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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 05, pp. 1--16.\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{8mm}}

\begin{document}
\title[\hfilneg EJDE-2010/05\hfil
Existence of positive and sign-changing solutions]
{Existence of positive and sign-changing solutions for
$p$-laplace equations with potentials in $\mathbb{R}^N$}

\author[M. Wu, Z. Yang\hfil EJDE-2010/05\hfilneg]
{Mingzhu Wu, Zuodong Yang}

\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 July 23, 2009. Published January 13, 2010.}
\thanks{Supported by grants 10871060 from the the NNSF of China and
08KJB110005 from \hfill\break\indent the NSF of
the Jiangsu Higher Education Institutions of China}
\subjclass[2000]{35J25, 35J60}
\keywords{Potential; critical point theory; $p$-Laplace;
\hfill\break\indent sign changing solution;
multiplicity of solutions; concentration-compactness}

\begin{abstract}
 We study the perturbed equation
 \begin{gather*}
 -\varepsilon^{p}\mathop{\rm div}(|\nabla u|^{p-2}\nabla
 u)+V(x)|u|^{p-2}u=h(x,u)+K(x)|u|^{p^*-2}u,\quad x\in \mathbb{R}^N\\
 u(x)\to 0\quad \text{as } |x|\to\infty\,.
 \end{gather*}
 where $2\leq p<N$, $p^*={\frac{pN}{N-p}}$, $p<q<p^*$.
 Under proper conditions on $V(x)$ and $h(x,u)$, we obtain
 the existence and multiplicity of solutions. We also study the
 existence of solutions which change sign.
\end{abstract}

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

\section{Introduction}

 In this article, we study the equation
\begin{equation} \label{NPE}
\begin{gathered}
-\varepsilon^{p}\mathop{\rm div}(|\nabla u|^{p-2}\nabla
u)+V(x)|u|^{p-2}u=h(x,u)+K(x)|u|^{p^*-2}u,\quad x\in \mathbb{R}^N\\
u(x)\to 0\quad \text{as } |x|\to\infty
\end{gathered}
\end{equation}
where $2\leq p<N$, $p^*={\frac{pN}{N-p}}$, $p<q<p^*$,
$K(x)$ is a bounded positive functions, and $h(x,u)$ is a
superlinear but subcritical function.


When $p=2$ and $\varepsilon=1$, this problem is a Schrodinger
equation which has been extensively studied; see for example
\cite{a1,a2,b1,b2,b4,b5,c1,d2,d4}. Authors have used different
methods to study this equation. In \cite{r1}, the authors
established many embedding results of weighted Sobolev spaces of
radially symmetric functions which be used to obtain ground state
solutions. In \cite{b4}, the authors studied the dependence upon
the local behavior of $V$ near its global minimum. In \cite{b1},
the authors used spectral properties of the Schrodinger operator
to study nonlinear Schrodinger equations with steep potential
well. In \cite{d2}, Ding and Szulkin used Rabinowitz's linking
theorem to study the equation. In \cite{d4}, Ding and Szulkin used
index theory obtain many solutions of the equation. In \cite{c1},
the author imposed on functions $k$ and $K$ conditions ensuring
that this problem can be written in a variational form. We know
that $W^{1,p}(\mathbb{R}^N)$ is not a Hilbert space for $1<p<N$,
except for $p=2$. The space $W^{1,p}(\mathbb{R}^N)$ with $p\neq 2$
does not satisfy the Lieb lemma (see for example \cite{s1}). Using
$\mathbb{R}^N$ results in the loss of compactness. So there are
many difficulties to overcome when we study \eqref{NPE} of $p\neq
2$ by the usual methods. There seems to be very little work on the
case $p\neq 2$, to the best of our knowledge. In this article, we
overcome these difficulties and study \eqref{NPE} of $p\geq 2$.

When $V(x)$ is a constant and $\varepsilon=1$, \eqref{NPE} becomes the
 quasilinear elliptic equation
\begin{equation}
\begin{gathered}
-\mathop{\rm 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{e1.1}
\end{equation}
where $1<p<N$, $N\geq 3$, $\lambda$ is a parameter, $\Omega$ is an
unbounded domain in $\mathbb{R}^N$. There are many results about it we
can see \cite{b3,b6,b7,c2,d1,y2}.
Because of the unboundedness of the
domain, the Sobolev compact embedding does not hold. There are many
methods to overcome this difficulty. In \cite{y2}, the author used 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 studied 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
$$
-\mathop{\rm div}(|\nabla u|^{p-2}\nabla u)+u^{p-1}=q(x)u^{\alpha}
\quad \text{in } \mathbb{R}^N,
$$
where $q(x)$ satisfies many conditions. We can see if $V(x)$
is not constant, then it can not be easily proved by the above
methods. In \cite{y1}, the authors used the concentration-compactness
principle posed by Lions and the mountain pass lemma to solve
problem with this situation.

 Tarantello \cite{t1} studied the equation
\begin{equation}
\begin{gathered}
-\Delta u=|u|^{2^*-2}+f(x,u),\quad  \text{in } \Omega\\
u=0,\quad  \text{on } \partial\Omega
\end{gathered}\label{e1.2}
\end{equation}
where $\Omega\subset \mathbb{R}^{N}$ is open bounded set. She showed
that for $f$ satisfying a suitable condition and $f\neq 0$, the
equation \eqref{e1.2} admits two solutions $u_0$ and $u_1$ in
$H^{1}_{0}(\Omega)$. She used suitable minimization and minimax
principles of mountain pass type. The author got the results when
$f$ satisfies the following condition
$$
\int_{\Omega}fu\leq C_{N}(\|\nabla u\|_{2})^{(N+2)/2}
$$
where $C_{N}={\frac{4}{N-2}}{(\frac{N-2}{N+2})^{(N+2)/4}}$.

Radulescu and Smets \cite{r1} proved existence
results for the  non autonomous perturbations of critical
singular elliptic boundary value problem
\begin{equation}
\begin{gathered}
-\mathop{\rm div}(|x|^{\alpha}\nabla u)=|u|^{2^*-2}+f(x,u),\quad
\text{in } \Omega\\
u=0,\quad  \text{on } \partial\Omega
\end{gathered} \label{e1.3}
\end{equation}
where $f$ satisfies suitable conditions. They proved a
corresponding multiplicity result for the degenerate problem \eqref{e1.3}.
In their case, $\Omega$ can be unbounded.

Silva and Xavier \cite{s2} used the symmetric
Mountain Pass Theorem and the concentration-compactness principle to
prove the multiplicity of solutions for the following equation under
the presence of symmetry
\begin{equation}
\begin{gathered}
-\mathop{\rm div}(|\nabla u|^{p-2}\nabla
u)=\mu|u|^{p^*-2}u+f(x,u),\quad \text{in } \Omega\\
u=0,\quad  \text{on } \partial\Omega
\end{gathered}\label{e1.4}
\end{equation}
where $f(x,s)$ is odd and also subcritical in $s$, and $\Omega$ is a
bounded smooth domain in $\mathbb{R}^N$. They used the
concentration-compactness principle to prove that the Palais-Smale
condition is satisfied below a certain level.

In this paper, we inspired by \cite{d3,r1,s2,t1,w1}
use critical point theory to study the equation \eqref{e1.1}.
We extend the equation in \cite{r1,s2,t1}
where  function $V(x)\neq 0$, $\varepsilon\neq 1$ ,
$K(x)\neq 1$ and $p\geq 2$. We will obtain the similar multiplicity
results with \cite{r1,s2,t1}.
However, our method has essential
differences with the methods used in \cite{r1,s2,t1}.
Also we obtain the
existence of sign-changing solutions.  Let us point out that
although the idea was used before for other problems, the adaptation
to the procedure to our problem is not trivial at all. Since we have
to overcome two main difficulties; one is that $\mathbb{R}^N$ results
in the loss of compactness; the other is that $W^{1,p}(\mathbb{R}^N)$
is not a Hilbert space for $1<p<N$ and it does not satisfy the Lieb
lemma, except for $p=2$. So we need more delicate estimates and
careful analysis. We obtain the existence and the multiplicity of
solutions in Theorems \ref{thmA} and \ref{thmB}.
By the Theorem \ref{thmC} we can obtain the
existence of sign-changing solutions.

