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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 47, 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-2015/47\hfil Existence of solutions]
{Existence of solutions for Kirchhoff type equations}

\author[Q.-L. Xie, X.-P. Wu, C.-L. Tang \hfil EJDE-2015/47\hfilneg]
{Qi-Lin Xie, Xing-Ping Wu, Chun-Lei Tang}

\address{Qi-Lin Xie \newline
 School of Mathematics and Statistics, Southwest University \\
Chongqing 400715, China.\newline
School of Mathematics Sciences LPMC, Nankai University \\
Tianjing 300071, China}
\email{xieqilinsxdt@163.com}

\address{Xing-Ping Wu (corresponding author) \newline
School of Mathematics and Statistics, Southwest University \\
Chongqing 400715, China}
\email{wuxp@swu.edu.cn}

\address{Chun-Lei Tang \newline
School of Mathematics and Statistics, Southwest University \\
Chongqing 400715, China}
\email{tangcl@swu.edu.cn, tangcl8888@sina.com}

\thanks{Submitted August 13, 2014. Published February 17, 2015.}
\subjclass[2000]{35J60, 47J30, 35J20}
\keywords{ Kirchhoff type equation; Dirichlet problem;
critical point; \hfill\break\indent Mountain Pass Theorem}

\begin{abstract}
In this article, we prove the existence of solutions for Kirchhoff type equations
with Dirichlet boundary-value condition. We use the Mountain Pass Theorem
in critical point theory, without the (PS) condition.
\end{abstract}

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

\section{Introduction and statement of main results}

Consider the Kirchhoff type problem
\begin{equation}\label{Eq.(5.1)} %\tag{$\mathcal{K}$}
\begin{gathered}
-(a+b\int_\Omega |\nabla u|^2dx)\Delta u=f(x,u), \quad\text{in } \Omega, \\
u=0,  \quad\text{on } \partial \Omega,
\end{gathered}
\end{equation}
where $\Omega $ is a smooth bounded domain in $ {\mathbb {R}}^N$, $a,b>0$,
and $f(x,t): \overline{\Omega}\times {\mathbb R}$ is a continuous real
function and satisfies the subcritical condition
\begin{equation}\label{8.1}
 |f(x,t)|\leq C(|t|^{p-1}+1)\quad \text{for some }
 2<p<2^*=\begin{cases}\frac{2N}{N-2},& N\geq 3,\\
\infty, & N=1,2,
\end{cases}
 \end{equation}
where $C$ denotes some positive constant.

It is pointed out in \cite{MB} that the similar nonlocal problems  model
several physical and biological systems where $u$ describes a process
which depends on the average of itself, for example that of the population density.

Problem \eqref{Eq.(5.1)} is related to the stationary analogue of the
 Kirchhoff equation
$$
u_{tt}-\Big(a+b\int_\Omega|\nabla u|^2dx\Big)\Delta u=g(x,u),
$$
proposed by Kirchhoff in \cite{GK} as an extension of the classical d'Alembert
wave equation for free vibrations of elastic strings.
Kirchhoff's model takes into account the changes in length of the string
produced by transverse vibrations. It received great attention only after
Lions \cite{JL} proposed an abstract framework for the problem.

Positive solutions are considered by authors, such as Alves et al \cite{CO},
Ma and Rivera \cite{TF}, Cheng and Wu \cite{CX}, Yang and Zhang \cite{YJ}.
Problems on the unbounded domain $ {\mathbb {R}}^3$ have also been considered
by authors, such as Jin and Wu \cite{JXW}, Nie and Wu \cite{NX}, Wu \cite{XW},
Liu and He \cite{LH}, Li et al \cite{LL}, Li and Ye \cite{GL}.
And more recently the concentration behavior of positive solutions has
 been studied by He and Zou \cite{HZ}, Wang et al \cite{WT}, He et al \cite{hl}.
A result with Hartree-type nonlinearities can be found in L\"{u} \cite{df}.
Ground state nonlinear with critical growth is considered in He and Zou \cite{HZ2}.
The readers may consult Bernstein \cite{SB} and Poho\v{z}aev \cite{SI},
Sun and Tang \cite{JJS}, Chen et al \cite{CY}, Cheng \cite{BC}, Perera
and Zhang \cite{KP, ZT}, Mao and Zhang \cite{AM}, Sun and Liu \cite{SJ}
and the references therein, for more information on this problem.

