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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 96, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/96\hfil Existence and concentration of solutions]
{Existence and concentration of solutions for a
$p$-laplace equation with potentials in $\mathbb{R}^N$}

\author[M. Wu, Z. Yang\hfil EJDE-2010/96\hfilneg]
{Mingzhu Wu, Zuodong Yang}  % in alphabetical order

\address{Mingzhu Wu \newline
Institute of Mathematics, School of Mathematical  Science, 
Nanjing Normal University,
Jiangsu Nanjing 210046, China}
\email{wumingzhu\_2010@163.com}

\address{Zuodong Yang \newline
Institute of Mathematics, School of
Mathematical  Science, Nanjing Normal University,
Jiangsu Nanjing 210046, China. \newline
College of Zhongbei, Nanjing Normal University, 
Jiangsu Nanjing 210046, China}
\email{zdyang\_jin@263.net}

\thanks{Submitted January 22, 2010. Published July 15, 2010.}
\thanks{Supported by grants 10871060 from the National Natural
Science Foundation of China, \hfill\break\indent and 08KJB110005
from the Natural Science Foundation of  the Jiangsu Higher
Education   \hfill\break\indent Institutions of China}

\subjclass[2000]{35J25, 35J60} 
\keywords{Potentials; critical point theory; concentration;
 existence; \hfill\break\indent concentration-compactness; $p$-Laplace}

\begin{abstract}
 We study the $p$-Laplace equation with Potentials
 $$
 -\operatorname{div}(|\nabla u|^{p-2}\nabla u)+\lambda
 V(x)|u|^{p-2}u=|u|^{q-2}u,
 $$
 $u\in W^{1,p}(\mathbb{R}^N)$, $x\in \mathbb{R}^N$ where
 $2\leq p$, $p<q<p^{*}$. Using a concentration-compactness principle
 from critical point theory, we obtain existence,
 multiplicity solutions, and concentration of solutions.
\end{abstract}

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

\section{Introduction}

 This article  concerns the existence and the multiplicity
of decaying solutions for the equation
\begin{equation}
-\operatorname{div}(|\nabla u|^{p-2}\nabla u)+V(x)|u|^{p-2}u
=|u|^{q-2}u,\quad x\in \mathbb{R}^N \label{e1.1}
\end{equation}
and for the related equations
\begin{equation}
-\operatorname{div}(|\nabla u|^{p-2}\nabla u)
+\lambda V(x)|u|^{p-2}u=|u|^{q-2}u,\quad x\in \mathbb{R}^N,
 \label{e1.2}
\end{equation}
and
\begin{equation}
-{\varepsilon^{p}}\operatorname{div}(|\nabla u|^{p-2}\nabla
u)+V(x)|u|^{p-2}u=|u|^{q-2}u,\quad x\in
\mathbb{R}^N \label{e1.3}
\end{equation}
respectively as
$\lambda\to\infty$, and $\varepsilon\to 0$. We also
consider concentration of solutions as $\lambda\to\infty$
or $\varepsilon\to 0$.

We assume throughout that $V$ and $p$, $q$ satisfy the following
conditions:
\begin{itemize}
\item[(V1)] $V\in C(\mathbb{R}^N)$ and $V$ is bounded.

\item[(V2)] There exists $b>0$ such that the set ${\{x\in
\mathbb{R}^N: V(x)<b}\}$ is nonempty and has finite measure.

\item[(P1)] $2\leq p$, $p<q<p^{*}$ where $p^{*}={\frac{pN}{N-p}}$
 if $N>p$ and $p^{*}=\infty$ if $1\leq N\leq p$.
\end{itemize}

Note that if $\varepsilon^{p}=\lambda^{-1}$, then $u$ is a solution
of \eqref{e1.2} if and only if  $v=\lambda^{\frac{-1}{q-p}}u$ is
a solution of \eqref{e1.3}, hence as far as the existence and the
number of solutions
are concerned, these two problems are equivalent.

$\|u\|_{p}$ will denote the usual $L^{p}(\mathbb{R}^N)$ norm and
$V^{\pm}(x)=\max{\{\pm V(x),0}\}$. $B_{\rho}$ and $S_{\rho}$ will
respectively denote the open ball and the sphere of radius $\rho$
and center at the origin.

It is well known that the functional
$$
\Phi(u)={\frac{1}{p}}\int_{\mathbb{R}^N}(|\nabla
u|^{p}+V(x)|u|^{p})dx-{\frac{1}{q}}\int_{\mathbb{R}^N}|u|^{q}dx
$$
is of class $C^{1}$ in the Sobolev space
\begin{equation}
E={\{u\in W^{1,p}(\mathbb{R}^N):\|u\|^{p}=\int_{\mathbb{R}^N}(|\nabla
u|^{p}+V^{+}(x)|u|^{p})dx<\infty}\} \label{e1.4}
\end{equation}
 and critical points of $\Phi$ correspond to solutions $u$
of \eqref{e1.1}. Moreover,
$u(x)\to 0$ as $|x|\to\infty$. It is easy to see
that if
\begin{equation}
M=\inf_{u\in E\backslash \{0\}}{\frac{\int_{\mathbb{R}^N}(|\nabla
u|^{p}+V(x)|u|^{p})dx}{\|u\|^{p}_{q}}}\; \label{e1.5}
\end{equation}
is attained
at some $\overline{u}$ and $M$ is positive, then
$u=M^{\frac{1}{q-p}}\overline{u}/\|\overline{u}\|_{q}$ is a solution
of \eqref{e1.1} and $u(x)\to 0$ as $|x|\to\infty$. Such $u$
is called a ground state. Because we have Poincar\'e
inequality
$$
\int_{\Omega}|u|^{p}dx\leq C\int_{\Omega}|\nabla u|^{p}dx,\quad
1\leq p<+\infty,\quad u\in W^{1,p}_{0}(\Omega)
$$
so  $E$ is continuously embedded in $W^{1,p}(\mathbb{R}^N)$.