This paper is organized as follows. In Section 2, we state some
condition and main results. Section 3 we obtain many lemmas which
will be used in the next section. Section 4 we give the proof of the
main result of the paper.

\section{Main Results}

We make the following assumptions
\begin{itemize}
\item[(V0)]
 $V\in C(\mathbb{R}^N)$; $\min V=0$; and there is $b>0$ such
that the set $\upsilon^{b}={\{x\in \mathbb{R}^N: V(x)<b}\}$ has finite
Lebesgue measure.

\item[(K0)] $K\in C(\mathbb{R}^N)$, $0<\inf K\leq \sup K<\infty$.

\item[(H0)]
\begin{itemize}
 \item  $h\in C(\mathbb{R}^N\times \mathbb{R})$ and
 $h(x,u)=o(|u|^{p-1})$ uniformly in $x$ as $u\to 0$;

 \item there are $C_0>0$ and $q<p^*$ such that
 $|h(x,u)|\leq C_0(1+|u|^{q-1})$ for all $(x,u)$;

 \item there are $a_0>0$, $s>p$ and $\mu>p$ such that
 $H(x,u)\geq a_0|u|^{s}$  and $\mu H(x,u)\leq h(x,u)u$,
 where $H(x,u)=\int^{u}_{0}h(x,s)ds$.
\end{itemize}

\item[(S)] V,K and h are Holder continuous, and there is an orthogonal
involution $\tau$ such that $V(\tau x)=V(x)$, $K(\tau x)=K(x)$ and
$H(\tau x,.)=H(x,.)$ for all $x\in \mathbb{R}^N$.

\end{itemize}

An example satisfying (H0) is the function $h(x,u)=P(x)|u|^{s-2}u$
with $p<s<p^*$ and $P(x)$ being positive and bounded.
Let $\lambda=\varepsilon^{-p}$. \eqref{NPE} reads then as
\begin{equation}
\begin{gathered}
-\mathop{\rm div}(|\nabla u|^{p-2}\nabla u)
 +\lambda V(x)|u|^{p-2}u=\lambda h(x,u)
 +\lambda K(x)|u|^{p^*-2}u,\quad x\in \mathbb{R}^N\\
u(x)\to 0\quad \text{as } |x|\to\infty
\end{gathered}\label{NPL}
\end{equation}
We introduce the space
$$
E={\{u\in W^{1,p}(\mathbb{R}^N):
\int_{\mathbb{R}^N}V(x)|u|^{p}dx<\infty}\}.
$$
It follows from (V0) and Poincare inequality that $E$ continuously
in $W^{1,p}(\mathbb{R}^N)$. It is thus clear that, for each
$s\in [p,p^*]$, there is $\upsilon_{s}>0$ independent of
$\lambda$ such that if $\lambda\geq 1$,
\begin{equation}
|u|_{s}\leq \upsilon_{s}\|u\|\leq \upsilon_{s}\|u\|_{\lambda}\quad
\text{for all } u\in E\label{e2.1}
\end{equation}
Set
\begin{equation} \label{e2.2}
\begin{gathered}
g(x,u)=K(x)|u|^{p^*-2}u+h(x,u), \\
G(x,u)=\int^{u}_{0}g(x,s)ds={\frac{1}{p^*}}K(x)|u|^{p^*}+H(x,u)
\end{gathered}
\end{equation}
Consider the functional
$$
\Phi_{\lambda}(u)={\frac{1}{p}}\int_{\mathbb{R}^N}(|\nabla
u|^{p}+\lambda V(x)|u|^{p})-\lambda\int_{\mathbb{R}^N}G(x,u)
={\frac{1}{p}}\|u\|^{p}_{\lambda}-\lambda\int_{\mathbb{R}^N}G(x,u).
$$
Under the assumptions, $\Phi_{\lambda}\in C^{1}(E,R)$ and its
critical points are solutions of $(NS)_{\lambda}$. Set
$g^{+}(x,u)=g(x,u^{+})$, $G^{+}(x,u)=G(x,u^{+})$ and define, on $E$,
$$
\Psi_{\lambda}(u)={\frac{1}{p}}\|u\|^{p}_{\lambda}
-\lambda\int_{\mathbb{R}^N}G^{+}(x,u)
$$
where as usual $u^{\pm}=\max{\{\pm u,0}\}$. Then
$\Psi_{\lambda}\in C^{1}(E,R)$ and critical points of
$\Psi_{\lambda}$ are positive solutions of $(NS)_{\lambda}$.

Let $\{u_n\}$ denote a $(PS)_c$-sequence. Let
$\eta:[0,\infty)\to [0,1]$ be a smooth function satisfying
$\eta(t)=1$ if $t\leq 1$, $\eta(t)=0$ if $t\geq 2$. Define
$\widetilde{u_j}(x)=\eta(2|x|/j)u(x)$. Then
\begin{equation}
\|u-\widetilde{u_j}\|\to 0\quad \text{as }
j\to\infty \label{e2.3}
\end{equation}
Set
$$
u^{1}_{n}=u_{n}-\widetilde{u_n}
$$
Then $u_{n}-u=u^{1}_{n}+(\widetilde{u_n}-u)$ and by \eqref{e2.3},
$u_n\to u$ if and only if $u^{1}_{n}\to 0$. If we
can shows that
$\lim_{n\to\infty}\Phi_{\lambda}(u^{1}_{n})\leq
c-\Phi_{\lambda}(u)$ and $\Phi_{\lambda}'(u^{1}_{n})\to 0$.
Note that
$$
\Phi_{\lambda}(u^{1}_{n})-{\frac{1}{p}}
 \Phi_{\lambda}'(u^{1}_{n}){u^{1}_{n}}
\geq {\frac{\lambda}{N}}\int_{\mathbb{R}^N}K(x)|u^{1}_{n}|^{p^*}
\geq {\frac{\lambda K_{\min}}{N}}\int_{\mathbb{R}^N}|u^{1}_{n}|^{p^*}
$$
where $K_{\min}=\inf_{x\in R^N}K(x)>0$, hence
\begin{equation}
|u^{1}_{n}|^{p^*}_{p^*}\leq {\frac{N(c-\Phi_{\lambda}(u))}{\lambda
K_{\min}}}+o(1)  \label{e2.4}
\end{equation}
 Let
\begin{equation}
V_{b}(x)=\max{\{V(x),b}\} \label{e2.5}
\end{equation}
where $b$ is the positive constant from the assumption (V0).

Since the set $\upsilon^{b}$ has finite measure and
$u^{1}_{n}\to 0$ in $L^{p}_{loc}$, we see that
$$
\int_{\mathbb{R}^N}V(x)|u^{1}_{n}|^{p}
=\int_{R^N}V_{b}(x)|u^{1}_{n}|^{p}+o(1).
$$
It follows from the definition \eqref{e2.2} of $g(x,u)$ and the
assumptions (K0) and (H0) that there exists a constant
$\gamma_{b}>0$ such that
\begin{equation}
g(x,u)u\leq b|u|^{p}+\gamma_{b}|u|^{p^*}\quad \text{for all }
(x,u) \label{e2.6}
\end{equation}
Let $S$ be the best Sobolev constant:
$$
S|u|^{p}_{p^*}\leq \int_{R^N}|\nabla u|^{p}\quad
\text{for all }  u\in W^{1,p}(\mathbb{R}^N)
$$

In the following we will find special finite-dimensional subspaces
by which we construct sufficiently small minimax levels.
Recall that the assumption (V0) implies that there is
$x_0\in \mathbb{R}^N$ such that
$V(x_0)=\min_{x\in \mathbb{R}^N}V(x)=0$. Without
loss of generality we assume  from now on that $x_0=0$.
Observe that, by (H0),
$$
G(x,u)\geq H(x,u)\geq a_0|u|^{s}
$$
Define the functional $J_{\lambda}\in C^{1}(E,R)$ by setting
$$
J_{\lambda}(u)={\frac{1}{p}}\int_{\mathbb{R}^N}(|\nabla u|^{p}+\lambda
V(x)|u|^{p})-a_{0}\lambda\int_{\mathbb{R}^N}|u|^{s}
$$
Then
$$
\Phi_{\lambda}(u)\leq J_{\lambda}(u)\quad \text{for all } u\in E
$$
 and it suffices to construct small minimax levels for
$J_{\lambda}$.

