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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 200, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/200\hfil Existence and multiplicity of solutions]
{Existence and multiplicity of solutions for semilinear
elliptic equations with Neumann boundary conditions}

\author[Q. Jiang, S. Ma \hfil EJDE-2015/200\hfilneg]
{Qin Jiang, Sheng Ma}

\address{Qin Jiang \newline
 Department of Mathematics, Huanggang Normal University,
 Hubei 438000, China}
\email{jiangqin999@126.com}

\address{Sheng Ma \newline
 Department of Mathematics, Huanggang Normal University,
 Hubei 438000, China}
\email{masheng666@126.com}

\thanks{Submitted May 29, 2015. Published August 4, 2015.}
\subjclass[2010]{35J20, 35J25}
\keywords{Elliptic equations;  Neumann boundary conditions;
 critical point; \hfill\break\indent 
least action principle; minimax methods}

\begin{abstract}
 This article shows the existence of solutions by the least action principle,
 for semilinear elliptic equations with Neumann boundary conditions, under
 critical growth and local coercive conditions. In the subcritical growth and
 local coercive case, multiplicity results are established by using the minimax
 methods together with a standard eigenspace decomposition.
\end{abstract}

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

\section{Introduction and statement of main results}
 
Since the 70s, several authors have studied the existence and multiplicity 
of solutions for the Neumann boundary-value problem
\begin{equation} \label{e1}
\begin{gathered}
-\Delta{u}=f(x,u)+h(x)\quad \text{for a.e. } x\in \Omega, \\
  \frac{\partial u}{\partial n}=0   \quad \text{on }\partial\Omega
\end{gathered}
\end{equation}
where $\Omega \subset  \mathbb{R}^N$ $(N \geq 1) $ is a bounded domain with 
smooth boundary and outer normal vector $n= n(x)$, 
$\partial u/\partial n = n(x)\cdot\nabla u$. The function 
$f : {\bar{\Omega}}\times R\longrightarrow R $
is a Caratheodory function with $F(x,u )=\int_0^{u}f(x,s )ds$
as its primitive. And then, for \eqref{e1}, a vast of  literature related
to the solvability conditions has been published. It has been showed that 
there is at least one solution for \eqref{e1}
under the assumptions of the periodicity condition, see\cite{PH}, or the monotonicity
condition, see\cite{JM1,JM2}, or the sign condition, see\cite{CP,RI1}, or the
Landesman-Lazer type condition, see\cite{RI2,Kuo}, or a new Landesman-Lazer 
type condition and sublinear condition,
see\cite{Tang1,Tang2}. At the same time, some authors studied
 multiplicity of solutions for \eqref{e1}, see\cite{Costa,Tang3,Tang4},
some authors obtained sign-changing solutions, see\cite{LiC,Li}.
In either case,  existence or  multiplicity of solutions, even  
sign-changing solutions, the main methods
 are the dual least action principle and the minimax methods respectively.

In this paper, under the critical growth  and
local coercive condition, we obtain the  existence theorem by the least
action principle for \eqref{e1}. What's more, in the subcritical growth and local
coercive case,  multiplicity results are
established by using the minimax methods, in particular, a three-critical-point 
theorem proposed by Brezis and Nirenberg \cite{H}. A contribution in this 
direction is \cite{Tang5}, where the authors use the  local coercive condition 
to study the second order Hamiltonian systems by variational method. 
We study \eqref{e1} under the following assumptions:
\begin{itemize}
\item[(H1)] There exist a constant $C_1 > 0$ and a real function
$\gamma \in  L^1(\Omega) $ such that
$$
|f(x,t)|\leq C_1 |t|^{2^*-1}+\gamma (x)
$$
for all $ t\in R$ and a.e. $x \in \Omega$, where
\[
2^*=\begin{cases}
\frac{2N}{N-2}, &  N\geq 3\\
\text{any value} & q\in (2,+\infty), \;  N=1,2
\end{cases}
\]

\item[(H1')] There exist $C_2 > 0$ and $ 2 <p <2^*$ such that
$$
|f (x, t)|\leq C_2( |t |^{p-1}+ 1)
$$
for all  $t\in R $ and a.e. $x \in \Omega$.

\item[(H2)] There exists a subset $E$ of $\Omega $ with
meas$(E)> 0 $ such that $ F(x,t)\to -\infty$
as $|t|\to \infty$,  uniformly for a.e. $x\in E$.

