\leq0
\end{align*}

(ii) For each $u\in M_{f_\eta,h_\eta}$, we have
$\|u\|^p=\int_{\mathbb{R}^N}f_\eta
|u|^{s}dx+\eta^{\frac{p(s-r)}{s-p}}\int_{\mathbb{R}^N}h_\eta |u|^{r}dx$.
Then by the H\"{o}lder and Young inequalities,
\begin{align*}
I_{f_\eta,h_\eta}(u)
&\geq \frac{s-p}{ps}\|u\|^p-\frac{s-r}{rs}\eta^\beta
 \|h\|_{L^{\frac{p}{p-r}}}\|u\|^r \\
&\geq (\frac{s-p}{ps}-\frac{s-r}{ps}\eta^\beta)\|u\|^p-
\eta^{\beta}\frac{(p-r)(s-r)}{prs}\|h\|^{\frac{p}{p-r}}_{L^{\frac{p}{p-r}}}.
\end{align*}
Thus, $I_{f_\eta,h_\eta}$ is coercive and bounded below on
$M_{f_\eta,h_\eta}$ for all $\eta\in
(0,(\frac{s-p}{s-r})^{\frac{1}{\beta}})$ and
$\alpha_{f_\eta,h_\eta}\to0$ as $\eta\to0$, where
$\beta=\frac{p(s-r)}{s-p}-\frac{p-r}{p}N>0$ as above.
\end{proof}


\section{Proofs of main results}

Now, we use the graph of the coefficient $f$ to find some
Palais-Smale sequences which are used to prove Theorem \ref{thm1.1}. For
$a>0$, let $C_a(x^i)$ denote the hypercube
$\Pi^N_{j=1}(x_j^i-a,x_j^i+a)$ centered at
$x^i=(x_1^i,x_2^i,\dots,x_N^i)$ for $i=1,2,\dots,k$. Let
$\overline{C_a(x^i)}$ and $\partial C_a(x^i)$ denote the closure
and the boundary of $C_a(x^i)$ respectively. By the conditions
(C1) and (C3), we can choose numbers $K,l>0$ such that $C_l(x^i)$
are disjoint, $f(x)<f(x^i)$ for $x\in\partial C_l(x^i)$ for all
$i=1,2,\dots,k$ and $\cup_{i=1}^kC_l(x^i)\subset
\Pi^N_{i=1}(-K,K)$.

Define $\phi_\eta\in C(R,R)$, $g_\eta\in(W^{1,p}(\mathbb{R}^N), \mathbb{R}^N)$ by
\begin{gather*}
\phi_\eta(t)= \begin{cases}
\frac{2K}{\eta} & t>\frac{2K}{\eta},\\
t & -\frac{2K}{\eta}\leq t\leq\frac{2K}{\eta},\\
-\frac{2K}{\eta}& t<-\frac{2K}{\eta}.
\end{cases}
\\
g_\eta^j(u)=\frac{\int_{\mathbb{R}^N}\phi_\eta(x_j)|u|^sdx}
{\int_{\mathbb{R}^N}|u|^sdx}\quad \text{for }j=1,2,\dots,N
\\
g_\eta(u)=(g^1_\eta(u),g^2_\eta(u),\dots,g^N_\eta(u))\in \mathbb{R}^N.
\end{gather*}
Let $C_{l/\eta}^i\equiv C_{l/\eta}(x^i/\eta)$,
\begin{gather*}
N_\eta^i=\{u\in M^-_{f_\eta,h_\eta}" u\geq0 \text{ and } g_\eta(u)\in
C_{l/\eta}^i\},\\
\partial N_\eta^i=\{u\in M^-_{f_\eta,h_\eta}:u\geq0 \text{ and }
 g_\eta(u)\in\partial C_{l/\eta}^i\}
\end{gather*}
for $i=1,2,\dots,k$. It is easy to
verify that $N_\eta^i$ and $\partial N_\eta^i$ are nonempty sets
for all $i=1,2,\dots,k$. Consider the minimization problems in
$N_\eta^i$ and $\partial N_\eta^i$ for $I_{f_\eta,h_\eta}$,
$$
\gamma_\eta^i=\inf_{u\in N_\eta^i}I_{f_\eta,h_\eta}(u),\quad
\overline{\gamma}_\eta^i=\inf_{u\in\partial
N_\eta^i}I_{f_\eta,h_\eta}(u).
$$ 
Using the results in \cite{s1}, we can
assume $w$ be a unique positive radial solution of
\begin{gather*}
 -\Delta_p u+|u|^{p-2}u=f_{\rm max} |u|^{s-2 }u\quad  x\in \mathbb{R}^N,\\
u>0\; \;x\in \mathbb{R}^N,\\
u\in W^{1,p}(\mathbb{R}^N)
\end{gather*}
and that $I_{f_{\rm max},0}(w)=\alpha_{f_{\rm max},0}$. By (C3)
and the routine computations, we have
$$
\alpha_{f_{\rm max},0}<\alpha_{f^\infty,0}.
$$
For small $\eta>0$ satisfying $2\sqrt{\eta}<1$, we define a 
function $\psi_\eta\in C^1(\mathbb{R}^N,[0,1])$ such that
\[
\psi_\eta(x)= \begin{cases}
1 & |x|<\frac{1}{2\sqrt{\eta}}-1,\\
0 & |x|>\frac{1}{2\sqrt{\eta}},
\end{cases}
\]
and $|\nabla\psi_\eta|\leq2$ in $\mathbb{R}^N$. Let
$x^\eta=\frac{1}{2\sqrt{\eta}}(1,1,\dots,1)\in \mathbb{R}^N$
and
$$
w_\eta(x)=t_\eta^-w(x-\frac{x^i}{\eta}+x^\eta)
\psi_\eta(x-\frac{x^i}{\eta}+x^\eta),
$$
where $t_\eta^->0$ are selected such that $w_\eta\in
M^-_{f_\eta,h_\eta}$. Then we have the following results.

 \begin{lemma} \label{lem3.1} 
As $\eta\to0$, we have
\begin{itemize}
\item[(i)] $\eta^{\frac{p(s-r)}{s-p}}\int_{\mathbb{R}^N}h_\eta
w^{r}(x-\frac{x^i}{\eta}+x^\eta)\psi_\eta^r(x-\frac{x^i}{\eta}+x^\eta)dx\to0$;

\item[(ii)] $t^-_\eta\to1$.
\end{itemize}
\end{lemma}

\begin{proof} 
(i) Since $\beta=\frac{p(s-r)}{s-p}-\frac{p-r}{p}N>0$
and $h_\eta(x)\geq0$, we have
\begin{align*}
0&\leq\ eta^{\frac{p(s-r)}{s-p}}\int_{\mathbb{R}^N}h_\eta
w^{r}(x-\frac{x^i}{\eta}+x^\eta)\psi_\eta^r(x-\frac{x^i}{\eta}+x^\eta)dx \\
&\leq \eta^\beta \|h\|_{L^{\frac{p}{p-r}}}
\|w(x-\frac{x^i}{\eta}+x^\eta)\psi_\eta(x-\frac{x^i}{\eta}+x^\eta)\|^r
\end{align*}
and
$$
\|w(x-\frac{x^i}{\eta}+x^\eta)\psi_\eta(x-\frac{x^i}{\eta}
+x^\eta)\|^p\to\frac{sp}{s-p}\alpha_{f_{\rm max},0}.
$$
Thus (i) holds.