In $W^{1,p}$ for $p>1$ the Sobolev constant is never achieved on any
domain $\Omega$ different from $\mathbb{R}^N$. Moreover, that for $u\in
C^{\infty}_{0}(\mathbb{R}^N)$ the support of $u$ lies in a fixed
compact set $\Omega$ different from $\mathbb{R}^N$. And combined with
Lions \cite{l1,l2}. It implies that
$$
\inf{\{\int_{\mathbb{R}^N}|\nabla \varphi|^{p}: \varphi\in
C^{\infty}_{0}(\mathbb{R}^N),\quad |\varphi|_{s}=1}\}=0.
$$
For any $\delta>0$ one can choose
$\varphi_{\delta}\in C^{\infty}_{0}(\mathbb{R}^N)$ with
$|\varphi_{\delta}|_{s}=1$ and
$\mathop{\rm supp}\varphi_{\delta}\subset B_{r_\delta}(0)$ so that
$|\nabla \varphi_{\delta}|^{p}_{p}<\delta$. Set
\begin{equation}
e_{\lambda}(x)=\varphi_{\delta}(\lambda^{1/p}x) \label{e2.7}
\end{equation}
Then
$\mathop{\rm supp} e_{\lambda}\subset
B_{\lambda^{1/p}{r_\delta}}(0)$.
For $t\geq 0$,
\begin{align*}
J_{\lambda}(t{e_{\lambda}})
&={\frac{t^p}{p}}\int_{\mathbb{R}^N}|\nabla e_{\lambda}|^{p}+\lambda
V(x)|e_{\lambda}|^{p}-{a_0}\lambda t^{s}
 \int_{\mathbb{R}^N}|e_{\lambda}|^{s}\\
&={\lambda}^{1-{\frac{N}{p}}}({\frac{t^p}{p}}\int_{\mathbb{R}^N}|\nabla
\varphi_{\delta}|^{p}+V(\lambda^{-1/p}x)|\varphi_{\delta}|^{p}-{a_0}
t^{s}\int_{\mathbb{R}^N}|\varphi_{\delta}|^{s})\\
&={\lambda}^{1-{\frac{N}{p}}}I_{\lambda}(t{\varphi_{\delta}}),
\end{align*}
where $I_{\lambda}\in C^{1}(E,R)$ defined by
$$
I_{\lambda}(u)={\frac{1}{p}}\int_{\mathbb{R}^N}|\nabla
u|^{p}+V(\lambda^{-1/p}x)|u|^{p}-{a_0}\int_{\mathbb{R}^N}|u|^{s}
$$
and
$$
\max_{t\geq 0}I_{\lambda}(t{\varphi_{\delta}})
={\frac{s-p}{sp(s{a_0})^{p/(s-p)}}}(\int_{\mathbb{R}^N}|\nabla
\varphi_{\delta}|^{p}+V(\lambda^{-1/p}x)|\varphi_{\delta}|^{p})^{s/(s-p)}.
$$
Since $V(0)=0$ and
$\mathop{\rm supp}\varphi_{\delta}\subset B_{r_{\delta}}(0)$,
there is $\widehat{\Lambda}_{\delta}>0$ such that
$$
V(\lambda^{-1/p}x)\leq
{\frac{\delta}{|\varphi_{\delta}|^{p}_{p}}}\quad \text{for all }
 |x|\leq {r_{\delta}}\quad \text{and}\quad\lambda\geq
\widehat{\Lambda}_{\delta}.
$$
This implies that
\begin{equation}
\max_{t\geq 0}I_{\lambda}(t{\varphi_{\delta}})\leq
{\frac{s-p}{sp(s{a_0})^{p/(s-p)}}}(2\delta)^{s/(s-p)}. \label{e2.8}
\end{equation}
Therefore, for all $\lambda\geq \widehat{\Lambda}_{\delta}$,
\begin{equation}
\max_{t\geq 0}\Phi_{\lambda}(t{e_{\lambda}})
\leq {\frac{s-p}{sp(s{a_0})^{p/(s-p)}}}
(2\delta)^{s/(s-p)}{\lambda}^{1-{\frac{N}{p}}}.\label{e2.9}
\end{equation}
In general, for any $m\in N$, one can choose $m$ functions
${\varphi}^{j}_{\delta}\in C^{\infty}_{0}(\mathbb{R}^N)$ such that
$\mathop{\rm supp}\;{\varphi}^{i}_{\delta}\cap
\mathop{\rm supp}\;{\varphi}^{k}_{\delta}=\emptyset$ if $i\neq k$,
$|{\varphi}^{i}_{\delta}|_{s}=1$ and $|\nabla
{\varphi}^{i}_{\delta}|_{p}^{p}<\delta$.

Let $r^{m}_{\delta}>0$ be such that $\mathop{\rm supp}
{\varphi}^{j}_{\delta}\subset B_{r^{m}_{\delta}}(0)$ for
$j=1,\dots,m$. Set
$$
e^{j}_{\lambda}(x)={\varphi}^{j}_{\delta}({\lambda}^{1/p}x)\quad
\text{for}\quad j=1,\dots,m
$$
and
$H^{m}_{\lambda\delta}=\text{span}{\{e^{1}_{\lambda},
\dots,e^{m}_{\lambda}}\}$.
Observe that for each $u=\Sigma^{m}_{j=1}C_{j}e^{j}_{\lambda}\in
H^{m}_{\lambda\delta}$,
\begin{gather*}
\int_{\mathbb{R}^N}|\nabla u|^{p}=\Sigma^{m}_{j=1}|C_{j}|^{p}
\int_{\mathbb{R}^N}|\nabla e^{j}_{\lambda}|^{p},\\
\int_{\mathbb{R}^N}V(x)|u|^{p}=\Sigma^{m}_{j=1}|C_{j}|^{p}
 \int_{\mathbb{R}^N}V(x)|e^{j}_{\lambda}|^{p},\\
\int_{\mathbb{R}^N}G(x,u)=\Sigma^{m}_{j=1}
 \int_{\mathbb{R}^N}G(x,C_{j}e^{j}_{\lambda})
\end{gather*}
Hence
$$
\Phi_{\lambda}(u)=\Sigma^{m}_{j=1}\Phi_{\lambda}(C_{j}e^{j}_{\lambda})
$$
and as before
$$
\Phi_{\lambda}(C_{j}e^{j}_{\lambda})\leq
{\lambda}^{1-{\frac{N}{p}}}I_{\lambda}(|C_{j}|e^{j}_{\lambda})
$$
Set
$$
\beta_{\delta}=\max{\{|\varphi^{j}_{\delta}|^{p}_{p}:j=1,\dots,m}\}
$$
and choose $\widehat{\Lambda}_{m\delta}$ so that
$$
V(\lambda^{-1/p}x)\leq {\frac{\delta}{\beta_{\delta}}}\quad
\text{for all }|x|\leq r^{m}_{\delta}
$$
and $\lambda\geq \widehat{\Lambda}_{m\delta}$.
As before, one obtains easily that
\begin{equation}
\sup_{u\in H^{m}_{\lambda\delta}}\Phi_{\lambda}(u)\leq
{\frac{s-p}{sp(s{a_0})^{p/(s-p)}}}
(2\delta)^{s/(s-p)}{\lambda}^{1-{\frac{N}{p}}} \label{e2.10}
\end{equation}
for all $\lambda\geq \widehat{\Lambda}_{m\delta}$.