Recently, there have been numerous works for the  eigenvalue
problem
\begin{equation}
\begin{gathered}
-\operatorname{div}(|\nabla u|^{p-2}\nabla u)=V(x)|u|^{p-2}u\\
u\in D^{1,p}_{0}(\Omega),\quad  u\neq 0
\end{gathered}
 \label{e1.6}
\end{equation}
where $\Omega\subseteq \mathbb{R}^N$. We can see \cite{b1,d2,r1,s1}
for different approaches.  Szulkin and Willem \cite{s1} generalized
several earlier results concerning the existence of an infinite
sequence of eigenvalues.

Consider the quasilinear elliptic equation
\begin{equation}
\begin{gathered}
-\operatorname{div}(|\nabla u|^{p-2}\nabla
u)+\lambda |u|^{p-2}u=f(x,u),\quad  \text{in } \Omega\\
u\in W^{1,p}_{0}(\Omega),\quad  u\neq 0
\end{gathered} \label{e1.7}
\end{equation}
where $1<p<N$, $N\geq 3$, $\lambda$ is a parameter, $\Omega$ is an
unbounded domain in $\mathbb{R}^N$. Existence of  solutions to
 \eqref{e1.7} has been  investigation in the previoius decade, see
for example \cite{c1,d1,l1,l2,y1,y2}.
Because of the unboundedness of the domain, the Sobolev compact
embedding do not hold. There are some methods to overcome
this difficulty. In \cite{y2},
 the authors used the concentration-compactness principle
posed by  Lions and the mountain pass lemma to solve problem
\eqref{e1.3}. In \cite{y1}, the author use that the projection $u\mapsto
f(x,u)$ is weak continuous in $W^{1,p}_{0}(\Omega)$ to consider
the problem. In \cite{b6,b7}, the authors study the problem in symmetric
Sobolev spaces which possess Sobolev compact embedding. By the
result and a min-max procedure formulated by Bahri and Li
\cite{b3}, they considered the existence of positive solutions of
$$
-\operatorname{div}(|\nabla u|^{p-2}\nabla u)
+u^{p-1}=q(x)u^{\alpha}\quad
\mbox{in}\quad \mathbb{R}^N,
$$
where $q(x)$ satisfies certain conditions.

When $p=2$,  problem \eqref{e1.1} has been studied in
\cite{a1,b2,b4,b5,c3,d3,d3,d5}.
 In \cite{g1}, a quasilinear problem in
bounded domains was considered with Hardy type potentials.
To the best of our knowledge, there is very little work on the
case $p\neq 2$ for problem \eqref{e1.1}.


 From its first appearance in the work by Lions \cite{l1,l2}, the
concentration-compactness principle in  calculus of variations
has been widely used and by many authors. In fact, one should
refer to the two concentration-compactness principles, as
``escape to infinity" and ``concentration around points"
as treated separately, originally.
This seemingly harmless dichotomy however
often leads to rather cumbersome and tricky calculations. To get
rid of these difficulties, some authors have developed variants
that encompass both possible loss of compactness in a whole; see
for instance Ben-Naoum et al. \cite{b8} and Bianchi et al. \cite{b9}
which seem to be the first works in this direction. When using the
original principle or its variants, it is necessary beforehand to
discover the so-called limiting problems that are responsible for
non-compactness. Often, these are related to the invariance of the
considered functional and constraint under a non-compact group;
translations and dilations being the two most studied.


Motivated by the results in
\cite{b4,b9,c2,c3,d3,d4,d5,g1,l2,s2,y1},
 we obtain the existence and the multiplicity of
solutions in Theorems \ref{thm1}--\ref{thm3}
 by using critical point theory.  By
Theorems \ref{thm4} and \ref{thm5}, we can obtain the concentration of
solutions.


This paper is organized as follows. In Section 2, we state some
condition and many lemmas which we need in the proof of the main
Theorem. In Section 3, we give the proof of the main result of the
paper.

\section{Preliminaries}

\begin{lemma} \label{lem1}
 Let $\Omega\subseteq \mathbb{R}^N$ be an open subset.
$(u_n)\subseteq W_{0}^{1,p}(\Omega)$ be a sequence such
that $u_n\rightharpoonup u$ in $W_{0}^{1,p}(\Omega)$ and
$p\geq 2$. Then
$$
\lim_{n\to\infty}\int_{\Omega}|\nabla u_n|^{p}dx
\geq \lim_{n\to\infty}\int_{\Omega}|\nabla u_n-{\nabla
u}|^{p}dx+\int_{\Omega}|\nabla u|^{p}dx.
$$
\end{lemma}