\item[(H3)] There exists $g \in L^1(\Omega )$ such that
$F(x,t) \leq g(x) $
for all $t\in R $ and a.e. $x \in \Omega$.

\item[(H4)] There exists $h\in  L^{2^{*'}}(\Omega)$ such that
$$
\int_\Omega h(x)dx = 0.
$$
where $2^{*'}$ is the conjugate exponent of $ 2^*$, that is,
$\frac{1}{2^{*'} }+ \frac{1}{2^{*}} = 1$.

\item[(H5)] There exist $ \delta>0$ and an integer $m \geq 1$ such that
$$   
\mu_{m}\leq \frac{f (x, t)}{t} \leq  \mu_{m+1} 
$$
for all   $ 0 < |t|\leq \delta $,  and a.e.
$x \in \Omega $, where
$$
0 =\mu_1 <\mu_2 \leq \dots \leq \mu_m \leq \mu_{m+1} \leq \dots, \quad
\mu_m\to \infty
$$
is the sequence of eigenvalues in $H^1(\Omega)$ for $ -\Delta $ 
with Neumann boundary condition.

\end{itemize}
Our main results read as follows.

\begin{theorem} \label{thm1}  
Under hypotheses {\rm (H1)--(H4)}, Problem \eqref{e1} has at least one solution in
the Sobolev space $H^1(\Omega)$.
\end{theorem}

\begin{theorem} \label{thm2}
 If $h= 0$, under hypotheses {\rm (H1'), (H2), (H3), (H5)}, Problem \eqref{e1} 
has at least  two nonzero solutions in $H^1(\Omega)$.
\end{theorem}

\begin{remark} \label{rmk1} \rm
Theorem \ref{thm1} generalizes \cite[Theorem 1]{Tang3} because that
conditions  (H2) and (H3) are  weaker than \cite[condition (3)]{Tang3}.
There are functions $f (x, t)$ and $h(x) $ satisfying our
Theorem \ref{thm1} and  not satisfying the corresponding results in 
\cite{Costa,CP,RI1,RI2,Kuo,LiC,Li,JM1,JM2,PH,Tang1,Tang2,Tang3,Tang4}.
 In fact, let
$$
f (x, t)=-(x-x_0)\frac{ 2t}{1+t^2} + 2^*|t |^{2^*-2}t \cos |t |^{2^*}
$$
and $h\in  L^{2^{*'}}(\Omega)$ satisfying (H4), where $x_0\in \bar{\Omega}$.
A direct computation shows that
$$
F(x, t)=-(x-x_0)\ln (1 + t^2)+ \sin |t |^{2^{*}}
$$
satisfies (H1), (H2) and (H3). But $f(x, t)$ does not satisfy the 
conditions in \cite{Costa,CP,RI1,RI2,Kuo,LiC,Li,JM1,JM2,PH,Tang1,Tang2,Tang3,Tang4}.
\end{remark}

\begin{remark} \label{rmk2}\rm
  Obviously, Theorem \ref{thm2} generalizes \cite[Theorem 2]{Tang3} 
because the local coercive condition (H2) and (H3) are weaker than  
\cite[condition (3)]{Tang3}
\eqref{e3},  and condition (H5) is weaker than  
\cite[condition (7)]{Tang3}.
Hence, we solve the open question posed in \cite[Remark 4]{Tang3}.
There are functions $f (x, t)$ satisfying our Theorem \ref{thm2}
and not satisfying the conditions in 
\cite{Costa,CP,RI1,RI2,Kuo,LiC,Li,JM1,JM2,PH,Tang1,Tang2,Tang3,Tang4}.
 For example,
$$ 
f(x, t)=\begin{cases}
-(x-x_0)\frac{ 2t}{1+t^2} + C_3 p|t |^{p-2}t \cos |t|^p, & |t|\geq \delta\\
[\mu_m \sin^2 t^{-2}+\mu_{m+1}(1-\sin^2 t^{-2})]t,
&|t|\leq \delta \\
 0, & t=0
\end{cases}
$$
where $x_0\in \bar{\Omega}$,  $C_3>0$ and $ 2 <p <2^*$.
\end{remark}

\section{Proof of main results}

The methods to prove the theorems are variational basically based upon minimization 
of coercive lower semicontinuous functionals for Theorem \ref{thm1}, and minmax methods 
together with a standard eigenspace decomposition for Theorem \ref{thm2}.