\noindent\textbf{Remark.}
 Let $h(x,u)$ is odd in $u$ and $\tau: \mathbb{R}^N\to
\mathbb{R}^N$ be an orthogonal involution. Then $\tau$ induces an
involution on E which we denote again by $\tau: E\to E$ as
follows $(\tau u)(x)=-u(\tau x)$. If (S) is satisfied, then
$\int_{\mathbb{R}^N}G(x,\tau u)=\int_{\mathbb{R}^N}G(x,u)$. This implies
that $\Phi_{\lambda}$ is $\tau$-invariant: $\Phi_{\lambda}(\tau
u)=\Phi_{\lambda}(u)$ and $\Phi_{\lambda}'$ is $\tau$-equivalent:
$\Phi_{\lambda}'(\tau u)=\tau\Phi_{\lambda}'(u)$. In particular, if
$\tau u=u$ then $\tau\Phi_{\lambda}'(u)=\Phi_{\lambda}'(u)$. Let
$E^{\tau}={\{u\in E:\tau u=u}\}$. It is known that critical points
of the restriction of $\Phi_{\lambda}$ on $E^{\tau}$ are solutions
of \eqref{NPL} satisfying $u(\tau x)=-u(x)$.

We modify the method developed in \cite{d3,r1,s2,t1,w1},
and obtain the following Theorems.

\begin{theorem} \label{thmA}
 Let {\rm (V0), (K0),  (H0)} be satisfied.
Then for any $\sigma>0$ there is $\omega_{\sigma}>0$ such that if
$\varepsilon\leq\omega_{\sigma}$, \eqref{NPE} has at least
one positive solution $u_{\varepsilon}$ of least energy satisfying
\begin{gather} \label{ethmAi}
\frac{\mu-p}{p}\int_{\mathbb{R}^N}H(x,u_{\varepsilon})
+{\frac{1}{N}}\int_{\mathbb{R}^N}K(x)|u_{\varepsilon}|^{p^*}dx\leq
\sigma{\varepsilon^{N}},\\
\label{ethmAii}
{\frac{\mu-p}{p\mu}}\int_{\mathbb{R}^N}(\varepsilon^{p}|\nabla
u_{\varepsilon}|^{p}+V(x)|u_{\varepsilon}|^{p})dx\leq
\sigma{\varepsilon^{N}}
\end{gather}
\end{theorem}

\begin{theorem} \label{thmB}
 Let {\rm (V0), (K0), (H0)} be satisfied. If
moreover $h(x,u)$ is odd in $u$, then for any $m\in N$ and
$\sigma>0$ there is $\omega_{m\sigma}>0$ such that if
$\varepsilon\leq \omega_{m\sigma}$, \eqref{NPE} has at
least $m$ pairs of solutions $u_{\varepsilon}$ which satisfy the
estimates \eqref{ethmAi} and \eqref{ethmAii}.
\end{theorem}

\begin{theorem} \label{thmC}
Let {\rm (V0), (K0), (H0), (S)} be
satisfied. If moreover $h(x,u)$ is odd in $u$, then for any $\sigma>0$
there exists $\omega_{\sigma}>0$ such that if
$\varepsilon\leq\omega_{\sigma}$, \eqref{NPE} has at least
one pair of solutions which change sign exactly once and satisfy the
estimates \eqref{ethmAi} and \eqref{ethmAii}.
\end{theorem}

\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
$$
\lim_{n\to\infty}\int_{\Omega}|\nabla u_n|^{p}dx
\geq \lim_{n\to\infty}\int_{\Omega}|\nabla u_n-{\nabla
u}|^{p}dx+\lim_{n\to\infty}\int_{\Omega}|\nabla u|^{p}dx
$$
\end{lemma}

\begin{proof}
 When $p=2$, from Brezis-Lieb Lemma (see \cite[lemma 1.32]{c2})
we have
$$
\lim_{n\to\infty}\int_{\Omega}|\nabla u_n|^{2}dx
=\lim_{n\to\infty}\int_{\Omega}|\nabla u_n-{\nabla
u}|^{2}dx+\lim_{n\to\infty}\int_{\Omega}|\nabla u|^{2}dx
$$
when $3\geq p>2$, 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
$$
\langle |\nabla u_n|^{p-2}{\nabla u_n}, {\nabla u_n}\rangle
\geq \langle|\nabla u|^{p-2}{\nabla u}, {\nabla
u}\rangle +o(1)
$$
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\\
&\geq 0=\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 $W^{1,p}(\Omega)$, it follows that
$$
\lim_{n\to\infty}\int_{\Omega}(|\nabla
u_{n}|^{p-2}|\nabla u|^{2}-|\nabla u|^{p-2}|\nabla u_{n}|^{2})dx=0\,.
$$
So that
\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*}
&\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 \langle |{\nabla u_n}-\nabla u|^{p-2}{\nabla u_n},
{\nabla u_n}\rangle + \langle|{\nabla u_n}-\nabla
u|^{p-2}{\nabla u},{\nabla u}\rangle\\
&\quad + \langle|{\nabla u}|^{p-2}{\nabla u},{\nabla u}\rangle +o(1).
\end{align*}
Then
\begin{align*}
&\langle |{\nabla u_n}|^{p-2}{\nabla u_n},{\nabla u_n}\rangle\\
&\geq \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  +o(1)
\end{align*}
and
\[
\lim_{n\to\infty}\int_{\Omega}|\nabla u_n|^{p}dx\\
\geq \lim_{n\to\infty}\int_{\Omega}|\nabla u_n-{\nabla u}|^{p}dx
+\lim_{n\to\infty}\int_{\Omega}|\nabla u|^{p}dx
\]
when $p>3$, there exist a $k\in N$ that $0<p-k\leq 1$. Then, we only
need to prove the following 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 is similar to the proof above, so we omit it. The
lemma is proved.
\end{proof}

Recall that  a sequence $\{u_n\}\subset E$ is a (PS) sequence
at level $c$ if $\Phi_{\lambda}(u_n)\to c$ and
${\Phi_{\lambda}}'(u_n)\to 0$.
$\Phi_{\lambda}$ is said to satisfy the $(PS)_c$ condition
if any $(PS)_c$-sequence contains a convergent subsequence.


\begin{lemma} \label{lem2}
 Assume that {\rm (V0), (K0), (H0)} be
satisfied. Let $\{u_n\}$ be a $(PS)_c$-sequence for
$\Phi_{\lambda}$. Then $c\geq 0$ and $\{u_n\}$ is bounded in $E$.
\end{lemma}

\begin{proof}
Let $\{u_n\}$ be a $(PS)_c$-sequence
$$
\Phi_{\lambda}(u_n)\to c\quad \text{and}\quad
{\Phi_{\lambda}}'(u_n)\to 0.
$$
By (H0) we have
\begin{equation}
\begin{aligned}
&d+\|u_n\|_{\lambda}+o(1)\\
&\geq \Phi_{\lambda}(u_n)-{\frac{1}{\mu}}{\Phi_{\lambda}}'(u_n){u_n}\\
&=({\frac{1}{p}}-{\frac{1}{\mu}})\int_{\mathbb{R}^N}(|\nabla
  u_n|^{p}+\lambda V(x)|u_n|^{p})\\
&\quad +\lambda\int_{\mathbb{R}^N}({\frac{1}{\mu}}h(x,u_n)u_n-H(x,u_n))
+({\frac{1}{\mu}}-{\frac{1}{p^*}})\lambda
 \int_{\mathbb{R}^N}K(x)|u_n|^{p^*}\\
&\geq ({\frac{1}{p}}-{\frac{1}{\mu}})\int_{\mathbb{R}^N}(|\nabla
u_n|^{p}+\lambda V(x)|u_n|^{p});
\end{aligned}\label{e3.1}
\end{equation}
hence for $n$ large,
$d+\|u_n\|_{\lambda}\geq \|u_n\|^{p}_{\lambda}$,
where $d$ is a positive constant. This implies that $\{u_n\}$ is
bounded. Taking the limit in \eqref{e3.1} shows that $c\geq 0$.
\end{proof}