\begin{proof}
 When $p=2$ from Lieb Lemma we have
$$
\lim_{n\to\infty}\int_{\Omega}|\nabla u_n|^{2}dx
=\lim_{n\to\infty}\int_{\Omega}|\nabla u_n-{\nabla
u}|^{2}dx+\int_{\Omega}|\nabla u|^{2}dx.
$$
For $3\geq p>2$, using
the lower semi-continuity of the $L^{p}$-norm with respect to the
weak convergence and $u_n\rightharpoonup u$ in $W^{1,p}(\Omega)$,
we deduce
$$
\lim_{n\to\infty}\langle|\nabla u_n|^{p-2}{\nabla u_n},
{\nabla u_n}\rangle\geq \langle|\nabla u|^{p-2}{\nabla u}, {\nabla
u}\rangle
$$
and
\begin{align*}
&\lim_{n\to\infty}\langle |{\nabla u_n}-\nabla
u|^{p-2}({\nabla u_n}-{\nabla u}),{\nabla u_n}-{\nabla
u}\rangle\\
&=0 \geq \lim_{n\to\infty}\langle |{\nabla u_n}-\nabla
u|^{p-2}({\nabla u}-{\nabla u}),{\nabla u}-{\nabla u}\rangle.
\end{align*}
 So
\begin{align*}
\lim_{n\to\infty}\langle|{\nabla u_n}-\nabla u|^{p-2}{\nabla
u_n},{\nabla u_n}\rangle
&\geq \lim_{n\to\infty}\langle|{\nabla u_n}-\nabla
u|^{p-2}{\nabla u_n},{\nabla u}\rangle \\
&=\lim_{n\to\infty}\langle|{\nabla u_n}-\nabla
u|^{p-2}{\nabla u},{\nabla u_n}\rangle\\
&=\lim_{n\to\infty}\langle|{\nabla u_n}-\nabla
u|^{p-2}{\nabla u},{\nabla u}\rangle.
\end{align*}
Then
\begin{align*}
&\lim_{n\to\infty}\int_{\Omega}(|\nabla u_{n}|^{p}-|\nabla
u|^{p})dx\\
& =\lim_{n\to\infty}\int_{\Omega}|\nabla
u_{n}|^{p-2}(|\nabla u_{n}|^{2}-|\nabla
u|^{2})dx+\lim_{n\to\infty}\int_{\Omega}(|\nabla
u_{n}|^{p-2}-|\nabla u|^{p-2})|\nabla u|^{2}dx\\
&=\lim_{n\to\infty}\int_{\Omega}(|\nabla
u_{n}|^{p-2}+|\nabla u|^{p-2})(|\nabla u_{n}|^{2}-|\nabla
u|^{2})dx\\
&\quad +\lim_{n\to\infty}\int_{\Omega}(|\nabla
u_{n}|^{p-2}|\nabla u|^{2}-|\nabla u|^{p-2}|\nabla
u_{n}|^{2})dx.
\end{align*}
  From $u_n\rightharpoonup u$ in $W^{1,p}(\Omega)$,
$$
\lim_{n\to\infty}\int_{\Omega}(|\nabla
u_{n}|^{p-2}|\nabla u|^{2}-|\nabla u|^{p-2}|\nabla
u_{n}|^{2})dx=0.
$$
So
\begin{align*}
\lim_{n\to\infty}\int_{\Omega}(|\nabla u_{n}|^{p}-|\nabla
u|^{p})dx
&=\lim_{n\to\infty}\int_{\Omega}(|\nabla
u_{n}|^{p-2}+|\nabla u|^{p-2})(|\nabla u_{n}|^{2}-|\nabla
u|^{2})dx\\
& \geq \lim_{n\to\infty}\int_{\Omega}|\nabla
u_{n}-\nabla u|^{p-2}(|\nabla u_{n}|^{2}-|\nabla u|^{2}).
\end{align*}
So we have
\begin{align*}
&\lim_{n\to\infty}\langle|{\nabla u_n}|^{p-2}{\nabla
u_n},{\nabla u_n}\rangle+\langle|{\nabla u_n}-\nabla
u|^{p-2}{\nabla u},{\nabla u_n}\rangle+\langle|{\nabla u_n}-\nabla
u|^{p-2}{\nabla
u_n},{\nabla u}\rangle\\
&\geq \lim_{n\to\infty}\langle|{\nabla u_n}-\nabla
u|^{p-2}{\nabla u_n},{\nabla u_n}\rangle\\
&\quad +\lim_{n\to\infty}\langle|{\nabla u_n}-\nabla
u|^{p-2}{\nabla u},{\nabla u}\rangle+\langle|{\nabla
u}|^{p-2}{\nabla u},{\nabla u}\rangle.
\end{align*}
Then
\begin{align*}
&\lim_{n\to\infty}\langle|{\nabla u_n}|^{p-2}{\nabla
u_n},{\nabla u_n}\rangle \\
&\geq
\lim_{n\to\infty}\langle|{\nabla u_n}-\nabla
u|^{p-2}{{\nabla u_n} -{\nabla u}},{\nabla u_n}-{\nabla u}\rangle
+\langle|{\nabla u}|^{p-2}{\nabla u},{\nabla u}\rangle.
\end{align*}
and
$$
\lim_{n\to\infty}\int_{\Omega}|\nabla u_n|^{p}dx
\geq \lim_{n\to\infty}\int_{\Omega}|\nabla u_n-{\nabla
u}|^{p}dx+\int_{\Omega}|\nabla u|^{p}dx.
$$


For $p>3$, there exist a $k\in N$ that $0<p-k\leq 1$. Then, we
only need to prove the  inequality
$$
\lim_{n\to\infty}\int_{\Omega}(|\nabla u_{n}|^{p}-|\nabla
u|^{p})dx \geq \lim_{n\to\infty}\int_{\Omega}|\nabla
u_{n}-\nabla u|^{p-k}(|\nabla u_{n}|^{k}-|\nabla u|^{k}).
$$
The proof of this iequality is similar to the above, so we omit it.
Therefore, the lemma is proved.
\end{proof}

Let
$V_{b}(x)=\max{\{V(x),b}\}$ and
\begin{equation}
M_b=\inf_{u\in E\setminus {\{0}\}}{\frac{\int_{\mathbb{R}^N}(|\nabla
u|^{p}+V_{b}(x)|u|^{p})dx}{\|u\|^{p}_{q}}}. \label{e2.1}
\end{equation}
Denote the spectrum of $-\Delta_{p}+V$ in $L^{p}(\mathbb{R}^N)$
by $\sigma(-\Delta_{p}+V)$ and recall the definition \eqref{e1.5} of
$M$.