(ii) Since $w_\eta\in M^-_{f_\eta,h_\eta}$, we have
\begin{align*}
&(t^-_\eta)^p[\int_{\mathbb{R}^N}|\nabla
(w(x-\frac{x^i}{\eta}+x^\eta)\psi_\eta(x-\frac{x^i}{\eta}+x^\eta))|^p\\
&+(w(x-\frac{x^i}{\eta}+x^\eta)\psi_\eta(x-\frac{x^i}{\eta}+x^\eta))^p] \\
&= (t^-_\eta)^s\int_{\mathbb{R}^N}f_\eta
w^s(x-\frac{x^i}{\eta}+x^\eta)\psi^s_\eta(x-\frac{x^i}{\eta}+x^\eta)dx \\
&\quad +\eta^{\frac{p(s-r)}{s-p}}(t^-_\eta)^r\int_{\mathbb{R}^N}h_\eta
w^r(x-\frac{x^i}{\eta}+x^\eta)\psi^r_\eta(x-\frac{x^i}{\eta}+x^\eta)dx.
\end{align*}
 When $\eta\to0$,
from part (i) it follows that
\begin{align*}
(t^-_\eta)^p(\|w\|^p+o(\eta))
&= (t^-_\eta)^p\|w(x-\frac{x^i}{\eta}+x^\eta)\psi_\eta(x-\frac{x^i}{\eta}
 +x^\eta)\|^p+o(\eta) \\
&= (t^-_\eta)^s\int_{\mathbb{R}^N}f_\eta
w^s(x-\frac{x^i}{\eta}+x^\eta)\psi^s_\eta(x-\frac{x^i}{\eta}+x^\eta)dx+o(\eta) \\
&= (t^-_\eta)^s\int_{\mathbb{R}^N}f(\eta x+x^i-\eta
x^\eta)w^sdx+o(\eta).
\end{align*}
Moreover, $\eta x^\eta\to0$ as $\eta\to0$, and
from $\|w\|^p=\int_{\mathbb{R}^N}f_{\rm max}w^sdx$, we
have
\begin{align*}
t^-_\eta
&> t_{\rm max}=(\frac{p-r}{s-r}\frac{\|w(x-\frac{x^i}{\eta}+x^\eta)\psi_\eta(x-\frac{x^i}{\eta}+x^\eta)\|^{p}}{\int_{\mathbb{R}^N}f_\eta
|w(x-\frac{x^i}{\eta}+x^\eta)\psi_\eta(x-\frac{x^i}{\eta}+x^\eta)|^{s}dx})^{\frac{1}{s-p}} \\
&\to (\frac{p-r}{s-r})^{\frac{1}{s-p}}>0.
\end{align*}
Thus,
$t^-_\eta\to1$ as $\eta\to0$ and (ii) holds.
\end{proof}

Let
$\eta_*=\min\{\eta_1,\eta_2,(\frac{s-p}{s-r})^{\frac{1}{\beta}}\}$,
then we have the following result.

\begin{lemma} \label{lem3.2} 
For each $\varepsilon>0$, there exists
$\eta_\varepsilon\in(0,\eta_*] $ such that
$$
\alpha^-_{f_\eta,h_\eta}\leq\gamma_\eta^i<\min\{\alpha_{f_{\rm max},0}
+\varepsilon,
\alpha_{f_\eta,h_\eta}+\alpha_{f^\infty,0}\},\quad
i=1,2,\dots,k,\;\eta\in(0,\eta_\varepsilon).
$$
\end{lemma}

\begin{proof} For $i=1,2,\dots,k$, obviously we have
$\alpha^-_{f_\eta,h_\eta}\leq\gamma_\eta^i$.

Now we show the second inequality hold. First, we prove that
$g_\eta(w_\eta)\in C^i_{l/\eta}$.
For $j=1,2,\dots,N$,
since
$$
g^j_\eta(w_\eta)=\frac{\int_{\mathbb{R}^N}\phi_\eta(x_j)
w^s(x-\frac{x^i}{\eta}+x^\eta)\psi^s_\eta(x-\frac{x^i}{\eta}+x^\eta)dx}
{\int_{\mathbb{R}^N}w^s(x-\frac{x^i}{\eta}+x^\eta)
\psi^s_\eta(x-\frac{x^i}{\eta}+x^\eta)dx}
$$
and
$$
\psi_\eta(x-\frac{x^i}{\eta}+x^\eta)=0\quad
\text{if }|x_j-\frac{x_j^i}{\eta}|>\frac{1}{\sqrt{\eta}}.
$$
By the definition of $\psi_\eta$, we have
$$
g^j_\eta(w_\eta)=\frac{\int_{C^i_{l/\eta}}\phi_\eta(x_j)w^s
(x-\frac{x^i}{\eta}+x^\eta)\psi^s_\eta(x-\frac{x^i}{\eta}+x^\eta)dx}
{\int_{C^i_{l/\eta}}w^s(x-\frac{x^i}{\eta}+x^\eta)
\psi^s_\eta(x-\frac{x^i}{\eta}+x^\eta)dx}
$$
provided $\frac{1}{\sqrt{\eta}}<\frac{l}{\eta}$. From the
definition of $\phi_\eta$ and $g_\eta$ we conclude that
$g_\eta(w_\eta)\in C^i_{l/\eta}$. Thus, $w_\eta\in
N^i_\eta$. Moreover, by Lemma \ref{lem3.1}, we obtain
\begin{align*}
I_{f_\eta,h_\eta}(w_\eta)
&= \frac{(t^-_\eta)^p}{p}[\int_{\mathbb{R}^N}|\nabla
(w(x-\frac{x^i}{\eta}+x^\eta)\psi_\eta(x-\frac{x^i}{\eta}+x^\eta))|^pdx \\
&\quad +\int_{\mathbb{R}^N}|w(x-\frac{x^i}{\eta}+x^\eta)
 \psi_\eta(x-\frac{x^i}{\eta}+x^\eta)|^pdx] \\
&\quad -\frac{(t^-_\eta)^s}{s}\int_{\mathbb{R}^N}f_\eta
w^s(x-\frac{x^i}{\eta}+x^\eta)\psi^s_\eta(x-\frac{x^i}{\eta}+x^\eta)dx \\
&\quad -\eta^{\frac{p(s-r)}{s-p}}\frac{(t^-_\eta)^r}{r}
 \int_{\mathbb{R}^N}h_\eta
w^r(x-\frac{x^i}{\eta}+x^\eta)\psi^r_\eta(x-\frac{x^i}{\eta}+x^\eta)dx \\
&= \frac{1}{p}\int_{\mathbb{R}^N}|\nabla
w|^p+|w|^pdx-\frac{1}{s}\int_{\mathbb{R}^N}f(\eta x+x^i-\eta
x^\eta)w^sdx+o(\eta).
\end{align*}
Since $\eta x^\eta\to0$ as $\eta\to0$ and from the
above, we have
$$
I_{f_\eta,h_\eta}(w_\eta)=I_{f_{\rm max},0}(w)+o(\eta)
=\alpha_{f_{\rm max},0}+o(\eta).
$$
Therefore, for any $\varepsilon>0$ there exists $\eta_3>0$ such that
$$
\gamma^i_\eta<\alpha_{f_{\rm max},0}+\varepsilon,\;i=1,2,\dots,k,\;\eta\in
(0,\eta_3).
$$ 
Moreover, $\alpha_{f_{\rm max},0}<\alpha_{f^\infty,0}$
and $\alpha_{f_\eta,h_\eta}\to0$ as $\eta\to0$,
then there exists $\eta_4>0$ such that
$$
\gamma^i_\eta<\alpha_{f_\eta,h_\eta}+\alpha_{f^\infty,0},\quad
i=1,2,\dots,k,\;\eta\in (0,\eta_4).
$$ 
We take $\eta_\varepsilon=\min\{\eta_3,\eta_4\}$, this implies
$$
\gamma^i_\eta<\min\{\alpha_{f_{\rm max},0}+\varepsilon,
\alpha_{f_\eta,h_\eta}+\alpha_{f^\infty,0}\},
$$
for $i=1,2,\dots,k$ and $\eta\in (0,\eta_\varepsilon)$. This
completes the proof.
\end{proof}


Since $W^{1,p}(\mathbb{R}^N)$ is not a Hilbert space in general, even
if the $(PS)$ sequence $\{u_n\}$ of $I_\lambda(u)$ is bounded,
hence there exists $u\in W^{1,p}(\mathbb{R}^N)$ such
that
$$
u_n\rightharpoonup u\quad\text{in } W^{1,p}(\mathbb{R}^N),
$$
we can not ensure
$$
|\nabla u_{n_k}|^{p-2}\nabla u_{n_k}\rightharpoonup |\nabla
u|^{p-2}\nabla u  \;\;\text{in} \;\;L^{\frac{p}{p-1}}(\mathbb{R}^N)
$$
for some subsequence $\{u_{n_k}\}$ of $\{u_n\}$, so we can not use
Brezis-Lieb lemma \cite{t1} directly. We use the following results.