Let $\{u_n\}$ denote a $(PS)_c$-sequence. By the above lemma, it is
bounded, hence, without loss of generality, we may assume
$u_n\rightharpoonup u$ in E, $L^{s}$ and $L^{p^*}$,
$u_n\to u$ in $L^{t}_{loc}$ for $1\leq t<p^*$, and
$u_n\to u$ a.e.for $x\in \mathbb{R}^N$.

\begin{lemma} \label{lem3}
Let $s\in [2, p^*)$. There is a subsequence
$(u_{n_j})$ such that for each $\varepsilon>0$, there exists
$r_{\varepsilon}>0$ with
$$
\limsup_{j\to\infty}\int_{B_{j}\setminus
B_{r}}|u_{n_j}|^{s}dx\leq \varepsilon
$$
for all $r\geq r_{\varepsilon}$, where
$B_{k}={\{x\in {\bf R }^N:|x|\leq k}\}$.
\end{lemma}

\begin{proof}
Note that for each $j\in N$,
$\int_{B_{j}}|u_n|^{s}\to\int_{B_{j}}|u|^{s}$ as
$n\to\infty$.
There exists $\widehat{n_j}\in N$ such that
$\int_{B_{j}}(|u_n|^{s}-|u|^{s})<{\frac{1}{j}}$ for all
$n=\widehat{n_j}+i$, $i=1,2,3,\dots$.
Without loss of generality we can assume $\widehat{n_{j+1}}\geq
\widehat{n_j}$. In particular, for $n_j=\widehat{n_j}+j$ we have
$$
\int_{B_{j}}(|u_{n_j}|^{s}-|u|^{s})<{\frac{1}{j}}
$$
Observe that there is $r_{\varepsilon}$ satisfying
\begin{equation}
\int_{\mathbb{R}^N\setminus B_r}|u|^{s}<\varepsilon \label{e3.2}
\end{equation}
for all $r\geq r_{\varepsilon}$. Since
\begin{align*}
\int_{B_{j}\setminus B_{r}}|u_{n_j}|^{s}
&=\int_{B_{j}} (|u_{n_j}|^{s}-|u|^{s})+\int_{B_{j}\setminus
B_{r}}|u|^{s}+\int_{B_{r}}(|u|^{s}-|u_{n_j}|^{s})\\
&\leq {\frac{1}{j}}+\int_{R^N\setminus
B_r}|u|^{s}+\int_{B_{r}}(|u|^{s}-|u_{n_j}|^{s})
\end{align*}
the lemma follows.
\end{proof}

Recall that, by (H0), $|h(x,u)|\leq C_{1}(|u|+|u|^{q-1})$ for all
$(x,u)$. Let firstly $\{u_{n_j}\}_{j\in N}$ be a subsequence of
$\{u_n\}_{n\in N}$ such that Lemma \ref{lem3} holds for $s=2$.
Repeating the argument we can then find a subsequence
$\{u_{n_{ji}}\}_{i\in N}$ of
$\{u_{n_j}\}_{j\in N}$ such that Lemma \ref{lem3} holds for $s=q$.
Therefore, for notational convenience, we can assume in the following that
Lemma \ref{lem3} holds for both $s=2$ and $s=q$ with the same subsequence.

\begin{lemma} \label{lem4} We have
$$
\lim_{j\to\infty}|\int_{\mathbb{R}^N}(h(x,u_{n_j})
-h(x,u_{n_j}-\widetilde{u_j})-h(x,\widetilde{u_j}))\varphi|=0
$$
uniformly in $\varphi\in E$ with $\|\varphi\|\leq 1$.
\end{lemma}

\begin{proof}
Note that \eqref{e2.3} and the local compactness of Sobolev
embedding imply that, for any $r>0$.
$$
\lim_{j\to\infty}|\int_{B_r}(h(x,u_{n_j})-h(x,u_{n_j}
-\widetilde{u_j})-h(x,\widetilde{u_j}))\varphi|=0
$$
uniformly in $\|\varphi\|\leq 1$. For any $\varepsilon>0$ it follows
from \eqref{e3.2} that
$$
\limsup_{j\to\infty}\int_{B_{j}\setminus
B_{r}}|\widetilde{u}_{j}|^{s}dx
\leq \varepsilon
\leq\int_{\mathbb{R}^N\setminus B_r}|u|^{s}<\varepsilon
$$
for all $r\geq r_{\varepsilon}$.
Using Lemma \ref{lem3} for $s=2,q$ we get
\begin{align*}
&\limsup_{j\to\infty}|\int_{\mathbb{R}^N}(h(x,u_{n_j})
 -h(x,u_{n_j}-\widetilde{u_j})-h(x,\widetilde{u_j}))\varphi|\\
&=\limsup_{j\to\infty}|\int_{B_{j}\setminus
B_{r}}(h(x,u_{n_j})-h(x,u_{n_j}-\widetilde{u_j})
 -h(x,\widetilde{u_j}))\varphi|\\
&\leq C_{2}\limsup_{j\to\infty}\int_{B_{j}\setminus
B_{r}}(|u_{n_j}|+|\widetilde{u}_{j}|)|\varphi|
+C_{3}\limsup_{j\to\infty}\int_{B_{j}\setminus
B_{r}}(|u_{n_j}|^{q-1}+|\widetilde{u}_{j}|^{q-1})|\varphi|\\
&\leq C_{2}\limsup_{j\to\infty}(|u_{n_j}|_{L^{2}(B_{j}\setminus
B_r)}+|\widetilde{u}_{j}|_{L^{2}(B_{j}\setminus B_r)})|\varphi|_{2}\\
&\quad +C_{3}\limsup_{j\to\infty}(|u_{n_j}|^{q}_{L^{q}(B_{j}\setminus
B_r)}+|\widetilde{u}_{j}|^{q}_{L^{q}(B_{j}\setminus
B_r)})|\varphi|_{q} \\
&\leq C_{4}\varepsilon^{\frac{1}{2}}
 +C_{5}\varepsilon^{\frac{(q-1)}{q}}
\end{align*}
the conclusion as required.
\end{proof}


\begin{lemma} \label{lem5}
One has along a subsequence:
(1) $\lim_{n\to\infty}\Phi_{\lambda}(u_n-\widetilde{u_n})\leq
c-\Phi_{\lambda}(u)$, and
(2) $\Phi_{\lambda}'(u_n-\widetilde{u_n})\to 0$.
\end{lemma}

\begin{proof} From Lemma \ref{lem1} we have
\begin{align*}
\Phi_{\lambda}(u_n-\widetilde{u_n})
&\leq \Phi_{\lambda}(u_n)-\Phi_{\lambda}(\widetilde{u_n})
 +{\frac{\lambda}{p^*}}\int_{R^N}K(x)(|u_n|^{p^*}
 -|u_n-\widetilde{u_n}|^{p^*}-|\widetilde{u_n}|^{p^*})\\
&\quad +\lambda\int_{\mathbb{R}^N}(H(x,u_n)
 -H(x,u_n-\widetilde{u_n})-H(x,\widetilde{u_n}))
\end{align*}
Using \eqref{e2.3} and the Lieb Lemma, we have
\begin{gather*}
\int_{\mathbb{R}^N}K(x)(|u_n|^{p^*}
 -|u_n-\widetilde{u_n}|^{p^*}-|\widetilde{u_n}|^{p^*})\to 0\,,\\
\int_{\mathbb{R}^N}(H(x,u_n)-H(x,u_n-\widetilde{u_n})
 -H(x,\widetilde{u_n}))\to 0
\end{gather*}
This, together with the facts $\Phi_{\lambda}(u_n)\to c$
and $\Phi_{\lambda}(\widetilde{u_n})\to\Phi_{\lambda}(u)$,
gives (1).