There are many solvability conditions for problem \eqref{Eq.(5.1)} with $f$,
like the asymptotical linear case (at infinity)  in \cite{YJ} and people are
more interested in the superlinear case (at infinity):
\begin{itemize}
\item[(S1)] there exists $\theta \geq 1$ such that $\theta G(t)\geq G(st)$
for all $t\in \mathbb {R}$ and $s \in [0, 1]$, where $G(t)=f(t)t-4F(t)$
(see \cite{JJS});

\item[(S2)] $\lim_{|t|\to\infty} G(t)=\infty$ and there exist
$ \sigma> \max \{1, N/2\}$ and $C>0$ such that
$|f (t)|^\sigma\leq CG(t)|t|^\sigma$ for $|t|$ large (see \cite{AM});
or some limitation forms,

\item[(S3)] $\lim_{|t|\to \infty} [f(t)t-4F(t)]=\infty$ (see \cite{YYJ});

\item[(S4)] $ \liminf_{|t|\to\infty}\frac{f(x,t)t-4F(x,t)}{|t|^\tau}
>-\alpha$ uniformly in $x\in \Omega$, where $\tau \in [0, 2]$ and
$0< \alpha<a\lambda_1$, $\lambda_1$ is the first eigenvalue of
$\big(-\Delta, H_0^1(\Omega)\big)$ (see \cite{BC}).

\end{itemize}
All kinds of conditions are mainly making sure the boundness of the Cerami
or Palais-Smale sequences. The following condition on $f$ which is
called Ambrosetti-Rabinowitz condition  is often used:
\begin{itemize}
\item[(S5)] there exists $\theta>4$ such that
$f(x, t)t\geq \theta F(x,t)$ for $|t|$ large, where $F (x, t)=\int_0^tf(x, s)ds$.
\end{itemize}

We consider the nonlinear eigenvalue problem
\begin{equation}
\begin{gathered}
-\Big(\int_{\Omega} |\nabla u|^2 dx\Big)\Delta u=\mu u^3, \quad\text{in }
  \Omega, \\
u=0,  \quad\text{on }  \partial\Omega,
\end{gathered}
\end{equation}
whose the eigenvalues are the critical values of the functional
\begin{equation}
J(u)=\|u\|^4 ,\quad
u\in S:=\big\{u \in H_0^1(\Omega): \int_\Omega |u|^4dx=1\big\},
\end{equation}
where $\|u\|=\big(\int_\Omega |\nabla u|^2dx\big)^{1/2}$.
We already know the first eigenvalue $\mu_1 > 0$ and the first eigenfunction
$\psi_1 > 0$ (see \cite{ZT}).

Now, we can state our main results.

\begin{theorem} \label{thm1.1}
Assume that $f\in C(\Omega\times{\mathbb {R}}, {\mathbb {R}})$ satisfies
\eqref{8.1} and
\begin{itemize}
\item[(F1)] ${\lim_{|t|\to\infty}\big(\frac{a\lambda_1}{2}t^2
+\frac{b\mu_1}{4}t^4-F(x,t)\big)=+\infty}$ uniformly in $x\in \Omega$;

\item[(F2)] there exists $\lambda>\lambda_1$ such that
$F(x,t)\geq\frac{a\lambda}{2}t^2$ for $|t|$ small.
\end{itemize}
Then  \eqref{Eq.(5.1)} has at least one nontrivial solution.
\end{theorem}

\begin{remark} \label{rmk1}  \rm
Theorem \ref{thm1.1} is a new for the case
 $\liminf_{|t|\to\infty}\frac{F(x, t)}{t^4}\leq\frac{b\mu_1}{4}$.
The condition (F1) is  weaker than  (F3) in \cite{YYJ}.
So our theorem is different from their theorems and obtains one nontrivial
solution by adding the condition (F2) near zero.
\end{remark}

\begin{theorem} \label{thm1.2}
Assume that  $f\in C(\Omega\times{\mathbb {R}}, {\mathbb {R}})$ satisfies
\eqref{8.1} and
\begin{itemize}
\item[(F3)] ${\liminf_{|t|\to\infty}\frac{F(x, t)}{t^4}>\frac{b\mu_1}{4}}$
 uniformly in $x\in \Omega$;