\begin{lemma} \label{lem3.3} 
If $\{u_n\}\subset W^{1,p}(\mathbb{R}^N)$ is a $(PS)_c$ sequence of 
$I_{f_\eta,h_\eta}$, then there exists a subsequence $\{u_k\}$ such 
that $u_k\rightharpoonup u_0$ in $W^{1,p}(\mathbb{R}^N)$ for some 
$u_0\in W^{1,p}(\mathbb{R}^N)$, and
$I'(u_0)=0$, $\nabla u_k\to\nabla u_0$ a.e. in $\mathbb{R}^N$.
\end{lemma}

The proof of the above lemma was given in \cite[Lemma 2.1]{c5}, also 
in \cite{l2}.  We omit it here.


\begin{lemma} \label{lem3.4} 
There are positive numbers $\delta$ and
$\eta_\delta\in (0,\eta_*]$ such that for $i=1,2,\dots,k$, we
have
$$
\widetilde{\gamma_\eta^i}>\alpha_{f_{\rm max},0}+\delta\quad
\text{for all }\eta\in(0,\eta_\delta).
$$
\end{lemma}

\begin{proof} Fix $i\in\{1,2,\dots,k\}$. Suppose the contrary that
there exists a sequence $\{\eta_n\}$ with $\eta_n\to0$ as
$n\to\infty$ such that
$\widetilde{\gamma_{\eta_n}^i}\to
c\leq\alpha_{f_{\rm max},0}$. Consequently, there exists a sequence
$\{u_n\}\subset\partial N^i_{\eta_n}$ such that
$g_{\eta_n}(u_n)\in\partial C^i_{\frac{l}{\eta_n}}$ and
\begin{gather*}
\langle I'_{f_{\eta_n},h_{\eta_n}}(u_n),u_n\rangle=0, \\
I_{f_{\eta_n},h_{\eta_n}}(u_n)\to c\leq\alpha_{f_{\rm max},0}.
\end{gather*}
By Lemma \ref{lem2.5}, $\{u_n\}$ is uniformly
bounded in $W^{1,p}(\mathbb{R}^N)$. For $u_n\in
M^-_{f_{\eta_n},h_{\eta_n}}$, we deduce from the Sobolev imbedding
theorem that there exists a constant $\nu>0$ such that
$\|u_n\|>\nu$ for all $n$. Applying the concentration-compactness
principle of  Lions \cite{l1} to $|u_n|^s$, there are positive
constants $R,\mu$ and $\{y_n\}\subset \mathbb{R}^N$ such that
$$
\int_{B^N(y_n,R)}|u_n|^sdx\geq\mu\quad \text{for all }n,
$$
where $B^N(y_n,R)=\{x\in\mathbb{R}^N||x-y_n|<R\}$. Let
$\widetilde{u}_n=u_n(x+y_n)$, and define
$$
\widetilde{f}_{\eta_n}(x)=f(\eta_nx+\eta_ny_n),\;\widetilde{h}_{\eta_n}(x)
=h(\eta_nx+\eta_ny_n).
$$
Then we have
\begin{equation} \label{e3.1}
\begin{gathered}
\langle I'_{\widetilde{f}_{\eta_n},\widetilde{h}_{\eta_n}}(\widetilde{u}_n),
 \widetilde{u}_n\rangle=0, \\
I_{\widetilde{f}_{\eta_n},\widetilde{h}_{\eta_n}}(\widetilde{u}_n)\to c.
\end{gathered}
\end{equation}
By Lemma \ref{lem3.3}, Sobolev imbedding theorem  and Riesz's theorem,
there is a $u_0\in W^{1,p}(\mathbb{R}^N)$ and a subsequence of
$\{\widetilde{u}_n\}$, still denoted by $\{\widetilde{u}_n\}$ such
that
\begin{gather*}
\widetilde{u}_n\rightharpoonup u_0 \quad\text{in }W^{1,p}(\mathbb{R}^N), \\
\widetilde{u}_n \to u_0\quad \text{a.e. in }\mathbb{R}^N, \\
\int_{B^N(0,R)}|\widetilde{u}_n|^sdx \to \int_{B^N(0,R)}|u_0|^sdx\geq\mu,
\end{gather*}
and
\begin{gather*}
\nabla \widetilde{u}_n \to \nabla u_0 \quad \text{a.e. in }\mathbb{R}^N, \\
|\nabla \widetilde{u}_{n}|^{p-2}\nabla
\widetilde{u}_{n} \rightharpoonup  |\nabla u_0|^{p-2}\nabla u_0
\quad\text{in } L^{\frac{p}{p-1}}(\mathbb{R}^N).
\end{gather*}
Set $w_n=\widetilde{u}_n-u_0$. By the Brezis-Lieb lemma \cite{t1}, we
have
\begin{equation}
\int_{\mathbb{R}^N}\widetilde{f}_{\eta_n}|\widetilde{u}_{n}|^sdx
=\int_{\mathbb{R}^N}\widetilde{f}_{\eta_n}|u_{0}|^sdx
+\int_{\mathbb{R}^N}\widetilde{f}_{\eta_n}|w_{n}|^sdx+o(1).\label{e3.2}
\end{equation}
Since $\{u_n\}$ is uniformly bounded and
$\widetilde{u}_n\rightharpoonup u_0$, we obtain
\begin{equation}
\eta_n^{\frac{p(s-r)}{s-p}}\int_{\mathbb{R}^N}h_{\eta_n}|u_n|^rdx
=\eta_n^{\frac{p(s-r)}{s-p}}\int_{\mathbb{R}^N}\widetilde{h}_{\eta_n}
|\widetilde{u}_n|^rdx\to0\quad\text{as }n\to\infty\label{e3.3}
\end{equation}
and
\begin{equation}
\|\widetilde{u}_n\|^p=\|u_0\|^p+\|w_n\|^p+o(1).\label{e3.4}
\end{equation}
Combining \eqref{e3.1}-\eqref{e3.4}, we have
\begin{equation}
\|w_n\|^p-\int_{\mathbb{R}^N}\widetilde{f}_{\eta_n}|w_{n}|^sdx
=-(\|u_0\|^p-\int_{\mathbb{R}^N}\widetilde{f}_{\eta_n}|u_{0}|^sdx)+o(1).
\label{e3.5}
\end{equation}
 We distinguish the two cases: (A) $\|w_n\|\to0$ and
(B) $\|w_n\|\to c>0$.

\noindent\textbf{Case (A):} 
From condition (C3) we can choose a positive
constant $\delta$ such that
$$
f(x)<f_{\max}\;\text{for}\;x\in
\overline{C}^i_{l+\delta}\backslash C^i_{l-\delta}.
$$ 
We complete the proof by establishing the contradiction
$$
\lim_{n\to\infty}I_{f_{\eta_n},h_{\eta_n}}(u_n)>\alpha_{f_{\rm max},0}.
$$
Consider the sequence $\{\eta_ny_n\}$. By passing to a subsequence
if necessary, we may assume that one of the following cases
occur:
\begin{itemize}
\item[(A1)] $\{\eta_ny_n\}\subset\overline{C}^i_{l+\delta}\backslash
C^i_{l-\delta}$,
\item[(A2)] $\{\eta_ny_n\}\subset\overline{C}^i_{l-\delta}$,
\item[(A3)] $\{\eta_ny_n\}\subset\mathbb{R}^N\backslash C^i_{l+\delta}$
and $\{\eta_ny_n\}$ is bounded;
\item[(A4)] $\{\eta_ny_n\}$ is unbounded.
\end{itemize}
Let $\epsilon>0$ and $R_\epsilon>0$ be such that
\begin{equation}
\frac{\int_{|x|\geq R_\epsilon}|\widetilde{u}_n|^sdx}{\int_{\mathbb{R}^N}
|\widetilde{u}_n|^sdx}\leq\epsilon.\label{e3.6}
\end{equation}

In case (A1), we assume $\eta_ny_n\to \widetilde{y}\in 
\overline{C}^i_{l+\delta}\backslash C^i_{l-\delta}$ and
 $f(\widetilde{y})<f_{\max}$. Consequently by
\eqref{e3.3} and \eqref{e3.4}, we have
\begin{align*}
\lim_{n\to\infty}I_{f_{\eta_n},h_{\eta_n}}(u_n) 
&= \lim_{n\to\infty}[\frac{1}{p}\|\widetilde{u}_n\|^p-\frac{1}{s}\int_{\mathbb{R}^N}\widetilde{f}_{\eta_n}(x)|\widetilde{u}_n|^sdx-\eta_n^{\frac{p(s-r)}{s-p}}\int_{\mathbb{R}^N}\widetilde{h}_{\eta_n}|\widetilde{u}_n|^rdx] \\
&= \frac{1}{p}\|u_0\|^p-\frac{1}{s}\int_{\mathbb{R}^N}f(\widetilde{y})|u_0|^sdx \\
&\geq \alpha_{f(\widetilde{y}),0} 
>\alpha_{f_{\max},0},
\end{align*}
we also have 
$$
\|u_0\|^p-\int_{\mathbb{R}^N}f(\widetilde{y})|u_0|^sdx=0,
$$ 
which is a contradiction.