To verify (2), observe that, as $\widetilde{u_n}\to u$ and
$u_n\rightharpoonup u$ in $W^{1,p}(\mathbb{R}^N)$ so
$u_n-\widetilde{u_n}\rightharpoonup 0$ in $W^{1,p}(\mathbb{R}^N)$, then
$$
\int_{\mathbb{R}^N}(|\nabla (u_n-\widetilde{u_n})|^{p-2}\nabla
(u_n-\widetilde{u_n})\nabla\varphi+\lambda
V(x)|u_n-\widetilde{u_n}|^{p-2}(u_n-\widetilde{u_n})\varphi)=o(1),
$$
for any $\varphi\in E$. So for any $\varphi\in E$,
\begin{align*}
&|\Phi_{\lambda}'(u_n-\widetilde{u_n})\varphi|\\
&\leq |\Phi_{\lambda}'(u_n)\varphi|
 +|\Phi_{\lambda}'(\widetilde{u_n})\varphi|\\
&\quad +\lambda\int_{\mathbb{R}^N}K(x)(|u_n|^{p^*-2}{u_n}
 -|u_n-\widetilde{u_n}|^{p^*-2}(u_n-\widetilde{u_n})
 -|\widetilde{u_n}|^{p^*-2}{\widetilde{u_n}})\varphi\\
&\quad +\lambda\int_{\mathbb{R}^N}(h(x,u_n)-h(x,u_n-\widetilde{u_n})
 -h(x,\widetilde{u_n}))\varphi
\end{align*}
It follows,  again from a standard argument, that
$$
\lim_{n\to\infty}\int_{\mathbb{R}^N}K(x)(|u_n|^{p^*}
-|u_n-\widetilde{u_n}|^{p^*}-|\widetilde{u_n}|^{p^*})\varphi=0
$$
uniformly in $\|\varphi\|\leq 1$. By Lemma \ref{lem4} we obtain
$$
\lim_{n\to\infty}\int_{\mathbb{R}^N}(h(x,u_n)
-h(x,u_n-\widetilde{u_n})-h(x,\widetilde{u_n}))\varphi=0
$$
uniformly in $\|\varphi\|\leq 1$, proving (2).
\end{proof}

\begin{lemma} \label{lem6}
Under the assumptions of Lemma \ref{lem2}, there is a
constant $\alpha_0>0$ independent of $\lambda$ such that, for any
$(PS)_c$-sequence $(u_n)$ for $\Phi_{\lambda}$ with
$u_n\rightharpoonup u$, either $u_n\to u$ or
$$
c-\Phi_{\lambda}(u)\geq \alpha_{0}\lambda^{1-{\frac{N}{p}}}
$$
where $\alpha_{0}=S^{N/p}{\gamma_{b}}^{-N/p}N^{-1}K_{\min}$.
\end{lemma}

\begin{proof}
Assume $u_n$ doesn't tend to $u$. Then
$\liminf_{n\to\infty}\|u^{1}_n\|_{\lambda}>0$ and
$c-\Phi_{\lambda}(u)>0$. By the Sobolev inequality, \eqref{e2.5} and
\eqref{e2.6},
\begin{align*}
S|u^{1}_n|^{p}_{p^*}
&\leq \int_{\mathbb{R}^N}|\nabla
 u^{1}_n|^{p}+\lambda V(x)|u^{1}_n|^{p}
 -\lambda\int_{\mathbb{R}^N}V(x)|u^{1}_n|^{p}\\
&=\lambda\int_{\mathbb{R}^N}g(x,u^{1}_n){u^{1}_n}
 -\lambda\int_{\mathbb{R}^N}V_{b}(x)|u^{1}_n|^{p}+o(1)\\
&\leq \lambda\int_{\mathbb{R}^N}g(x,u^{1}_n){u^{1}_n}
 -\lambda b\int_{\mathbb{R}^N}|u^{1}_n|^{p}+o(1) \\
&\leq \lambda \gamma_{b}|u^{1}_n|^{p^*}_{p^*}+o(1).
\end{align*}
Thus by \eqref{e2.4}
\begin{align*}
S&\leq \lambda \gamma_{b}|u^{1}_n|^{p^*-p}_{p^*}+o(1)\\
&\leq \lambda \gamma_{b}({\frac{N(c-\Phi_{\lambda}(u))}{\lambda
K_{\min}}})^{p/N}+o(1)\\
&=\lambda^{1-{\frac{p}{N}}}\gamma_{b}({\frac{N}{
K_{\min}}})^{p/N}(c-\Phi_{\lambda}(u))^{p/N}+o(1)
\end{align*}
or
$$
\alpha_{0}\lambda^{1-{\frac{p}{N}}}\leq c-\Phi_{\lambda}(u)+o(1)
$$
where
$$
\alpha_{0}=S^{N/p}{\gamma_{b}}^{-N/p}N^{-1}K_{\min}
$$
The proof is complete.
\end{proof}

\begin{lemma} \label{lem7}
Under the assumptions of Lemma \ref{lem2}, $\Psi_{\lambda}$
satisfies the $(PS)_c$ condition for all
$c<\alpha_{0}\lambda^{1-{\frac{p}{N}}}$.
\end{lemma}

\begin{proof}
Assume $(u_n)$ is a $(PS)_c$ sequence for
$\Psi_{\lambda}$. Then $o(1)\|u^{-}_n\|_{\lambda}\geq
{\Psi_{\lambda}}'(u_n)u^{-}_{n}=\|u^{-}_n\|_{\lambda}^{p}$ which
implies $\|u^{-}_n\|_{\lambda}\to 0$. In addition,
$$
\Psi_{\lambda}(u_n)-{\frac{1}{p}}{\Psi_{\lambda}}'(u_n)u_{n}\geq
{\frac{\lambda}{N}}\int_{\mathbb{R}^N}K(x)|u^{+}_n|^{p^*}
$$
and
$$
o(1)\|u^{+}_n\|_{\lambda}
\geq {\Psi_{\lambda}}'(u_n)u^{+}_{n}
=\|u^{+}_n\|^{p}_{\lambda}-\int_{\mathbb{R}^N}g(x,u^{+}_{n}){u^{+}_{n}}
$$
Using the above argument, it is not difficult to check that under
the assumptions of Lemma \ref{lem2}, $\Psi_{\lambda}$ satisfies the $(PS)_c$
condition for all $c<\alpha_{0}\lambda^{1-{\frac{p}{N}}}$.
\end{proof}

We consider $\lambda\geq 1$. The following two Lemmas imply that
$\Phi_{\lambda}$ possesses the mountain-pass structure.

\begin{lemma} \label{lem8}
Assume {\rm (V0), (K0), (H0)} hold. There exist
$\alpha_{\lambda}$, $\rho_{\lambda}>0$ such that
$\Phi_{\lambda}(u)>0$ if $u\in B_{\rho_{\lambda}}\setminus {\{0}\}$
and $\Phi_{\lambda}(u)\geq\alpha_{\lambda}$ if
$u\in \partial{B_{\rho_{\lambda}}}$, where
$$
B_{\rho_{\lambda}}={\{u\in E:\|u\|_{\lambda}\leq\rho_{\lambda}}\}.
$$
\end{lemma}

\begin{proof}
By (H0), for $\delta\leq (2p\lambda
\upsilon^{p}_{p})^{-1}$ there is $C_{\delta}>0$ such that
$G(x,u)\leq \delta|u|^{p}+C_{\delta}|u|^{p^*}$ for all $(x,u)$,
where $\upsilon_{p}$ is the embedding constant of \eqref{e2.1}. Thus
$$
\Phi_{\lambda}(u)\geq
{\frac{1}{p}}\|u\|^{p}_{\lambda}-\lambda\delta|u|^{p}_{p}-\lambda
C_{\delta}|u|^{p^*}_{p^*} \geq
{\frac{1}{2p}}\|u\|^{p}_{\lambda}-\lambda
C_{\delta}{\upsilon}^{p^*}_{p^*}\|u\|^{p^*}_{\lambda}.
$$
Consequently the conclusion follows because $p^*>p$.
\end{proof}