\begin{lemma} \label{lem2}
 Suppose {\rm (V1), (V2), (P1)} are satisfied and
$\sigma(-\Delta_{p}+V)\subset (0,\infty)$. If $M<M_{b}$, then
each minimizing sequence for $M$ has a convergent subsequence. So in
particular, $M$ is attained at some $u\in E\setminus {\{0}\}$.
\end{lemma}

\begin{proof}
Let ${\{u_m}\}$ be a minimizing sequence. We may
assume $\|u_m\|_{q}=1$. Since $V<0$ on a set of finite measure,
${\{u_m}\}$ is bounded in the norm of $E$ given by $\eqref{e1.4}$.
Passing to a subsequence we may assume $u_m\rightharpoonup u$ in $E$
and by the continuity of the embedding $E\hookrightarrow
W^{1,p}(\mathbb{R}^N)$, $u_m\to u$ in
$L^{p}_{\rm loc}(\mathbb{R}^N)$, $L^{q}_{\rm loc}(\mathbb{R}^N)$ and
a.e. in $\mathbb{R}^N$. Let $u_m=v_m+u$. Then by Lemma \ref{lem1},
 we have
\begin{equation}
\begin{aligned}
&\lim_{m\to\infty}\int_{\mathbb{R}^N}(|\nabla u_m|^{p}+V(x)|u_m|^{p})dx\\
&\geq\lim_{m\to\infty}\int_{\mathbb{R}^N}(|\nabla
v_m|^{p}+V(x)|v_m|^{p})dx+\int_{\mathbb{R}^N}(|\nabla
u|^{p}+V(x)|u|^{p})dx,
\end{aligned}\label{e2.2}
\end{equation}
by the Lieb Lemma,
\begin{equation}
\lim_{m\to\infty}\int_{\mathbb{R}^N}|u_m|^{p}dx
=\lim_{m\to\infty}\int_{\mathbb{R}^N}|v_m|^{p}dx
+\int_{\mathbb{R}^N}|u|^{p}dx. \label{e2.3}
\end{equation}
 Moreover, by (V2) and since $v_m\rightharpoonup 0$ as
$m\to\infty$,
\begin{equation}
\lim_{m\to\infty}\int_{\mathbb{R}^N}(V(x)-V_{b}(x))|v_m|^{p}dx\to
0. \label{e2.4}
\end{equation}
  Using \eqref{e2.2}-\eqref{e2.4} and the definitions of
$M$, $M_{b}$, we obtain
\begin{align*}
&\int_{\mathbb{R}^N}(|\nabla u|^{p}+V(x)|u|^{p})dx
+\lim_{m\to\infty}\int_{\mathbb{R}^N}(|\nabla v_m|^{p}
+V(x)|v_m|^{p})dx \\
& \leq M\lim_{m\to\infty}\|u_m\|^{p}_{q}\\
&=M\lim_{m\to\infty}(\|u\|^{q}_{q}+\|v_m\|^{q}_{q})^{\frac{p}{q}}\\
&\leq M\lim_{m\to\infty}(\|u\|^{p}_{q}+\|v_m\|^{p}_{q})\\
&\leq \int_{\mathbb{R}^N}(|\nabla
u|^{p}+V(x)|u|^{p})dx+MM^{-1}_{b}
 \lim_{m\to\infty}\int_{\mathbb{R}^N}(|\nabla
v_m|^{p}+V_{b}(x)|v_m|^{p})dx\\
& \leq \int_{\mathbb{R}^N}(|\nabla u|^{p}+V(x)|u|^{p})dx
 +MM^{-1}_{b}\lim_{m\to\infty} \int_{\mathbb{R}^N}(|\nabla
v_m|^{p}+V(x)|v_m|^{p})dx.
\end{align*}
 Since $MM^{-1}_{b}<1$ and
$\int_{\mathbb{R}^N}V^{-1}(x)|v_m|^{p}dx\to 0$ as
$m\to\infty$, it follows that $v_m\to 0$ and
therefore $u_m\to u$ as $m\to\infty$. It is clear
that $u\neq 0$.
\end{proof}

 From the above lemma it follows that if
$\sigma(-\Delta_{p}+V)\subset (0,\infty)$ and $M<M_{b}$, then
there exists a ground state solution of \eqref{e1.1}. Recall that
${\{u_m}\}$ is called a Palais-Smale sequence at the level $c$ (a
$(PS)_{c}$-sequence) if $\Phi'(u_m)\to 0$ and
$\Phi(u_m)\to c$. If each $(PS)_{c}$-sequence has a
convergent subsequence, then $\Phi$ is said to satisfy the
$(PS)_{c}$-condition.


\begin{lemma} \label{lem3}
If {\rm (V1), (V2), (P1)} hold, then $\Phi$
satisfies $(PS)_{c}$ for all
$$
c<({\frac{1}{p}}-{\frac{1}{q}})M_{b}^{\frac{q}{(q-p)}}\,.
$$
\end{lemma}

\begin{proof}
 Let ${\{u_m}\}$ be a $(PS)_{c}$-sequence with $c$
satisfying the inequality above. First we show that ${\{u_m}\}$ is
bounded. We have
\begin{equation}
2c+d\|u_m\|\geq{\Phi(u_m)-{\frac{1}{p}}\langle\Phi'(u_m),u_m\rangle}
=({\frac{1}{p}}-{\frac{1}{q}})\|u_m\|^{q}_{q} \label{e2.5}
\end{equation}
and
\begin{equation}
\begin{aligned}
&2c+d\|u_m\|\geq{\Phi(u_m)-{\frac{1}{q}}\langle\Phi'(u_m),u_m\rangle}\\
&=({\frac{1}{p}}-{\frac{1}{q}})\|u_m\|^{p}-({\frac{1}{p}}
-{\frac{1}{q}})\int_{\mathbb{R}^N}V^{-}(x)|u_m|^{p}dx
\end{aligned}\label{e2.6}
\end{equation}
for some constants $d>0$. Suppose $\|u_m\|\to\infty$ as
$m\to\infty$ and let $w_m=u_m/\|u_m\|$. Dividing \eqref{e2.5} by
$\|u_m\|^{q}$ we see that $w_m\to 0$ in
$L^{q}(\mathbb{R}^N)$ as $m\to\infty$ and therefore
$w_m\rightharpoonup 0$ in $E$ as $m\to\infty$ after passing
to a subsequence. Hence
$\int_{\mathbb{R}^N}V^{-}(x)|w_m|^{p}dx\to 0$ as
$m\to\infty$. So dividing \eqref{e2.6} by $\|u_m\|^{p}$, it
follows that $w_m\to 0$ in $E$ as $m\to\infty$, a
contradiction. Thus ${\{u_m}\}$ is bounded.