 To make the  statements precise, let us introduce some notation. 
The Sobolev space  $H^1(\Omega)$ is the usual space of $L^2(\Omega)$ 
functions with weak derivative in $L^2(\Omega)$, endowed with the norm
 $$
\|u\|_*=(|\bar{u}|^2+\int_\Omega |\nabla  u(x)|^{2} dx)^{1/2}
$$
where
$$
\bar{u}=(\operatorname{meas} \Omega)^{-1}\int_\Omega u(x)dx,
$$
or the norm defined by
$$
\|u\|=\Big(\int_\Omega
| u(x)|^{2} dx+\int_\Omega |\nabla  u(x)|^{2} dx\Big)^{1/2}
$$
for all $u \in H^1(\Omega)$.
The two norms  $\|u\|$ and $\|u\|_*$ are equivalent. In fact,  
Poincar\'e-Wirtinger's inequality asserts that
$$
\int_\Omega |u-\bar{u}|^2 dx\leq c_1\int_\Omega |\nabla  u|^{2} dx
$$
for some constant $c_1>0$. Hence, one has
$$
\int_\Omega |u|^2 dx\leq c_2(|\bar{u}|^2 +\int_\Omega |\nabla  u|^{2} dx)
$$
for some constant $c_2>0$, which implies
$\|u\|\leq c_3\|u\|_*$ for some constant $c_3>0$. 
On the other hand, H\"older inequality leads to
 $$
\bar{u}=(\operatorname{meas} \Omega)^{-1}\int_\Omega u(x)dx\leq \|u\|_{L^2} $$
 Thus, we obtain
$\|u\|_*\leq c_4\|u\|$
for some constant $c_4>0$. 
That is, the two norms  $\|u\|$ and $\|u\|_*$ are equivalent.

 It is well known that, by
Sobolev's inequality, there exists a constant  $C>0$ such that
\begin{equation} \label{e2}
\|u\|_{L^1(\Omega)}\leq C\|u\|, \quad
\|u\|_{L^{2^*}(\Omega)}\leq C\|u\|,\quad  \|u\|_{L^p(\Omega)}\leq C\|u\|
\end{equation}
where $p$ is the same as in Theorem \ref{thm2}.
Now, the functional $\varphi$ on
$H^1(\Omega)$ is given by
\[
\varphi(u)=\frac{1}{2}\int_\Omega |\nabla u(x)|^2\,dx -\int_\Omega
F(x,u(x))dx-\int_\Omega h u \,dx
\]
for all $u\in H^1(\Omega)$. By the critical growth conditions
(H1) or subcritical growth condition (H1'), we can easy prove
that $\varphi$ is continuously differentiable in $H^1(\Omega)$ ,
in a way similar to \cite[Theorem 1.4]{J}.
  It is well known that finding solutions of \eqref{e1} is equivalent to
finding critical points of $\varphi$ in $H^1(\Omega)$.

For the sake of convenience, we show $C_i\ (i=1,2,\dots,8)$ be positive constants.
Before giving the proof of Theorem \ref{thm1}, we show the following lemmas.


\begin{lemma}[{The least action principle, \cite[Theorem 1.1]{J}}]  \label{lem1}
Suppose that $ X$ is a reflexive Banach space and $\varphi: X
\to R$ is weakly lower semi-continuous. Assume that
$\varphi$ is coercive; that is,
$\varphi(u)\to +\infty$  as $\|u\|\to \infty$
for $u\in X$. Then $\varphi$ has at least one minimum.
\end{lemma}

\begin{lemma} \label{lem2}
 Suppose that $F$ satisfies assumption {\rm (H1)} and
{\rm (H2)}. Then there exist a real function
$\beta \in  L^1(\Omega)$, and $G \in  C(R,R)$ which is subadditive,
that is,
$$ 
G(s + t)\leq G(s)+ G(t) 
$$
for all $s,t\in R$, and coercive, that is,
$ G(t)\to +\infty$  as $|t|\to \infty$ and satisfies
 $$ 
G(t) \leq |t| + 4 
$$
for all $t\in R$, such
that
 $$
F(x,t) \leq -G(t)+ \beta(x) 
$$
for all $t \in R$ and a.e. $t\in E$.
\end{lemma}

 The proof of Lemma \ref{lem2} is essentially the same one as the 
introductory part of the proof of \cite[Theorem 1]{Tang3}.