\begin{lemma} \label{lem9}
 Under the assumptions of Lemma \ref{lem8}, for any finite
dimensional subsequence $F\subset E$,
$\Phi_{\lambda}(u)\to -\infty$  as $u\in F$,
$\|u\|_{\lambda}\to\infty$.
\end{lemma}

\begin{proof}
 By (H0),
$$
\Phi_{\lambda}(u)\leq {\frac{1}{p}}\|u\|^{p}_{\lambda}
-\lambda_{0}a_{0}|u|^{s}_{s}
$$
for all $u\in E$. Since all norms in a finite-dimensional space are
equivalent and $s>p$, one obtains easily the desired conclusion.
\end{proof}

\begin{lemma} \label{lem10}
 Under the assumptions of Lemma \ref{lem8}, for any $\sigma>0$
there exists $\Lambda_{\sigma}>0$, such that, for each $\lambda\geq
\Lambda_{\sigma}$, there is $\overline{e}_{\lambda}\in E$ with
$\|\overline{e}_{\lambda}\|>{\sigma}_{\lambda}$,
$\Phi_{\lambda}(\overline{e}_{\lambda})\leq 0$ and
$$
\max_{t\in [0,1]}\Phi_{\lambda}(t\overline{e}_{\lambda})\leq
\sigma{\lambda}^{1-{\frac{N}{p}}},
$$
where $\rho_{\lambda}$ is from Lemma \ref{lem8}.
\end{lemma}

\begin{proof}
Choose $\delta>0$ so small that
$$
{\frac{s-p}{sp(s{a_0})^{p/(s-p)}}}(2\delta)^{s/(s-p)}\leq \sigma
$$
and let $e_{\lambda}\in E$ be the function defined by \eqref{e2.7}.
Take $\Lambda_{\sigma}=\widehat{\Lambda}_{\delta}$. Let
$\overline{t}_{\lambda}>0$ be such that
$\overline{t}_{\lambda}\|e_{\lambda}\|_{\lambda}>{\rho}_{\lambda}$
and $\Phi_{\lambda}(t{e_{\lambda}})\leq 0$ for all
$t>{\overline{t}_{\lambda}}$. Then by \eqref{e2.9},
$\overline{e}_{\lambda}={\overline{t}_{\lambda}}{e_{\lambda}}$
satisfies the requirements.
 \end{proof}

\begin{lemma} \label{lem11}
Under the assumptions of Lemma \ref{lem8}, for any $m\in N$
and $\sigma>0$ there exist $\Lambda_{m\sigma}>0$, such that, for
each $\lambda\geq \Lambda_{m\sigma}$, there exists an $m$-dimensional
subspace $F_{\lambda m}$ satisfying
$$
\sup_{u\in F_{\lambda m}}\Phi_{\lambda}(u)\leq
\sigma{\lambda}^{1-{\frac{N}{p}}}.
$$
\end{lemma}

\begin{proof}
Choose $\delta>0$ small so that
$$
{\frac{s-p}{sp(s{a_0})^{p/(s-p)}}}(2\delta)^{s/(s-p)}\leq \sigma
$$
and take $F_{\lambda m}=H^{m}_{\lambda\delta}$.
Then \eqref{e2.10} yields the conclusion as required.
\end{proof}

\section{Proof of Main Theorems}

\begin{theorem} \label{thm1}
Let {\rm (V0), (K0), (H0)} be satisfied.
Then for any $\sigma>0$ there is $\Lambda_{\sigma}>0$ such that if
$\lambda\geq\Lambda_{\sigma}$, then \eqref{NPL} has at least
one positive solution $u_{\lambda}$ of least energy satisfying
\begin{gather} \label{ethm1i}
{\frac{\mu-p}{p}}\int_{\mathbb{R}^N}H(x,u_{\lambda})
+{\frac{1}{N}}\int_{\mathbb{R}^N}K(x)|u_{\lambda}|^{p^*}dx\leq
\sigma{\lambda^{-{\frac{N}{p}}}}, \\
{\frac{\mu-p}{p\mu}}\int_{\mathbb{R}^N}(\varepsilon^{p}|\nabla
u_{\lambda}|^{p}+V(x)|u_{\lambda}|^{p})dx\leq
\sigma{\lambda^{1-{\frac{N}{p}}}} \label{ethm1ii}
\end{gather}
\end{theorem}

\begin{proof}
Consider the functional $\Psi_{\lambda}$. For any
$0<\sigma<{a_0}$, we choose $\Lambda_{\sigma}$ and define for
$\lambda\geq \Lambda_{\sigma}$ the minimax value
$$
c_{\lambda}=\inf_{\gamma\in \Gamma_{\lambda}}\max_{t\in
[0,1]}\Psi_{\lambda}(\gamma(t))
$$
where $\Gamma_{\lambda}={\{\gamma\in C([0,1],E):\gamma(0)=0,\;
\gamma(1)=\overline{e}_{\lambda}}\}$.
By Lemma \ref{lem8},
$$
\alpha_{\lambda}\leq c_{\lambda}\leq
\sigma{\lambda}^{1-{\frac{N}{p}}}
$$
Since by Lemma \ref{lem7},
$\Psi_{\lambda}$ satisfies the $(PS)_{c_{\lambda}}$-condition, the
mountain-pass theorem implies that there is $u_{\lambda}\in E$ such
that $\Psi_{\lambda}'(u_{\lambda})=0$ and
$\Psi_{\lambda}(u_{\lambda})=c_{\lambda}$.
Then $u_{\lambda}$ is a
positive solution of \eqref{NPL}. Moreover, it is well known
that such a Mountain-Pass solution is a least energy solution of
\eqref{NPL}.

Since $u_{\lambda}$ is a critical point of $\Psi_{\lambda}$, for
$\nu\in [p,p^*]$,
\begin{align*}
\sigma{\lambda}^{1-{\frac{N}{p}}}\\
&\geq \Psi_{\lambda}(u_{\lambda})\\
&=\Psi_{\lambda}(u_{\lambda})-{\frac{1}{\nu}}
 \Psi_{\lambda}'(u_{\lambda})u_{\lambda}\\
&\geq ({\frac{1}{p}}-{\frac{1}{\nu}})\int_{\mathbb{R}^N}(|\nabla
u_{\lambda}|^{p}+\lambda V(x)|u_{\lambda}|^{p})\\
&\quad +\lambda({\frac{1}{\nu}}-{\frac{1}{p^*}})
 \int_{\mathbb{R}^N}K(x)|u_{\lambda}|^{p^*}
 +\lambda({\frac{\mu}{\nu}}-1)\int_{R^N}H(x,u_{\lambda}),
\end{align*}
where $\mu$ is the constant in (H0). Taking $\nu=p$ yields the
estimate \eqref{ethm1i}, and taking $\nu=\mu$ gives the estimate \eqref{ethm1ii}.
The proof is complete.
\end{proof}

\begin{theorem} \label{thm2}
Let {\rm (V0), (K0), (H0)} be satisfied. If
moreover $h(x,u)$ is odd in $u$, then for any $m\in N$ and $\sigma>0$
there is $\Lambda_{m\sigma}>0$ such that if $\lambda\geq
\Lambda_{m\sigma}$, \eqref{NPL} has at least m pairs of
solutions $u_{\lambda}$ which satisfy the estimates
\eqref{ethm1i}  and \eqref{ethm1ii}.
\end{theorem}

\begin{proof}
 Consider the functional $\Phi_{\lambda}$. By virtue of
Lemma \ref{lem11}, for any $m\in N$ and $\sigma\in (0, a_0)$ there is
$\Lambda_{m\sigma}$ such that for each $\lambda\geq
\Lambda_{m\sigma}$, we can choose a m-dimensional subspace
$F_{\lambda m}$ with $\max \Phi_{\lambda}(F_{\lambda m})\leq
\sigma{\lambda}^{1-{\frac{N}{p}}}$. By Lemma \ref{lem9}, there is $R>0$ which
depending on $\lambda$ and m such that $\Phi_{\lambda}(u)\leq 0$ for
all $u\in F_{\lambda m}\setminus B_{R}$.