As in the preceding proof, we may assume $u_m\rightharpoonup u$ in
$E$ and $u_m\to u$ in $L^{p}_{\rm loc}(\mathbb{R}^N)$. Set
$u_m=v_m+u$. Since $\Phi'(u)=0$ and
$$
\Phi(u)=\Phi(u)-{\frac{1}{p}}\langle\Phi'(u),u\rangle
=({\frac{1}{p}}-{\frac{1}{q}})\|u\|^{q}_{q}\geq 0,
$$
it follows from \eqref{e2.2}, \eqref{e2.3} that
$$
\lim_{m\to\infty}(|\|v_m\|^{p}-\|v_m\|^{q}_{q}|)\leq
\lim_{m\to\infty}(|\|u_m\|^{p}-\|u_m\|^{q}_{q}|
+|\|u\|^{p}-\|u\|^{q}_{q}|)=0
$$
so
\begin{equation}
\lim_{m\to\infty}(\|v_m\|^{p}-\|v_m\|^{q}_{q})=0 \label{e2.7}
\end{equation}
and
\begin{equation}
c=\lim_{m\to\infty}\Phi(u_m)
 \geq \lim_{m\to\infty}(\Phi(v_m)+\Phi(u))
 \geq \lim_{m\to\infty}\Phi(v_m). \label{e2.8}
\end{equation}
By \eqref{e2.7}, we have
\begin{equation}
\lim_{m\to\infty}\int_{\mathbb{R}^N}(|\nabla
v_m|^{p}+V(x)|v_m|^{p})dx=\lim_{m\to\infty}
\int_{\mathbb{R}^N}|v_m|^{q}dx=\gamma \label{e2.9}
\end{equation}
possibly after passing to a subsequence, and therefore it follows
from \eqref{e2.8} that
\begin{equation}
c\geq ({\frac{1}{p}}-{\frac{1}{q}})\gamma.  \label{e2.10}
\end{equation}
By \eqref{e2.4},
$$
\lim_{m\to\infty}\int_{\mathbb{R}^N}(|\nabla
v_m|^{p}+V_{b}(x)|v_m|^{p})dx
=\lim_{m\to\infty}\int_{\mathbb{R}^N}(|\nabla
v_m|^{p}+V(x)|v_m|^{p})dx=\gamma\,.
$$
On the other hand,
$$
\|v_m\|^{p}_{q}\leq M^{-}_{b}\lim_{m\to\infty}\int_{\mathbb{R}^N}(|\nabla
v_m|^{p}+V_{b}(x)|v_m|^{p})dx;
$$
therefore, $\gamma^{\frac{p}{q}}\leq
M^{-}_{b}\gamma$. Combining this with \eqref{e2.10}, we see that either
$\gamma=0$, or
$$
c\geq ({\frac{1}{p}}-{\frac{1}{q}})M_{b}^{\frac{q}{(q-p)}}
$$
hence $\gamma$ must be $0$ by the assumption on $c$. So according
to \eqref{e2.9}, we have
$$
\lim_{m\to\infty}\int_{\mathbb{R}^N}(|\nabla
v_m|^{p}+V^{+}(x)|v_m|^{p})dx
=\lim_{m\to\infty}\int_{\mathbb{R}^N}(|\nabla
v_m|^{p}+V(x)|v_m|^{p})dx=0.
$$
Therefore, $v_m\to 0$ and
$u_m\to u$ in $E$ as $m\to\infty$.
\end{proof}
Next we recall a usual critical point theory which will be used in
the below Theorem. Here $\gamma(A)$ is the Krasnoselskii genus of
$A$.

\begin{theorem} \label{thmA}
Suppose $E\in C^{1}(M)$ is an even functional on a
complete symmetric $C^{1,1}$-manifold $M\subset V\setminus {\{0}\}$
in some Banach space $V$. Also suppose $E$ satisfies $(PS)$ and is
bounded below on $M$. Let
$\widetilde{\gamma}(M)=\sup\{\gamma(K);
K\subset M \text{ and symmetric}\}$.
Then the functional $E$ possesses at least
$\widetilde{\gamma}(M)\leq \infty$
pairs of critical points.
\end{theorem}

\section{Proof of Main Theorems}

\begin{theorem} \label{thm1}
Suppose Assumptions {\rm (V1), (P1)} are satisfied,
$\sigma(-\Delta_{p}+V)\subset (0,\infty)$,
$\sup_{x\in \mathbb{R}^N}V(x)=b>0$ and the measure of the set
${\{x\in \mathbb{R}^N:V(x)<b-\varepsilon}\}$ is finite for all
$\varepsilon>0$. Then the infimum in \eqref{e1.5} is attained at some
$u\geq 0$. If $V\geq 0$, then $u>0$ in $\mathbb{R}^N$.
\end{theorem}