\begin{proof}[Proof of Theorem \ref{thm1}]
 First, we prove that the functional
$\varphi$ is coercive. By Lemma \ref{lem2}, (H3) and \eqref{e2} we obtain
\begin{align*}
 \int_\Omega F(x,u)dx
&=  \int_E F(x,u)dx+ \int_{\Omega\setminus E} F(x,u)dx\\
&\leq   -\int_E G(u)dx+ \int_E \beta(x) dx+ \int_{\Omega\setminus E} g(x)dx\\
&\leq   -\int_E G(\bar{u})dx+ \int_E
G(-\tilde{u})dx + \int_{E} \beta(x) dx+ \int_{\Omega\setminus E} g(x)dx\\
&\leq  -\operatorname{meas}{E} \cdot G(\bar{u})+ \int_E G(-\tilde{u})dx
+ \int_\Omega |\beta(x)| dx+ \int_{\Omega} |g(x)|dx\\
&\leq  -\operatorname{meas}{E} G(\bar{u})+ \int_E(|\tilde{u}|+4)dx
+ C_4\\
&\leq  -\operatorname{meas}{E} G(\bar{u})+ \|\tilde{u}\|_{L^1(\Omega)}+4 \operatorname{meas}{E}
+ C_4\\
&\leq  \operatorname{meas}{E} (4-G(\bar{u}))+ C\|\tilde{u}\|
+ C_4
\end{align*}
for all $u \in H^1(\Omega)$, where 
$C_4=\int_\Omega |\beta(x)| dx+ \int_{\Omega} |g(x)|dx$ and
$$
\tilde{u}(x) = u(x)-\bar{u}.
$$
Hence by the inequality above, H\"older inequality and \eqref{e2} we have
\begin{align*}
\varphi(u)
&= \frac{1}{2}\int_\Omega |\nabla u|^2dx -\int_\Omega F(x,u)dx-\int_\Omega h u dx\\
&\geq \frac{1}{2}\int_\Omega |\nabla \tilde{u}|^2dx+
\operatorname{meas}{E} (G(\bar{u})-4)- C\|\tilde{u}\|
-C_4-\int_\Omega h \tilde{u }dx\\
&\geq \frac{1}{2}\int_\Omega |\nabla \tilde{u}|^2dx+
(G(\bar{u})-4)\operatorname{meas}{E} - C\|\tilde{u}\|
-C_4-\|h\|_{L^{2^{*'}}(\Omega)} \|\tilde{u }\|_{L^{2^*}(\Omega)}\\
&\geq \frac{1}{2} \|\tilde{u}\|^2+(G(\bar{u})-4) \operatorname{meas}{E}
- C(1+\|h\|_{L^{2^{*'}}(\Omega)})\|\tilde{u}\|-C_4
\end{align*}
for all  $u \in H^1(\Omega)$. By Lemma \ref{lem2}, we know that $ G(t)\to +\infty$
 as $|t|\to \infty$, together with the
fact that
$$ 
\|\tilde{u }\|^2+ \|\bar{u }\|^2=\|u\|^2,
$$
it is easy to obtain  $\varphi$ is coercive.

Next, by (H3), in a way similar to the first part of the proof of 
\cite[Theorem 1]{G} or the part of the proof of 
\cite[Theorem 1]{Tang3}, we can easily prove the functional $\varphi$ 
is weakly lower semicontinuous.
Derived by the least action
principle (see, Lemma \ref{lem1}),  $\varphi$ has a minimum. Hence
\eqref{e1} has at least one solution, which
completes the proof.
\end{proof}

Next, we prove Theorem \ref{thm2} by using the following
three-critical-point theorem proposed by Brezis-Nirenberg \cite{H}.


\begin{lemma}[\cite{H}] \label{lem3} 
Let $X$ be a Banach space with a direct sum decomposition
$$
X=X_1\oplus X_2
$$
with $\operatorname{dim}X_2<\infty$ and let $\varphi$ be a $C^1$ function on $X$
with $\varphi(0)=0$, satisfying the $(PS)$ condition. Assume that, for some 
$\delta_0>0$,
\begin{gather*}
\varphi(v)\geq 0, \quad \text{for $v\in X_1$  with } \|v\|\leq \delta_0,\\
\varphi(v)\leq 0, \quad \text{for $v\in X_2$  with }  \|v\|\leq \delta_0\,.
\end{gather*}
Assume also that $\varphi$ is bounded from below and $\inf_X
\varphi<0$. Then $\varphi$ has at least two nonzero
critical points.
\end{lemma}
 
\begin{proof}[Proof of Theorem \ref{thm2}]
Let $X=H^1(\Omega)=X_1\oplus X_2$, where 
$X_2=\oplus_{1\leq i \leq m}\ker(\Delta +\mu_i)$ 
is a finite dimension subspace and $X_1=X_2^\perp$.