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 gen(A) be the
Krasnoselski genus and
$$
i(A)=\min_{h\in \Gamma_{m}}\text{gen}(h(A)\cap\partial
B_{\rho_{\lambda}})
$$
where $\Gamma_{m}$ is the set of all odd
homeomorphisms $h\in C(E,E)$ and $\rho_{\lambda}$ is the number from
Lemma \ref{lem8}. Then \eqref{ethm1i} is a version of
 Benci's pseudoindex. Let
$$
c_{\lambda_{j}}=\inf_{i(A)\geq j}\sup_{u\in
A}\Phi_{\lambda}(u),\quad 1\leq j\leq m
$$
Since $\Phi_{\lambda}(u)\geq \alpha_{\lambda}$ for all
$u\in\partial B_{\rho_{\lambda}}$ and since
$i(F_{\lambda m})=\dim  F_{\lambda m}=m$,
$$
\alpha_{\lambda}\leq c_{\lambda_1}\leq\dots\leq
c_{\lambda_m}\leq\sup_{u\in F_{\lambda m}}\Phi_{\lambda}(u)\leq
\sigma{\lambda}^{1-{\frac{N}{p}}}.
$$
It follows from Lemma \ref{lem6} that $\Phi_{\lambda}$ satisfies the
$(PS)_c$-condition at all levels
$c<\lambda^{1-{\frac{N}{p}}}\alpha_{0}$. By the critical point
theory, all $e_{\lambda_{j}}$ are critical levels and
$\Phi_{\lambda}$ has at least m pairs of nontrivial critical points
satisfying
$$
\alpha_{\lambda}\leq \Phi_{\lambda}(u_{\lambda})
\leq \sigma{\lambda}^{1-{\frac{N}{p}}}
$$
Therefore, $(NS)_{\lambda}$ has at least $m$ pairs of solutions.
Finally, as in the proof of Theorem \ref{thm1} one sees that these solutions
satisfy the estimates (i) and (ii).
\end{proof}

\begin{theorem} \label{thm3}
Let {\rm (V0), (K0), (H0), (S)} be satisfied.
If moreover $h(x,u)$ is odd in u, then for any $\sigma>0$
there exists $\Lambda_{\sigma}>0$ such that if
$\Lambda\geq\omega_{\sigma}$, \eqref{NPL} has at least one pair
of solutions which change sign exactly once and satisfy the
estimates \eqref{ethm1i} and \eqref{ethm1ii}.
\end{theorem}

\begin{proof}
We say that a function $u:\mathbb{R}^N\to \mathbb{R}$
changes sign $n$ times if the set ${\{x\in \mathbb{R}^N:u(x)\neq 0}\}$
has $n+1$ connected components. If $u$ is a solution of
\eqref{NPL} then it is of class $C^{2}$ and $\tau$ induces
a bijection between the connected components of
${\{x\in \mathbb{R}^N: u(x)>0}\}$ and those of
${\{x\in \mathbb{R}^N: u(x)<0}\}$. So $u$ changes sign an
odd number of times.
Define the $\tau$-Nehari manifold
$$
N^{\tau}_{\lambda}={\{u\in E^{\tau}: u\neq 0,\;\;
\Phi_{\lambda}'(u)u=0}\}.
$$
Then critical points of the restriction of $\Phi_{\lambda}$ on
$N^{\tau}_{\lambda}$ are solutions of \eqref{NPL}. Set
$$
c^{\tau}_{\lambda}=\inf{\{\Phi_{\lambda}(u):u\in
N^{\tau}_{\lambda}}\}.
$$
Assume (S) holds. If $u\in E$ then the function
$\widetilde{u}=(u+\tau u)/2$ satisfies
$\tau\widetilde{u}=\widetilde{u}$; i.e.,
$\widetilde{u}\in E^{\tau}$.
It is clear that if $(\varphi_{j})\subset C^{\infty}_{0}(\mathbb{R}^N)$,
 $|\varphi_{j}|_{s}=1$ and $|\nabla
\varphi_{j}|_{p}\to 0$, then
$\widetilde{\varphi_{j}}=({\varphi_{j}}+\tau {\varphi_{j}})/2\in
E^{\tau}$ and $|\nabla \widetilde{\varphi_{j}}|_{p}\to 0$.
Arguing as before, we see the conclusion:
Assume (V0), (K0), (H0) and (S) be satisfied. Then for
any $\sigma>0$ there exists
$\Lambda_{\sigma}>0$ such that for each $\lambda\geq
\Lambda_{m\sigma}$ there exists $0\neq \overline{e}_{\lambda}\in
E^{\tau}$ such that
$\Phi_{\lambda}'(\overline{e}_{\lambda}){\overline{e}_{\lambda}}=0$
and
$$
\Phi_{\lambda}(\overline{e}_{\lambda})\leq
\sigma{\lambda}^{1-{\frac{N}{p}}}.
$$
So for any $\sigma\in(0,a_0)$,
there is $\Lambda_{\sigma}>0$ such that
$$
0<C^{\tau}_{\lambda}\leq \sigma{\lambda}^{1-{\frac{N}{p}}}\quad
\text{if }\lambda\geq \Lambda_{\sigma}.
$$
By Lemma \ref{lem6}, $\Phi_{\lambda}$ satisfies the
$(PS)_{c^{\tau}_{\lambda}}$ condition. Thus $c^{\tau}_{\lambda}$ is
a critical value of $\Phi_{\lambda}$. Let $u_{\lambda}\in E^{\tau}$
be the relative critical point which is a solution of
\eqref{NPL} with $u_{\lambda}(\tau x)=-u_{\lambda}(x)$. It
remains to show that $u_{\lambda}$ changes sign exactly once.

Observe that if $u\in N^{\tau}_{\lambda}$ is a solution of
\eqref{NPL} which changes sign $2m-1$ times, then
$\Phi_{\lambda}(u)\geq m c^{\tau}_{\lambda}$. Indeed, the set
${\{x\in \mathbb{R}^N: u(x)>0}\}$ has m connected components
$X_{1},\dots,X_{m}$. Let $u_{i}(x)=u(x)$ if $x\in X_{i}\cup\tau X_{i}$
and $u_{i}(x)=0$ otherwise. Since u is a critical point of
$\Phi_{\lambda}$,
$$
\Phi_{\lambda}'(u)u_{i}
=\|u_{i}\|^{p}_{\lambda}-\int_{\mathbb{R}^N}g(x,u_{i})u_{i}=0.
$$
Thus $u_{i}\in N^{\tau}_{\lambda}$ for $i=1,\dots,m$, and
$$
\Phi_{\lambda}(u)=\Phi_{\lambda}(u_1)+\dots+\Phi_{\lambda}(u_m)\geq
m c^{\tau}_{\lambda}.
$$
Now since $\Phi_{\lambda}(u_{\lambda})=c^{\tau}_{\lambda}$,
one concludes that $u_{\lambda}$ changes sign only $m=1$ time.
Final, as before one
sees that $u_{\lambda}$ satisfies (i) and (ii). The proof is
complete.
\end{proof}

\noindent\textbf{Remark.}
 Clearly we can see that the Theorems \ref{thmA}, \ref{thmB}
and \ref{thmC} also be proofed.
Indeed, \eqref{NPE}$\sim$\ref{NPL}. Our
methods and results can also be applicable to subcritical nonlinear
problems \eqref{e1.1}.


\subsection*{Acknowledgments}
The authors want to thank the anonymous the
referees for their comments and suggestions.

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\end{document}