\begin{proof}
 Since $V^{+}$ is bounded, $E=W^{1,p}(\mathbb{R}^N)$
here. Let $u_b$ be the radially symmetric positive solution of the
equation
$$
-\operatorname{div}(|\nabla u|^{p-2}\nabla u)
+b|u|^{p-2}u=|u|^{q-2}u,\quad x\in \mathbb{R}^N.
$$
It is well known that such $u_b$ exists, is
unique and minimizes
\begin{equation}
N_b=\inf_{u\in E\setminus {\{0}\}}{\frac{\int_{\mathbb{R}^N}(|\nabla
u|^{p}+b|u|^{p})dx}{\|u\|^{p}_{q}}} \label{e3.1}
\end{equation}
(see \cite{c1}). So if $V\equiv b$, the proof is complete.
Otherwise we may assume without loss of generality that $V(0)<b$.
Then
\begin{align*}
M&=\inf_{u\in E\setminus {\{0}\}}{\frac{\int_{\mathbb{R}^N}(|\nabla
u|^{p}+V(x)|u|^{p})dx}{\|u\|^{p}_{q}}}\\
&\leq {\frac{\int_{\mathbb{R}^N}(|\nabla
u_b|^{p}+V(x)|u_b|^{p})dx}{\|u_b\|^{p}_{q}}}\\
&<{\frac{\int_{\mathbb{R}^N}(|\nabla
u_b|^{p}+b|u_b|^{p})dx}{\|u_b\|^{p}_{q}}}\\
&=N_b=M_b,
\end{align*}
where the last equality follows from the fact that $V_b=b$.
To apply Lemma \ref{lem2} we need to show that $M<M_{b-\varepsilon}$
for some $\varepsilon>0$. A simple computation shows that
if $\lambda>0$,
then $N_{\lambda b}$ is attained at
$$
u_{\lambda b}(x)=\lambda^{\frac{1}{(q-p)}}u_{b}
 (\lambda^{\frac{1}{p}}x)\quad \mbox{and}\quad
N_{\lambda b}=\lambda^{r}N_b,
$$
where $r=1-{\frac{N}{p}}+{\frac{N}{q}}$.

Choosing $\lambda=(b-\varepsilon)/b$ we see that
$N_{b-\varepsilon}<N_b$ and $N_{b-\varepsilon}\to N_b$ as
$\varepsilon\to 0$. So for $\varepsilon$ small enough we
have
\begin{equation} \label{e3.2}
\begin{aligned}
M&<N_{b-\varepsilon}
=\inf_{u\in E\setminus {\{0}\}}{\frac{\int_{\mathbb{R}^N}(|\nabla u|^{p}
+(b-\varepsilon)|u|^{p})dx}{\|u\|^{p}_{q}}}\\
& \leq \inf_{u\in E\setminus {\{0}\}}{\frac{\int_{\mathbb{R}^N}(|\nabla
u|^{p}+V_{b-\varepsilon}(x)|u|^{p})dx}{\|u\|^{p}_{q}}}\\
&=M_{b-\varepsilon}.
\end{aligned}
\end{equation}
Hence $M$ is attained at some $u$. If $u$ is replaced by $|u|$,
the expression on the right-hand side of \eqref{e1.5} does not change,
we may assume $u\geq 0$. By the maximum principle, if $V\geq 0$, then
$u>0$ in $\mathbb{R}^N$.
\end{proof}

\begin{theorem} \label{thm2}
Suppose $V\geq 0$ and {\rm (V1), (V2), (P1)} are
satisfied. Then there exists $\Lambda>0$ such that for each
$\lambda\geq \Lambda$ the infimum in \eqref{e1.5} is attained at some
$u_{\lambda}>0$. Here $V(x)$ replaced by $\lambda V(x)$.
\end{theorem}

\begin{proof}
 Here $V=V^{+}$. Let $b$ be as in (V2) and
\begin{equation}
\begin{gathered}
M^{\lambda}=\inf_{u\in E\setminus {\{0}\}}{\frac{\int_{\mathbb{R}^N}(|\nabla
u|^{p}+\lambda V(x)|u|^{p})dx}{\|u\|^{p}_{q}}},\\
M^{\lambda}_{b}=\inf_{u\in E\setminus {\{0}\}}{\frac{\int_{\mathbb{R}^N}(|\nabla
u|^{p}+\lambda
V_{b}(x)|u|^{p})dx}{\|u\|^{p}_{q}}}.
\end{gathered}\label{e3.3}
\end{equation}
It suffices to show that $M^{\lambda}<M^{\lambda}_{b}$ for all
$\lambda$ large enough. We may assume $V(0)<b$ and choose
$\varepsilon, \delta>0$ so that $V(x)<b-\varepsilon$ whenever
$|x|<2\delta$. Let $\varphi\in C^{\infty}_{0}(\mathbb{R}^N,[0,1])$
be a function such that $\varphi(x)=1$ for $|x|\leq \delta$ and
$\varphi(x)=0$ for $|x|\geq 2\delta$.
Set $w_{\lambda b}(x)=\varphi(x)u_{\lambda b}(x)
=\lambda^{\frac{1}{q-p}}u_{b}(\lambda^{\frac{1}{p}}x)\varphi(x)$,
where $u_b$ is as in the proof of Theorem \ref{thm1}. Then for all
sufficiently large $\lambda$ and some $C_0>0$,
\begin{align*}
M^{\lambda}
&\leq {\frac{\int_{\mathbb{R}^N}(|\nabla w_{\lambda
b}|^{p}+\lambda V(x)|w_{\lambda b}|^{p})dx}{\|w_{\lambda
b}\|^{p}_{q}}}\\
& \leq {\frac{\int_{\mathbb{R}^N}(|\nabla w_{\lambda
b}|^{p}+\lambda (b-\varepsilon)|w_{\lambda
b}|^{p})dx}{\|w_{\lambda b}\|^{p}_{q}}}\\
&\leq \lambda^{r}({\frac{\int_{\mathbb{R}^N}(|\nabla u_b|^{p}
+\lambda b|u_b|^{p})dx-\varepsilon\int_{\mathbb{R}^N}|u_b|^{p}dx}
{\|u_b\|^{p}_{q}}}+\varepsilon)\\
& \leq\lambda^{r}(N_b-C_{0}\varepsilon)
\end{align*}


where $N_b$ is defined in
\eqref{e3.1} and $r$ in \eqref{e3.2}. Using \eqref{e3.2}
and \eqref{e3.3} we also see that
\begin{equation}
M^{\lambda}_{b}\geq \inf_{u\in E\setminus
{\{0}\}}{\frac{\int_{\mathbb{R}^N}(|\nabla u|^{p}+\lambda
b|u|^{p})dx}{\|u\|^{p}_{q}}}=N_{\lambda
b}=\lambda^{r}N_b, \label{e3.4}
\end{equation}
hence
$M^{\lambda}<M^{\lambda}_{b}$. By the argument at the end of the
proof of Theorem \ref{thm1}, the infimum is attained at some
$u_{\lambda}>0$.
\end{proof}