Obviously, $\varphi$ is  a $C^1$ function
on $H^1(\Omega)$ with $\varphi(0)=0$. Similar to  the proof of the coercivity of
$\varphi$ in Theorem \ref{thm1}, by condition (H2), (H3) and (H1'), 
the subcritical growth condition, we can easily obtain that $\varphi$ is coercive and
bounded from below. Therefore, the functional $\varphi$ satisfies the $(PS)$
condition; that is, $\{u_n\}$ possesses a convergent subsequence if $\{u_n\}$ 
is a sequence of $X$ such that $\{\varphi(u_n)\}$ is bounded and 
$\varphi'(u_n)\to 0 $ as $n\to \infty$.

 Firstly, we obtain that
\begin{equation} \label{e3}
\varphi(u)\leq 0, \quad \text{for $u\in X_2$  with }  \|u\|\leq \delta_0
\end{equation}
By (H5), we have
$$
\mu_mt^2\leq tf(x,t)\leq \mu_{m+1}t^2
$$
for all $|t|\leq \delta$ and a.e.$x\in \Omega$.
 Hence, the following inequality holds
$$
\mu_mt^2s\leq tf(x,ts)\leq \mu_{m+1}t^2s
$$
for all $0<s\leq 1$, $|t|\leq \delta$ and a.e. $x\in \Omega$.
It follows from the fact that $F(x,t)=\int_0^1 tf(x,st)ds$,
\begin{equation} \label{e4}
\frac{1}{2}\mu_mt^2\leq F(x,t)\leq \frac{1}{2}\mu_{m+1}t^2
\end{equation}
for all $|t|\leq \delta$ and a.e. $x\in \Omega$. 
$X_2$ is a finite dimensional space, hence there is a positive constant $C_5$ 
such that
$\|u\|_\infty\leq C_5\|u\|$
for all $u\in X_2$. Therefore, by \eqref{e4}, we have
\begin{align*}
\varphi(u) 
& = \frac{1}{2}\int_\Omega |\nabla u(x)|^2dx -\int_\Omega
F(x,u(x))dx\\
&\leq  \frac{1}{2}\int_\Omega |\nabla u(x)|^2dx 
-\frac{1}{2}\mu_m\int_\Omega |u(x)|^2dx,
\end{align*}
for all $u\in X_2$ with $|u|\leq \delta$, which implies that
$$
\varphi(u) \leq 0, \quad \text{with }  \|u\|\leq \frac{\delta}{C_5}\,.
$$

Secondly, we prove that
\begin{equation} \label{e5}
\varphi(u)\geq 0, \quad \text{for $u\in X_1$  with } \|u\|\leq \delta_0\,.
\end{equation}
In fact, by (H1'), one has
$$|F(x,t)|\leq C_2(\frac{|t|^p}{p}+|t|)$$
 for all $t\in R$ and a.e.$x\in \Omega$.
Thus, we have
\begin{equation} \label{e6}
|F(x,t)|\leq C_2(p^{-1}+\delta^{1-p})|t|^p=C_6|t|^p
\end{equation}
 for all $|t|\geq \delta$ and a.e.$x\in \Omega$, 
where $C_6=C_2(p^{-1}+\delta^{1-p})$.