In case (A2),
\begin{align*}
 g_{\eta_n}^j(u_n)
&= \frac{\int_{\mathbb{R}^N}\phi_{\eta_n}(x_j+(y_n)_j)
 |\widetilde{u}_n|^sdx}{\int_{\mathbb{R}^N}|\widetilde{u}_n|^sdx} \\
&= \frac{\int_{|x|\leq
R_\epsilon}\phi_{\eta_n}(x_j+(y_n)_j)|\widetilde{u}_n|^sdx
 +\int_{|x|\geq R_\epsilon}\phi_{\eta_n}(x_j+(y_n)_j)|\widetilde{u}_n|^sdx}
 {\int_{\mathbb{R}^N}|\widetilde{u}_n|^sdx}.
\end{align*}
In the region $|x|\leq R_\epsilon$, when $n$ is sufficiently
large, we have
\[
x_j+(y_n)_j\in(\frac{x^i_j-(l-\delta)}{\eta_n}-R_\epsilon,
\frac{x^i_j+(l-\delta)}{\eta_n}+R_\epsilon) 
\subset (-\frac{2K}{\eta_n},\frac{2K}{\eta_n}).
\]
It follows from \eqref{e3.6} and the definition of $\phi_{\eta_n}$ that
\begin{gather*}
g_{\eta_n}^j(u_n)>(\frac{x^i_j-(l-\delta)}{\eta_n}-R_\epsilon)(1-\epsilon)
-\frac{2K}{\eta_n}\epsilon,\\
g_{\eta_n}^j(u_n)<(\frac{x^i_j+(l-\delta)}{\eta_n}
+R_\epsilon)(1-\epsilon)+\frac{2K}{\eta_n}\epsilon.
\end{gather*}
It is clear from the above inequalities that we can choose
$\epsilon>0$, $\delta>\epsilon$ sufficiently small such that
$$
g_{\eta_n}^j(u_n)\in(\frac{x^i_j-l}{\eta_n},\frac{x^i_j+l}{\eta_n})
$$
for $n$ large enough, which contradicts $g_{\eta_n}(u_n)\in
\partial C^i_{l/\eta_n}$.

In case (A3), we  assume that $\eta_ny_n\to
\widetilde{y} \not\in C^i_{l+\delta}$ as
$n\to\infty$, then for some $j\in\{1,2,\dots,N\}$, we
have $\widetilde{y}_j\geq x_j^i+(l+\delta)$ or
$\widetilde{y}_j\leq x_j^i-(l+\delta)$.

 First, we assume
$\widetilde{y}_j\geq x_j^i+(l+\delta)$ occurs, then
$(y_n)_j>\frac{x^i_j+(l+\frac{\delta}{2})}{\eta_n}$ for all $n$.
When $|x_j|\leq R_\epsilon$, we have
$$
x_j+(y_n)_j>\frac{x^i_j+(l+\frac{\delta}{2})}{\eta_n}-R_\epsilon
$$
and
$$
g_{\eta_n}^j(u_n)>(\frac{x^i_j+(l+\frac{\delta}{2})}{\eta_n}-R_\epsilon)
(1-\epsilon)-\frac{2K}{\eta_n}\epsilon>\frac{x^i_j+l}{\eta_n},
$$
for sufficiently small $\epsilon>0$, $\delta>\epsilon$ and $n$
large enough. This contradicts to 
$g_{\eta_n}(u_n)\in \partial C^i_{\frac{l}{\eta_n}}$.
When $\widetilde{y}_j\leq x_j^i-(l+\delta)$, the argument is
similar.

In case (A4), we assume $\eta_ny_n\to\infty$ as
$n\to\infty$, using a similar argument to case (A1) and
condition (C3), we can also obtain a contradiction.

\noindent\textbf{Case (B):} 
Set 
$$
\|u_0\|^p-\int_{\mathbb{R}^N}\widetilde{f}_{\eta_n}|u_{0}|^sdx=A+o(1),
$$
then by \eqref{e3.5},
$$
\|w_n\|^p-\int_{\mathbb{R}^N}\widetilde{f}_{\eta_n}|w_{n}|^sdx=-A+o(1).
$$ 
Without loss of generality, we may assume that $A>0$($A<0$ can be considered
similarly). We can choose a sequence $\{t_n\}$ with
$t_n\to1$ as $n\to\infty$ such that $v_n=t_nw_n$
satisfies
$$
\|v_n\|^p-\int_{\mathbb{R}^N}\widetilde{f}_{\eta_n}|v_{n}|^sdx=-A.
$$ 
Since
 $u_0\in M_{\widetilde{f}_{\eta_n},0}(A+o(1))$, by \eqref{e3.2}-\eqref{e3.4} 
and Lemma \ref{lem2.1} we have
\begin{align*}
I_{f_{\eta_n},h_{\eta_n}}(u_n)
&= \frac{1}{p}\|u_0\|^p-\frac{1}{s}\int_{\mathbb{R}^N}\widetilde{f}_{\eta_n}(x)
 |u_0|^sdx \\
&\quad +\frac{1}{p}\|w_n\|^p-\frac{1}{s}
 \int_{\mathbb{R}^N}\widetilde{f}_{\eta_n}(x)|w_n|^sdx+o(1) \\
&\geq \frac{A+o(1)}{p}+\frac{1}{p}\|v_n\|^p-\frac{1}{s}
 \int_{\mathbb{R}^N}\widetilde{f}_{\eta_n}(x)|v_n|^sdx+o(1) \\
&\geq \alpha_{\widetilde{f}_{\eta_n},0}(A)
 +\alpha_{\widetilde{f}_{\eta_n},0}(-A)+o(1) \\
&> \alpha_{\widetilde{f}_{\eta_n},0}+\frac{s-p}{2sp}A+o(1) \\
&\geq \alpha_{f_{\max},0}+\frac{s-p}{2sp}A+o(1),
\end{align*}
which is a contradiction. If $A=0$, we can find two sequences
$\{t_n\}$ and $\{s_n\}$ with $t_n\to1$, $s_n\to1$
as $n\to\infty$ such that
$\overline{w}_n=t_nw_n,\;\overline{v}_n=s_nu_0$ satisfy
\begin{gather*}
\|\overline{w}_n\|^p-\int_{\mathbb{R}^N}\widetilde{f}_{\eta_n}|
\overline{w}_{n}|^sdx=0,\\
\|\overline{v}_n\|^p-\int_{\mathbb{R}^N}\widetilde{f}_{\eta_n}
|\overline{v}_{n}|^sdx=0.
\end{gather*} 
Thus
\begin{align*}
&\lim_{n\to\infty}I_{f_{\eta_n},h_{\eta_n}}(u_n)\\
&= \lim_{n\to\infty}\Big[\frac{1}{p}\|\overline{w}_n\|^p-\frac{1}{s}
 \int_{\mathbb{R}^N}\widetilde{f}_{\eta_n}|\overline{w}_{n}|^sdx 
 +\frac{1}{p}\|\overline{v}_n\|^p-\frac{1}{s}
 \int_{\mathbb{R}^N}\widetilde{f}_{\eta_n}|\overline{v}_{n}|^sdx\Big] \\
&>\alpha_{f_{\max},0},
\end{align*}
which is a contradiction. This completes the proof.
\end{proof}