Next we consider the existence of multiple solutions under the
hypothesis that $V^{-1}(0)$ has nonempty interior.

\begin{theorem} \label{thm3}
 Suppose $V\geq 0$, $V^{-1}(0)$ has nonempty
interior and {\rm (V1), (V2), (P1)} are satisfied. For each $k\geq
1$ there exists $\Lambda_{k}>0$ such that if $\lambda\geq
\Lambda_{k}$, then \eqref{e1.2} has at the least $k$ pairs of nontrivial
solutions in $E$.
\end{theorem}

\begin{proof}
For a fixed $k$ we can find
$\varphi_1$,\dots,$\varphi_{k}\in C^{\infty}_{0}(\mathbb{R}^N)$ such
that $\operatorname{supp}\varphi_{j}$, $1\leq j\leq k$, is contained in
the interior of $V^{-1}(0)$ and
$\operatorname{supp}\varphi_{i}\cap{\operatorname{supp}\varphi_{j}}
=\emptyset$
whenever $i\neq j$. Let
$$
F_{k}=\operatorname{span}{\{\varphi_{1},\dots,\varphi_{k}}\}.
$$
Since $V\geq 0$,
$\Phi(u)={\frac{1}{p}}\|u\|^{p}-{\frac{1}{q}}\|u\|^{q}_{q}$ and
therefore there exist $\alpha,\quad  \rho>0$ such that
$\Phi|_{S_{\rho}}\geq \alpha$. Denote the set of all symmetric (in
the sense that $-A=A$) and closed subsets of $E$ by $\Sigma$, for each
$A\in \Sigma$ let $\gamma(A)$ be the Krasnoselski genus and
$$
i(A)=\min_{h\in \Gamma}\gamma(h(A)\cap S_{\rho})
$$
where $\Gamma$ is the set of all odd homeomorphisms $h\in C(E,E)$.
Then $i$ is a version of Benci's pseudoindex. Let
$$
\Phi_{\lambda}(u)={\frac{1}{p}}\int_{\mathbb{R}^N}(|\nabla
u|^{p}+\lambda
V(x)|u|^{p})dx-{\frac{1}{q}}\int_{\mathbb{R}^N}|u|^{q}dx,\quad
 \lambda\geq 1
$$
and
$$
c_{j}=\inf_{i(A)\geq j}\sup_{u\in A}\Phi_{\lambda}(u),\quad 1\leq
j\leq k.
$$
Since $\Phi_{\lambda}(u)\geq \Phi(u)\geq \alpha$ for
all $u\in S_{\rho}$ and since $\mbox{i}(F_k)=\dim {F_{k}}=k$,
$$
\alpha\leq c_1\leq \dots \leq c_k\leq \sup_{u\in
F_k}\Phi_{\lambda}(u)=C.
$$
It is clear that $C$ depends on $k$ but
not on $\lambda$. As in \eqref{e3.4}, we have
$$
M^{\lambda}_{b}\geq N_{\lambda b}=\lambda^{r}N_{b}
$$
where $r>0$, and therefore $M^{\lambda}_{b}\to\infty$. Hence
$C<({\frac{1}{p}}-{\frac{1}{q}})(M^{\lambda}_{b})^{\frac{q}{(q-p)}}$
whenever $\lambda$ is large enough and it follows from
Lemma \ref{lem3} that
for such $\lambda$ the Palais-Smale condition is satisfied at all
levels $c\leq C$. By the usual critical point theory Theorem \ref{thmA},
all $c_j$ are critical levels and $\Phi_{\lambda}$ has at least $k$
pairs of nontrivial critical points.
\end{proof}

\begin{theorem} \label{thm4}
Suppose {\rm (V1), (V2), (P1)} are satisfied and
$V^{-1}(0)$ has nonempty interior $\Omega$. Let $u_m\in E$ be a
solution of the equation
\begin{equation}
-\operatorname{div}(|\nabla u|^{p-2}\nabla
u)+\lambda_{m} V(x)|u|^{p-2}u=|u|^{q-2}u,\quad x\in
\mathbb{R}^N. \label{e3.5}
\end{equation}
If $\lambda_{m}\to\infty$
and $\|u_m\|_{\lambda_m}\leq C$ for some $C>0$, then, up to a
subsequence, $u_m\to\overline{u}$ in
$L^{q}(\mathbb{R}^N)$, where $\overline{u}$ is a weak solution of
the equation
\begin{equation}
-\operatorname{div}(|\nabla u|^{p-2}\nabla
u)=|u|^{q-2}u,\quad x\in\Omega, \label{e3.6}
\end{equation}
and $\overline{u}=0$ a.e. in $\mathbb{R}^N\setminus V^{-1}(0)$. If
moreover $V\geq 0$, then $u_m\to\overline{u}$ in $E$ as
$m\to\infty$.
\end{theorem}

\begin{proof}
 Since $\lambda_{m}\geq 1$, $\|u_m\|\leq
\|u_m\|_{\lambda_m}\leq C$. Passing to a subsequence,
$u_m\rightharpoonup\overline{u}$ in $E$ and
$u_m\to\overline{u}$ in $L^{q}_{\rm loc}(\mathbb{R}^N)$ as
$m\to\infty$. Since
$\langle{\Phi_{\lambda_m}}'(u_m),\varphi\rangle=0$, we see that
$$
\lim_{m\to\infty}\int_{\mathbb{R}^N}V(x)|u_m|^{p-2}u_{m}\varphi dx
= 0,\quad
\int_{\mathbb{R}^N}V(x)|\overline{u}|^{p-2}\overline{u}\varphi
dx=0
$$
and for all $\varphi\in C^{\infty}_{0}(\mathbb{R}^N)$.
Therefore, $\overline{u}=0$ a.e. in $\mathbb{R}^N\setminus
V^{-1}(0)$.