For $u \in X_1$, let $u = v + w$, where 
$v \in  E(\mu_{m+1}), w \in W=(X_2 +E(\mu_{m+1}))^\perp$. 
For $\|u\|\leq \frac{\delta}{2C_5}$, and  $|u(x)| >\delta$, we have
\begin{align*}
|w(x)|
&\geq  |u(x)|-|v(x)|\geq |u(x)|-\|v\|_\infty \\
&\geq  |u(x)|-C_5\|v\| \geq  |u(x)|-C_5\|u\|\\
&\geq \frac{1}{2}|u(x)|
\end{align*}
Moreover,
\[
\mu_{m+2}\int_\Omega|w(x)|^2dx\leq\int_\Omega |\nabla w(x)|^2dx
\]
Hence, we obtain
\[
\|w\|^2=\int_\Omega |\nabla w(x)|^2dx +\int_\Omega |w(x)|^2dx
\leq (1+\frac{1}{ \mu_{m+2}})\int_\Omega |\nabla w(x)|^2dx\,;
\]
that is,
\begin{equation} \label{e7}
\int_\Omega |\nabla w(x)|^2dx\geq  \frac{\mu_{m+2}}{1+\mu_{m+2}}\|w\|^2
\end{equation}
By \eqref{e4}, \eqref{e6}, \eqref{e2} and \eqref{e7}, one has
\begin{align*} %\label{e8}
&\varphi(u)\\
&= \frac{1}{2}\int_\Omega |\nabla u(x)|^2dx -\int_\Omega
F(x,u(x))dx   \\
& = \frac{1}{2}\int_\Omega |\nabla u(x)|^2dx -\int_{\{x\in \Omega:|u(x)|> \delta\}}
F(x,u(x))dx 
 -\int_{\{x\in \Omega:|u(x)|\leq \delta\}} F(x,u(x))dx  \\
& = \frac{1}{2}\int_\Omega |\nabla u(x)|^2dx 
 - \int_{\{x\in \Omega:|u(x)|\leq \delta\}}
\frac{1}{2}\mu_{m+1}|u|^2dx  \\
&\quad -\int_{\{x\in \Omega:|u(x)|> \delta\}}F(x,u(x))dx
 - \int_{\{x\in \Omega:|u(x)|\leq \delta\}}
\Big(F(x,u)-\frac{1}{2}\mu_{m+1}|u|^2\Big)dx \\
& \geq \frac{1}{2}\int_\Omega |\nabla w(x)|^2dx 
 +\frac{1}{2}\int_\Omega |\nabla v(x)|^2dx- \int_{\Omega}
\frac{1}{2}\mu_{m+1}|u|^2dx \\
& \quad -\int_{\{x\in \Omega:|u(x)|> \delta\}}
|F(x,u(x))|dx  \\
& \geq \frac{1}{2}\int_\Omega |\nabla w(x)|^2dx 
 +\frac{1}{2}\int_\Omega |\nabla v(x)|^2dx- \int_{\Omega}
\frac{1}{2}\mu_{m+1}w^2dx \\
&\quad - \int_{\Omega}
\frac{1}{2}\mu_{m+1}v^2dx
-\int_{\{x\in \Omega:|u(x)|> \delta\}}
C_6|u|^pdx \\
&\geq  \frac{1}{2}\int_\Omega |\nabla w(x)|^2dx 
 -\frac{1}{2}\int_\Omega \mu_{m+1}|w(x)|^2dx-\int_{\Omega}C_6 |2w|^pdx \\
&=  \frac{1}{2}\int_\Omega |\nabla w(x)|^2dx 
 -\frac{1}{2}\int_\Omega \mu_{m+1}|w(x)|^2dx-C_6\|2w\|_{L^p(\Omega)}^p  \\
&\geq  \frac{1}{2}(1-\frac{\mu_{m+1}}{\mu_{m+2}})
 \int_\Omega |\nabla w(x)|^2dx-C_6C^p\|2w\|^p   \\
&\geq \frac{\mu_{m+2}-\mu_{m+1}}{2(1+\mu_{m+2})}\|w\|^2-C_7\|w\|^p   
=  C_8\|w\|^2-C_7\|w\|^p
\end{align*}
for all $u\in X_1$ with  $\|u\|\leq \frac{\delta}{2C_5}$. 
From the above inequality,  we can conclude that
$$
\varphi(u)\geq 0, \quad \text{for $u\in X_1$  with } 
\|u\|\leq \delta_1= \big(\frac{C_8}{C_7}\big)^{\frac{1}{p-2}}
$$
Let $\delta_0=\min\{ \frac{\delta}{2C_5},\delta_1\}$, 
hence \eqref{e3} and \eqref{e5} hold.

In the case $\inf_X \varphi<0$, the proof of Theorem \ref{thm2} is complete directly 
by Lemma \ref{lem3}.

In the case $\inf_X \varphi\geq 0$, it follows from \eqref{e3} that
$$
\varphi(u)=\inf_X \varphi=0\  \text{ for all $u\in X_2$  with }  \|u\|\leq \delta
$$
Hence all $u \in  X_2$ with $ \|u\|\leq \delta$ are solutions of \eqref{e1}. 
Therefore, Theorem \ref{thm2} is proved.
\end{proof}

\subsection*{Acknowledgments}
 This research was supported by  the Science Foundation of Hubei
Provincial Department of Education, China (No.Q20132902) and by the
Science Foundation of Huanggang Normal University (2014018703).
The authors would like to thank the anonymous referees for their valuable 
suggestions.

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