From now on, taking $\delta>0$ as in Lemma \ref{lem3.4}, and fixing 
$\varepsilon>0$ such that $\varepsilon\leq \delta$, consider
 $\eta_\varepsilon$ as in Lemma \ref{lem3.2}, $\eta_\delta$ as in
 Lemma \ref{lem3.4}, 
 and denote
$\eta_0=\min\{\eta_\varepsilon,\eta_\delta\}$.

 \begin{lemma} \label{lem3.5} 
If $\eta\in (0,\eta_0)$, then for each $u\in M_{f_\eta,h_\eta}$, there exist
$\varepsilon_u>0$ and a differentiable function
$\xi_u:B(0,\varepsilon_u)\subset W^{1,p}(\mathbb{R}^N)\to R^+$
such that $\xi_u(0)=1,\;\xi_u(v)(u-v)\in M_{f_\eta,h_\eta}$, and
\begin{align*}
\langle\xi_u'(0),v\rangle
&=\Big[p\int_{\mathbb{R}^N}|\nabla u|^{p-2}\nabla u\nabla v
+|u|^{p-2}uvdx\\
&\quad -s\int_{\mathbb{R}^N}f_\eta|u|^{s-2}uvdx
 -\eta^{\frac{p(s-r)}{s-p}}r\int_{\mathbb{R}^N}h_\eta|u|^{r-2}uvdx\Big]\\
&\quad\div\Big[(p-r)\|u\|^p-(s-r)\int_{\mathbb{R}^N}f_\eta|u|^sdx\Big]
\end{align*}
for all $v\in W^{1,p}(\mathbb{R}^N)$.
\end{lemma}

\begin{proof} For $u\in M_{f_\eta,h_\eta}$, define a function
$F:R\times W^{1,p}(\mathbb{R}^N)\to R$ by
\begin{align*}
F_u(\xi_u,w)
&= \langle I'_{f_\eta,h_\eta}(\xi_u(u-w)),\xi_u(u-w)\rangle \\
&= \xi_u^p\int_{\mathbb{R}^N}|\nabla(u-w)|^p+|u-w|^pdx
 -\xi_u^s\int_{\mathbb{R}^N}f_\eta|u-w|^sdx \\
&\quad -\eta^{\frac{p(s-r)}{s-p}}\xi_u^r\int_{\mathbb{R}^N}h_\eta|u-w|^rdx.
\end{align*}
Then $F_u(1,0)=\langle I'_{f_\eta,h_\eta}(u),u\rangle=0$ and
\begin{align*}
\frac{d}{d\xi_u}F_u(1,0)
&= p\|u\|^p-s\int_{\mathbb{R}^N}f_\eta|u|^sdx
 -\eta^{\frac{p(s-r)}{s-p}}r\int_{\mathbb{R}^N}h_\eta|u|^rdx \\
&= (p-r)\|u\|^p-(s-r)\int_{\mathbb{R}^N}f_\eta|u|^sdx\neq 0.
\end{align*}
According to the implicit function theorem, there exist
$\varepsilon_u>0$ and a differentiable function
$\xi_u:B(0,\varepsilon_u)\subset W^{1,p}(\mathbb{R}^N)\to R^+$
such that $\xi_u(0)=1$, and
\begin{align*}
\langle\xi_u'(0),v\rangle
&=\Big[p\int_{\mathbb{R}^N}|\nabla u|^{p-2}\nabla
u\nabla v+|u|^{p-2}uvdx\\
&\quad -s\int_{\mathbb{R}^N}f_\eta|u|^{s-2}uvdx
-\eta^{\frac{p(s-r)}{s-p}}r\int_{\mathbb{R}^N}h_\eta|u|^{r-2}uvdx\Big]\\
&\quad \div \Big[(p-r)\|u\|^p-(s-r)\int_{\mathbb{R}^N}f_\eta|u|^sdx\Big],
\end{align*}
and
$F_u(\xi_u(v),v)=0$ for all $v\in B(0,\varepsilon_u)$,
which is equivalent to
$$
\langle
I'_{f_\eta,h_\eta}(\xi_u(v)(u-w)),\xi_u(v)(u-w)\rangle=0,\quad
\forall v\in B(0,\varepsilon_u).
$$ 
That is, $\xi_u(v)(u-v)\in M_{f_\eta,h_\eta}$.
\end{proof}


\begin{lemma} \label{lem3.6} 
If $\eta\in (0,\eta_0)$, then for each $u\in
N^i_{\eta}$, there exist $\varepsilon_u>0$ and a differentiable
function $\xi_u^-:B(0,\varepsilon_u)\subset W^{1,p}(\mathbb{R}^N)\to R^+$ 
such that $\xi_u^-(0)=1,\;\xi_u^-(v)(u-v)\in N^i_{\eta}$ for all 
$v\in B(0,\varepsilon_u)$, and
\begin{align*}
\langle(\xi_u^-)'(0),v\rangle
&=\Big[p\int_{\mathbb{R}^N}|
\nabla u|^{p-2}\nabla u\nabla v+|u|^{p-2}uvdx\\
&\quad -s\int_{\mathbb{R}^N}f_\eta|u|^{s-2}uvdx
-\eta^{\frac{p(s-r)}{s-p}}r\int_{\mathbb{R}^N}h_\eta|u|^{r-2}uvdx\Big]\\
&\quad\div \Big[(p-r)\|u\|^p-(s-r)\int_{\mathbb{R}^N}f_\eta|u|^sdx\Big]
\end{align*}
for all $v\in W^{1,p}(R^N)$.
\end{lemma}

\begin{proof} 
Similar to the argument in Lemma \ref{lem3.5}, there exist
$\varepsilon_u>0$ and a differentiable function
$\xi_u^-:B(0,\varepsilon_u)\subset W^{1,p}(\mathbb{R}^N)\to R^+$
such that $\xi_u^-(0)=1$, $\xi_u^-(v)(u-v)\in M_{f_\eta,h_\eta}$ for
all $v\in B(0,\varepsilon_u)$, Since
$$
(p-r)\|u\|^p-(s-r)\int_{\mathbb{R}^N}f_\eta |u|^{s}dx<0,
$$ 
thus, if $\varepsilon_u$ small enough, by the continuity of the functions
$\xi_u^-$ and $g_\eta$, we have
\begin{align*}
&\langle\psi'(\xi_u^-(v)(u-v))),\xi_u^-(v)(u-v)\rangle \\
&=  (p-r)\|\xi_u^-(v)(u-v)\|^p-(s-r)\int_{\mathbb{R}^N}f_\eta
|\xi_u^-(v)(u-v)|^{s}dx <0.
\end{align*}
and $g_\eta(\xi_u^-(v)(u-v))\in C^i_{l/\eta}$.
\end{proof}


\begin{proposition} \label{prop3.7} 
(i) If $\eta\in (0,\eta_0)$, then there
exists a $(PS)_{\alpha_{f_\eta,h_\eta}}$ sequence $\{u_n\}\subset
M_{f_\eta,h_\eta}$ in $W^{1,p}(\mathbb{R}^N)$ for
$I_{f_\eta,h_\eta}$.