We claim that $u_m\to \overline{u}$ in
$L^{q}(\mathbb{R}^N)$ as $m\to\infty$. Assuming the
contrary, it follows from Lion vanishing lemma that
$$
\int_{B_{\rho}(x_m)}|u_m-\overline{u}|^{p}dx\geq \gamma
$$
for some $\{x_m\}\subset \mathbb{R}^N$, $\rho$, $\gamma>0$ and
almost all $m$, where $B_{\rho}(x)$ denotes the open ball of
radius $\rho$ and center $x$.

Since $u_m\to\overline{u}$ in $L^{q}_{\rm loc}(\mathbb{R}^N)$,
$|x_m|\to\infty$. Therefore, the measure of the set
$B_{\rho}(x_m)\cap{\{x\in \mathbb{R}^N:V(x)<b}\}$ tends to $0$ and
\begin{align*}
\lim_{m\to\infty}\|u_m\|^{p}_{\lambda_m}
&\geq \lim_{m\to\infty}{\lambda_m}b\int_{B_{\rho}(x_m)\cap{\{V\geq
b}\}}|u_m|^{p}dx\\
&=\lim_{m\to\infty}{\lambda_m}b(\int_{B_{\rho}(x_m)}
|u_m-\overline{u}|^{p}dx)=\infty,
\end{align*}
 which is a contradiction.

Let now $V\geq 0$. Since $u_m$ satisfies \eqref{e3.5},
$\langle{\Phi_{\lambda_m}'(u_m),\overline{u}}\rangle =0$ and
$\overline{u}=0$ whenever $V>0$, it follows that
$$
\|u_m\|^{p}\leq \|u_m\|^{p}_{\lambda_m}=\|u_m\|^{q}_{q}
$$
and
$$
\|\overline{u}\|^{p}= \|\overline{u}\|^{p}_{\lambda_m}
=\|\overline{u}\|^{q}_{q}.
$$
Hence $\limsup_{m\to\infty}\|u_m\|^{p}\leq
\|\overline{u}\|^{q}_{q}=\|\overline{u}\|^{p}$; therefore,
$u_m\to\overline{u}$ in $E$ as $m\to\infty$.
\end{proof}

\begin{theorem} \label{thm5}
Suppose {\rm (V1), (V2), (P1)} are satisfied and
$V^{-1}(0)$ has nonempty interior, $V\geq 0$, $u_m\in E$ is a
solution of \eqref{e3.5}, $\lambda_{m}\to\infty$ and
$\Phi_{\lambda_m}(u_m)$ is bounded and bounded away from $0$. Then
the conclusion of Theorem \ref{thm4} is satisfied and $\overline{u}\neq 0$.
\end{theorem}

\begin{proof} We have
$$
\Phi_{\lambda_m}(u_m)={\frac{1}{p}}\|u_m\|^{p}_{\lambda_m}
-{\frac{1}{q}}\|u_m\|^{q}_{q}
$$
and
$$
\Phi_{\lambda_m}(u_m)=\Phi_{\lambda_m}(u_m)
 -{\frac{1}{p}}\langle{\Phi_{\lambda_m}'(u_m),u_m}\rangle
=({\frac{1}{p}}-{\frac{1}{q}})\|u_m\|^{q}_{q}
$$
Hence $\|u_m\|_{q}$, and therefore also $\|u_m\|_{\lambda_m}$ is
bounded. So the conclusion of Theorem \ref{thm4} holds. Moreover, as
$\|u_m\|_{q}$ is bounded away from $0$, $\overline{u}\neq 0$.
\end{proof}

As a consequence of this corollary, if $k$ is fixed, then any
sequence of solutions $u_m$ of \eqref{e1.2} with
$\lambda=\lambda_{m}\to\infty$ obtained in Theorem \ref{thm3}
contains a subsequence concentrating at some $\overline{u}\neq 0$.
Moreover, it is possible to obtain a positive solution for each
$\lambda$, either via Theorem \ref{thm1} or by the mountain pass theorem. It
follows that each sequence ${\{u_m}\}$ of such solutions with
$\lambda_{m}\to\infty$ has a subsequence concentrating at
some $\overline{u}$ which is positive in $\Omega$. Corresponding to
$u_m$ are solutions $v_m=\varepsilon^{p/(q-p)}_{m}u_{m}$
of \eqref{e1.3},
where $\varepsilon^{p}_{m}=\lambda^{-1}_{m}$. Then $v_m\to
0$ and $\varepsilon^{-p/(q-p)}_{m}v_{m}\to\overline{u}$.


subsection*{Remark} In the proof of Lemmas \ref{lem2} and \ref{lem3}
 and Theorems  \ref{thm1}--\ref{thm3}, the
condition (V1) can be replaced by
\begin{itemize}
\item[(V1')] $v\in L^{1}_{\rm loc}(\mathbb{R}^N)$ and
$V^{-}=\max{\{-V,0}\}\in L^{q}(\mathbb{R}^N)$,
where $q=N/p$ if $N\geq p+1$, $q>1$ if $N=p$ and $q=1$ if $N<p$.
\end{itemize}
Meanwhile in Theorems \ref{thm4} and \ref{thm5} we also need
$V\in L^{q}_{\rm loc}(\mathbb{R}^N)$.

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\end{document}