(ii) If $\eta\in (0,\eta_0)$, then there exists a
$(PS)_{\gamma_\eta^i}$ sequence $\{u_n\}\subset N_\eta^i$ in
$W^{1,p}(\mathbb{R}^N)$ for $I_{f_\eta,h_\eta}$, $i=1,2,\dots,k$.
\end{proposition}

\begin{proof} 
Since the proof of (i) is similar to that of (ii),
but simpler, we only prove (ii) here.
We denote by $\overline{N_\eta^i}$ the closure of $N_\eta^i$, then
we note that
$$
\overline{N_\eta^i}=N_\eta^i\cup\partial N_\eta^i,\quad
\text{for each }i=1,2,\dots,k.
$$ 
From Lemma \ref{lem3.2} and Lemma \ref{lem3.4}, we obtain
\begin{equation}
\gamma_\eta^i<\min\{\alpha_{f_\eta,h_\eta}
+\alpha_{f^\infty,0},\widetilde{\gamma_\eta^i}\},\quad
i=1,2,\dots,k,\;\eta\in(0,\eta_0).\label{e3.7}
\end{equation}
Hence
$$
\gamma_\eta^i=\inf\{I_{f_\eta,h_\eta}(u):u\in \overline{N_\eta^i}\}
\quad\text{for }i=1,2,\dots,k.
$$
Fix some $i\in\{1,2,\dots,k\}$. Applying the Ekeland variational principle
\cite{l2} there exists a minimizing sequence
$\{u_n\}\subset\overline{N_\eta^i}$ such that
\begin{gather}
I_{f_\eta,h_\eta}(u_n)<\gamma_\eta^i+\frac{1}{n},\label{e3.8} \\
I_{f_\eta,h_\eta}(u_n)<I_{f_\eta,h_\eta}(w)+\frac{1}{n}\|w-u_n\|\quad
\text{for all }w\in \overline{N_\eta^i}.\label{e3.9}
\end{gather}
From \eqref{e3.7} we may assume that $u_n\in N_\eta^i$ for $n$ sufficiently
large. Applying Lemma \ref{lem3.6} with $u=u_n$ we obtain the functional
$\xi^-_{u_n}:B(0,\varepsilon_{u_n})\to R$ for some $\varepsilon_{u_n}>0$
such that $\xi^-_{u_n}(w)(u_n-w)\in N^i_{\eta}$. Choose
$0<\rho<\varepsilon_{u_n}$ and $u\in W^{1,p}(\mathbb{R}^N)$ with
$u\not\equiv0$. Set $w_\rho=\frac{\rho u}{\|u\|}$ and
$z^n_\rho=\xi^-_{u_n}(w_\rho)(u_n-w_\rho)$. Since $z^n_\rho\in N^i_{\eta}$,
we deduce from \eqref{e3.9} that
$$
I_{f_\eta,h_\eta}(z^n_\rho)-I_{f_\eta,h_\eta}(u_n)
\geq-\frac{1}{n}\|z^n_\rho-u_n\|.
$$
By the mean value theorem, we have
$$
\langle I'_{f_\eta,h_\eta}(u_n),z^n_\rho-u_n\rangle+o(\|z^n_\rho-u_n\|)
\geq-\frac{1}{n}\|z^n_\rho-u_n\|.
$$
Thus,
\begin{equation}
\langle I'_{f_\eta,h_\eta}(u_n),-w_\rho\rangle+(\xi^-_{u_n}(w_\rho)-1)\langle
I'_{f_\eta,h_\eta}(u_n),u_n-w_\rho\rangle
\geq-\frac{1}{n}\|z^n_\rho-u_n\|+o(\|z^n_\rho-u_n\|).\label{e3.10}
\end{equation}
Since $\xi^-_{u_n}(w_\rho)(u_n-w_\rho)\in N^i_{\eta}$ and consequently
from \eqref{e3.10} we obtain
\begin{align*}
&-\rho\langle I'_{f_\eta,h_\eta}(u_n),\frac{u}{\|u\|}\rangle
 +(\xi^-_{u_n}(w_\rho)-1)\langle
I'_{f_\eta,h_\eta}(u_n)-I'_{f_\eta,h_\eta}(z^n_\rho),u_n-w_\rho\rangle \\
&\geq -\frac{1}{n}\|z^n_\rho-u_n\|+o(\|z^n_\rho-u_n\|).
\end{align*}
Thus,
\begin{equation}
\begin{aligned}
 \langle I'_{f_\eta,h_\eta}(u_n),\frac{u}{\|u\|}\rangle
&\leq\frac{(\xi^-_{u_n}(w_\rho)-1)}{\rho}\langle
I'_{f_\eta,h_\eta}(u_n)-I'_{f_\eta,h_\eta}(z^n_\rho),u_n-w_\rho\rangle\\
&\quad +\frac{\|z^n_\rho-u_n\|}{n\rho}+\frac{o(\|z^n_\rho-u_n\|)}{\rho}.
\end{aligned}\label{e3.11}
\end{equation}
Since
$$
\|z^n_\rho-u_n\|\leq\rho|\xi^-_{u_n}(w_\rho)|+|\xi^-_{u_n}(w_\rho)-1|\|u_n\|
$$
and
$$
\lim_{\rho\to0}\frac{|\xi^-_{u_n}(w_\rho)-1|}{\rho}\leq\|(\xi^-_{u_n})'(0)\|,
$$
if we let $\rho\to0$ in \eqref{e3.11} for a fixed $n$, and by
Lemma \ref{lem2.5} (ii) we can find a constant $C>0$, independent of $\rho$,
such that
$$
\langle I'_{f_\eta,h_\eta}(u_n),\frac{u}{\|u\|}\rangle
\leq\frac{C}{n}(1+\|(\xi^-_{u_n})'(0)\|).
$$
We are done once we show that $\|(\xi^-_{u_n})'(0)\|$ is uniformly bounded
in $n$. By Lemma \ref{lem2.5} (ii), Lemma \ref{lem3.6} and the H\"{o}lder inequality,
we have
$$
\langle{(\xi^-_{u_n})}'(0),v\rangle\leq\frac{b\|v\|}
{|(p-r)\|u_n\|^p-(s-r)\int_{\mathbb{R}^N}f_\eta|u_n|^sdx|}\quad
\text{for some }b>0.
$$
We only need to show that
$$
|(p-r)\|u_n\|^p-(s-r)\int_{\mathbb{R}^N}f_\eta|u_n|^sdx|>C
$$
for some $C>0$ and $n$ large. We argue by way of contradiction.
Assume that there exists a subsequence $\{u_n\}$ satisfy
\begin{equation}
(p-r)\|u_n\|^p-(s-r)\int_{\mathbb{R}^N}f_\eta|u_n|^sdx=o(1).\label{e3.12}
\end{equation}
By the fact that $u_n\in M^-_{f_\eta,h_\eta}(u_n)$ and \eqref{e3.12}, we
obtain that
$$
\int_{\mathbb{R}^N}f_\eta|u_n|^sdx>0.
$$
So we have
\begin{gather}
\|u_n\|\leq[\frac{s-r}{s-p}\eta^\beta\|h\|_{L^{\frac{p}{p-r}}}]^{\frac{1}{p-r}}
+o(1)\label{e3.13} \\
\|u_n\|>(\frac{p-r}{s-r}\frac{S^s}{f_{\max}})^{\frac{1}{s-p}}+o(1).\label{e3.14}
\end{gather}
Then
\begin{equation}
\begin{aligned}
K(u_n)&=c(s,r)\Big[\frac{(\frac{s-r}{p-r}\int_{\mathbb{R}^N}f_\eta
|u_n|^{s}dx+o(1))^{\frac{s-1}{p-1}}}{\int_{\mathbb{R}^N}f_\eta
|u_n|^{s}dx}\Big]^{\frac{p-1}{s-p}}
 -\frac{s-p}{p-r}
\int_{\mathbb{R}^N}f_\eta |u_n|^{s}dx\\
&=o(1).
\end{aligned}\label{e3.15}
\end{equation}
However, by
\eqref{e3.13}-\eqref{e3.14}, the H\"{o}lder and Sobolev inequalities, combining
with $\beta>0$ and $\eta\in (0,\eta_0)$, we have
\begin{align*}
K(u_n)
&\geq  c(s,r)(\frac{\|u_n\|^{p\frac{s-1}{p-1}}}{\int_{\mathbb{R}^N}f_\eta
|u_n|^{s}dx})^{\frac{p-1}{s-p}}-\eta^\beta\|h\|_{L^{\frac{p}{p-r}}}\|u_n\|^r+o(1) \\
&\geq \|u_n\|^r[c(s,r)(\frac{S^s}{f_{\rm max}})^{\frac{p-1}{s-p}}\|u_n\|^{1-r}-\eta^\beta\|h\|_{L^{\frac{p}{p-r}}}]+o(1) \\
&\geq \|u_n\|^r[c(s,r)(\frac{S^s}{f_{\rm max}})^{\frac{p-1}{s-p}}(\eta^\beta\frac{s-r}{s-p}\|h\|_{L^{\frac{p}{p-r}}})^{\frac{1-r}{p-r}}-\eta^\beta\|h\|_{L^{\frac{p}{p-r}}}]+o(1) \\
&\geq d
\end{align*}
for some $d>0$ and $n$ large enough. This is a contradiction to
\eqref{e3.15}.
So we have
$$
I_{f_\eta,h_\eta}(u_n)=\gamma_\eta^i+o(1)
$$
and
$I'_{f_\eta,h_\eta}(u_n)=0\;\text{in}\;W^{-1}(\mathbb{R}^N)$.
Thus we complete the proof of (ii).
\end{proof}

 \begin{theorem} \label{thm3.8} 
For each $\eta\in(0,\eta_0)$, Equation \eqref{Ee} has a positive
solution $u_\eta\in M^+_{f_\eta,h_\eta}$ such that
$I_{f_\eta,h_\eta}(u_\eta)=\alpha_{f_\eta,h_\eta}=\alpha^+_{f_\eta,h_\eta}$.
\end{theorem}

\begin{proof} By Proposition \ref{prop3.7} (i), there exists a
$(PS)_{\alpha_{f_\eta,h_\eta}}$ sequence $\{u_n\}\subset
M_{f_\eta,h_\eta}$, by Lemma \ref{lem2.5} (ii) and Lemma \ref{lem3.3}, there exist a
subsequence $\{u_n\}$ and $u_\eta$ in $W^{1,p}(\mathbb{R}^N)$ such
that
\begin{gather*}
u_n\rightharpoonup u_\eta\quad \text{weakly in } W^{1,p}(\mathbb{R}^N), \\
u_n\to u_\eta\quad\text{a.e. in }\mathbb{R}^N, \\
u_n\to u_\eta\quad \text{in $L^q(\mathbb{R}^N)$ for} 1\leq q\leq p^*, \\
\nabla u_n\to\nabla u_\eta \quad\text{a.e. in }\mathbb{R}^N, \\
|\nabla u_{n}|^{p-2}\nabla u_{n}\rightharpoonup |\nabla
u_\eta|^{p-2}\nabla u_\eta \quad\text{in }L^{\frac{p}{p-1}}(\mathbb{R}^N),
\end{gather*}
It is easy to see that $u_\eta$ is a solution of \eqref{Ee}

 Moreover, by
the Egorov theorem and the H\"{o}lder inequality and condition
$h\in L^{\frac{p}{p-r}}(\mathbb{R}^N)$, we obtain
$$
\int_{\mathbb{R}^N}h_\eta|u_n|^rdx\to\int_{\mathbb{R}^N}h_\eta|u_\eta|^rdx.
$$
 We claim that $\int_{\mathbb{R}^N}h_\eta|u_\eta|^rdx\neq0$. If not,
$$
\|u_n\|^p=\int_{\mathbb{R}^N}f_\eta|u_n|^sdx+o(1),
$$
and
\begin{align*}
&(\frac{1}{p}-\frac{1}{s})\int_{\mathbb{R}^N}f_\eta|u_n|^sdx\\
&= \frac{1}{p}\|u_n\|^p-\frac{1}{s}\int_{\mathbb{R}^N}f_\eta
|u_n|^{s}dx-\eta^{\frac{p(s-r)}{s-p}}\frac{1}{r}\int_{\mathbb{R}^N}h_\eta
|u_n|^{r}dx+o(1) \\
&= \alpha_{f_\eta,h_\eta}+o(1),
\end{align*}
this contradicts $\alpha_{f_\eta,h_\eta}<0$. Thus, $u_\eta$ is
a nontrivial solution of \eqref{Ee}. Now we show that
$u_n\to u_\eta$ strongly in $W^{1,p}(\mathbb{R}^N)$. If not,
$\|u_\eta\|<\liminf_{n\to\infty}\|u_n\|$, so we have
\begin{align*}
\alpha_{f_\eta,h_\eta}
&\leq I_{f_\eta,h_\eta}(u_\eta)
 =(\frac{1}{p}-\frac{1}{s})\|u_\eta\|^p-(\frac{1}{r}-\frac{1}{s})
 \eta^{\frac{p(s-r)}{s-p}}\int_{\mathbb{R}^N}h_\eta
|u_\eta|^{r}dx \\
&< \lim_{n\to\infty}I_{f_\eta,h_\eta}(u_n)=\alpha_{f_\eta,h_\eta},
\end{align*}
this is a contradiction. Thus $I_{f_\eta,h_\eta}(u_\eta)=\alpha_{f_\eta,h_\eta}$. 
At last, we show $u_\eta\in M^+_{f_\eta,h_\eta}$. If not, 
 by Lemma \ref{lem2.2},
we know that $u_\eta\in M^-_{f_\eta,h_\eta}$, by Lemma \ref{lem2.4}, there exist
unique $t_0^+$ and $t_0^-$ such that $t_0^+u_\eta\in
M^+_{f_\eta,h_\eta}$ and $t_0^-u_\eta\in M^-_{f_\eta,h_\eta}$, and
$t_0^+<t_0^-=1$.
Since
$$
\frac{d}{dt}I_{f_\eta,h_\eta}(t_0^+u_\eta)=0,\quad
\frac{d^2}{dt^2}I_{f_\eta,h_\eta}(t_0^+u_\eta)>0,
$$
there exists $\widetilde{t}\in (t_0^+,t_0^-]$ such that
$I_{f_\eta,h_\eta}(t_0^+u_\eta)<I_{f_\eta,h_\eta}(\widetilde{t}u_\eta)$.
By Lemma \ref{lem2.4},
$$
I_{f_\eta,h_\eta}(t_0^+u_\eta)<I_{f_\eta,h_\eta}(\widetilde{t}u_\eta)\leq
I_{f_\eta,h_\eta}(t_0^-u_\eta)=I_{f_\eta,h_\eta}(u_\eta),
$$ 
which is a contradiction. Thus,
$I_{f_\eta,h_\eta}(u_\eta)=\alpha_{f_\eta,h_\eta}=\alpha^+_{f_\eta,h_\eta}$.
Since $I_{f_\eta,h_\eta}(u_\eta)=I_{f_\eta,h_\eta}(|u_\eta|)$ and
$|u_\eta|\in M^+_{f_\eta,h_\eta}$, by Lemma \ref{lem2.3} and the maximum
principle, we may assume that $u_\eta$ is a positive solution of
\eqref{Ee}.
\end{proof}

\begin{proposition} \label{prop3.9}
Assume that $\{u_n\}\subset M^-_{f_\eta,h_\eta}$ is a $(PS)_c$ sequence, where
$c<\alpha_{f_\eta,h_\eta}+\alpha_{f^\infty,0}$. Then there exists
a subsequence, still denoted by $\{u_n\}$, and $u_0$ in
$W^{1,p}(\mathbb{R}^N)$ such that $u_n\to u_0$ strongly in
$W^{1,p}(\mathbb{R}^N)$ and $I_{f_\eta,h_\eta}(u_0)=c$.
\end{proposition}

\begin{proof} 
By Lemma \ref{lem2.5} (ii), there exists a subsequence
$\{u_n\}$ and $u_0$ in $W^{1,p}(\mathbb{R}^N)$ such
that
$$
u_n\rightharpoonup u_0\quad \text{weakly in }W^{1,p}(\mathbb{R}^N).
$$ 
First, we claim that $u_0\equiv0$ is impossible. If not,
by $h\in L^{\frac{p}{p-r}}(\mathbb{R}^N)$, the Egorov theorem and the
H\"{o}lder inequality, we have 
\begin{equation}
\|u_n\|^p=o(1)\label{e3.16}.
\end{equation}
Moreover, $\{u_n\}\subset M^-_{f_\eta,h_\eta}$, we deduce
from the Sobolev imbedding theorem that
$$
\|u_n\|>C\quad \text{for some }C>0, \;n=1,2,\dots.
$$
which contradicts to \eqref{e3.16}. Thus, by Lemma \ref{lem3.3},
$u_0$ is a nontrivial solution of \eqref{Ee} and
$I_{f_\eta,h_\eta}(u_0)\geq\alpha_{f_\eta,h_\eta}$. We write
$u_n=u_0+v_n$ with $v_n\rightharpoonup0$ weakly in
 $W^{1,p}(\mathbb{R}^N)$. By the Brezis-Lieb lemma \cite{l1}, we have
\begin{align*}
\int_{\mathbb{R}^N}f_\eta|u_n|^pdx
&= \int_{\mathbb{R}^N}f_\eta|u_0|^pdx+\int_{\mathbb{R}^N}f_\eta|v_n|^pdx+o(1) \\
&= \int_{\mathbb{R}^N}f_\eta|u_0|^pdx+\int_{\mathbb{R}^N}f^\infty|v_n|^pdx+o(1).
\end{align*}
Since $\{u_n\}$ is a bounded sequence in $W^{1,p}(\mathbb{R}^N)$, we
have $\{v_n\}$ is also a bounded sequence in $W^{1,p}(\mathbb{R}^N)$.
Moreover, by $h\in L^{\frac{p}{p-r}}(\mathbb{R}^N)$,  the Egorov
theorem and the H\"{o}lder inequality, we have
$$
\int_{\mathbb{R}^N}h_\eta|v_n|^rdx
=\int_{\mathbb{R}^N}h_\eta|u_n|^rdx-\int_{\mathbb{R}^N}h_\eta|u_0|^rdx+o(1)=o(1).
$$
Hence, for $n$ large enough, we can conclude that
\begin{align*}
\alpha_{f_\eta,h_\eta}+\alpha_{f^\infty,0}
&> I_{f_\eta,h_\eta}(u_0+v_n) \\
&\geq I_{f_\eta,h_\eta}(u_0)+\frac{1}{p}\|v_n\|^p
-\frac{1}{s} \int_{\mathbb{R}^N}f^\infty|v_n|^sdx+o(1) \\
&\geq \alpha_{f_\eta,h_\eta}+\frac{1}{p}\|v_n\|^p
-\frac{1}{s}\int_{\mathbb{R}^N}f^\infty|v_n|^sdx+o(1),
\end{align*}
we obtain
\begin{equation}
\frac{1}{p}\|v_n\|^p-\frac{1}{s}\int_{\mathbb{R}^N}f^\infty|v_n|^sdx
<\alpha_{f^\infty,0}+o(1).\label{e3.17}
\end{equation}
Also from $I'_{f_\eta,h_\eta}(u_n)=o(1)$ in $W^{-1}(\mathbb{R}^N)$,
 $\{u_n\}$ is uniformly bounded and $u_0$ is a solution of \eqref{Ee}, we
obtain
\begin{equation}
\langle I'_{f_\eta,h_\eta}(u_n),u_n\rangle
=\|v_n\|^p-\int_{\mathbb{R}^N}f^\infty|v_n|^sdx+o(1)=o(1).\label{e3.18}
\end{equation}
We claim that \eqref{e3.17} and \eqref{e3.18} can be hold simultaneously
only if $\{v_n\}$ admits a subsequence which converges strongly to zero. If not,
then $\|v_n\|$ is bounded away from zero; that is,
$$
\|v_n\|\geq C \quad \text{for some }C>0.
$$
From \eqref{e3.18}, it follows that
$$
\int_{\mathbb{R}^N}f^\infty|v_n|^sdx
\geq\frac{sp}{s-p}\alpha_{f^\infty,0}+o(1).
$$
By \eqref{e3.17} and \eqref{e3.18}, for $n$ large enough
\begin{align*}
\alpha_{f^\infty,0}
&\leq (\frac{1}{p}-\frac{1}{s})\int_{\mathbb{R}^N}f^\infty|v_n|^sdx+o(1) \\
&= \frac{1}{p}\|v_n\|^p-\frac{1}{s}\int_{\mathbb{R}^N}f^\infty|v_n|^sdx+o(1) 
< \alpha_{f^\infty,0},
\end{align*}
which is a contradiction. Therefore, $u_n\to u_0$ strongly
in $W^{1,p}(R^N)$ and $I_{f_\eta,h_\eta}(u_0)=c$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.1}]
 By Lemma \ref{lem3.2}, Proposition \ref{prop3.7} and 
Proposition \ref{prop3.9}, 
for each $\eta\in(0,\eta_0)$ and $i\in \{1,2,\dots,k\}$, 
there exist a sequence $\{u^i_n\}\subset N^i_\eta$ and 
$u^i_0\in W^{1,p}(\mathbb{R}^N)\backslash\{0\}$ such
that
\begin{gather*}
I_{f_\eta,h_\eta}(u^i_n)=\gamma_\eta^i+o(1),\\
I'_{f_\eta,h_\eta}(u^i_n)=o(1)
\end{gather*}
and $u_n^i\to u_0^i$ strongly in $W^{1,p}(\mathbb{R}^N)$.
Obviously, the function $u_0^i$ is a solution of the equation
\eqref{Ee} and $I_{f_\eta,h_\eta}(u^i_0)=\gamma_\eta^i$.
Similar to the argument in Theorem \ref{thm3.8}, we have $u_0^i$ is
positive. Since $g^i_\eta(u_0^i)\in
\overline{C_{l/\eta}(x^i)}$, $u_\eta\in
M^+_{f_\eta,h_\eta}$ and $u_0^i\in M^-_{f_\eta,h_\eta}$, where
$u_\eta$ is a positive solution of Eq.\eqref{Ee} as in 
Theorem \ref{thm3.8}. This implies $u_\eta$, $u_0^i$ and $u_0^j$ are different for
$i\neq j$.

Letting $\lambda_0=\eta_0^{-p}$,
$U_\lambda(x)=\lambda^{\frac{1}{s-p}}u_\eta(\lambda^{1/p}x)$
and $U_i(x)=\lambda^{\frac{1}{s-p}}u_0^i(\lambda^{1/p}x)$.
We obtain $U_\lambda$ and $U_i$ are positive solutions of the
\eqref{El} with $i=1,2,\dots,k$. This completes the
proof.
\end{proof}

\begin{remark} \label{rmk3.10} \rm
 It is easy to see from the proof of Theorem \ref{thm1.1}
that the solutions $U_\lambda,\;U_i(i=1,2,\dots,k)$ satisfy
\begin{enumerate}
\item $\|U_\lambda\|_{L^\infty(\mathbb{R}^N)}, \;
\|U_i\|_{L^\infty(\mathbb{R}^N)}\to\infty$ as $\lambda\to\infty$;

\item $\|U_\lambda\|_{L^p(\mathbb{R}^N)},\;
\|U_i\|_{L^p(\mathbb{R}^N)}\to\infty$ as $\lambda\to\infty$ if
$p<s<\frac{p^2}{N}+p$;

\item $\|U_\lambda\|_{L^p(\mathbb{R}^N)},\;
\|U_i\|_{L^p(\mathbb{R}^N)}\to0$ as $\lambda\to\infty$ if
$\frac{p^2}{N}+p<s<p^*$.
\end{enumerate}
\end{remark}

\begin{lemma} \label{lemA} When
$1\leq r<p<s<p^*$ and $N\geq 1$, we have
$\frac{p(s-r)}{s-p}-\frac{(p-r)N}{p}>0$.
\end{lemma}

\begin{proof} When $N\leq p$ and $1\leq r<p<s<p^*$,
obviously, we have
$$
\frac{p(s-r)}{s-p}-\frac{(p-r)N}{p}>0.
$$ 
We consider only the case $N>p$.
Set 
$$
L(s)=p^2(s-r)-(p-r)N(s-p),\quad s\in(p,p^*).
$$ 
Then it is easy to see that
$$
L(s)\geq\min\{L(p),L(p^*)\}=\min\{p^2(p-r),\frac{p^3r}{N-p}\}>0.
$$
This completes the proof.
\end{proof}

\subsection*{Acknowledgements}
The authors wish to thank the anonymous reviewers and the editor for 
their helpful comments. We would like to thank Professot D. M. Cao and 
T. F. Wu for their help and advice for completing this article. 
This research was supported by the
National Natural Science Foundation of China (No. 11171092); the
Natural Science Foundation of the Jiangsu Higher Education
Institutions of China (No. 08KJB110005); 
the Natural Science Foundation of Jiangsu Education Office (No. 12KJB110002)

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\end{document}