\item[(F4)] ${\lim_{|t|\to \infty}\big(\frac{1}{4}f(x,t)t-F(x,t)
+\frac{a\lambda_1}{4}t^2\big)=+\infty}$ uniformly in $x\in \Omega$;

\item[(F5)] there exists $\mu<\mu_1$ such that
$F(x,t)\leq\frac{a\lambda_1}{2}t^2+\frac{b\mu}{4}t^4$ for $|t|$ small.
\end{itemize}
Then \eqref{Eq.(5.1)} has at least one nontrivial solution.
\end{theorem}

\begin{remark} \label{rmk2} \rm
Condition (F4) is a new condition for a class of function $f(x,t)$ and is
 weaker than (S1)--(S5).
For example, let
$$
f(x,t)=\frac{a\lambda_1}{8}\Big(8t^3\ln(1+t^2)+\frac{4t^5}{1+t^2}
 +4t^3\cos t^4\Big).
$$
A simple computation shows that
$$
\frac{1}{4}f(x,t)t-F(x,t)+\frac{a\lambda_1}{4}t^2
= \frac{a\lambda_1}{8}\Big(\frac{t^6 (1+\cos t^4)}{1+t^2}
+ \frac{t^4 (2+\cos t^4)+2t^2}{1+t^2}- \sin t^4\Big)
$$
and
$$
\lim_{|t|\to \infty}\Big(\frac{1}{4}f(x,t)t-F(x,t)+\frac{a\lambda_1}{4}t^2\Big)
=+\infty.
$$
Hence, $f(x,t)$ satisfies all the assumptions of Theorem \ref{thm1.2}, but
it does not satisfy any conditions of (S1)--(S5).
\end{remark}


\section{Preliminaries}

We consider  $H:=H_0^1(\Omega)$ endowed with the norm
$\|u\|=\big(\int_\Omega |\nabla u|^2dx\big)^{1/2}$. We denote the usual
$L^p(\Omega)$-norm by $|\cdot|_p$. Since $\Omega $ is a bounded domain,
it is well known that $H \hookrightarrow L^p(\Omega)$ continuously for
 $p\in[1, 2^*]$, and compactly for $p\in[1, 2^*)$. Moreover there
exists $\gamma_p>0$ such that
\begin{equation}\label{eq6.1}
|u|_p\leq\gamma_p\|u\|, \quad  u\in H.
\end{equation}

Seeking a weak solution of problem \eqref{8.1} is equivalent to finding a
critical point of the $C^1$ functional
\begin{equation}\label{eq6.2}
I(u):=\frac{a}{2}\|u\|^2+\frac{b}{4}\|u\|^4-\int_\Omega F(x,u)dx, \quad u\in H,
\end{equation}
which implies that
\begin{equation}\label{eq6.3}
\langle I'(u),v \rangle =(a+b\|u\|^2)\int_\Omega \nabla u \cdot\nabla v dx-\int_\Omega f(x,u)vdx, \ \ u,v\in H.
\end{equation}
Let
$$
{E_j:=\oplus_{i\leq j} \ker(-\Delta-\lambda_i)},
$$
where $0<\lambda_1\leq\lambda_2\leq\lambda_3 \leq\dots\leq\lambda_i\leq\dots$
are the eigenvalues of $(-\Delta,H)$.
We denote a subsequence of a sequence $\{u_n\}$ as $\{u_n\}$ to simplify
the notation unless specified. We need the following
concept, which was introduced by Cerami \cite{GC} and is a weak version of
 the (PS) condition.

\begin{definition}[\cite{GC}] \label{def2.1} \rm
 Let $J\in C^1(X,\mathbb {R})$, we say that $J$ satisfies the Cerami condition
at the level $c\in \mathbb {R}$ ($(Ce)_c$ for short), if any sequence
$\{u_n\}\subset X$ with
$$
J(u_n)\to c, \quad (1+\|u_n\|)J'(u_n)\to 0\quad\text{as } n\to \infty,
$$
possesses a convergence subsequence in $X$; $J$ satisfies the $(Ce)$
condition if $J$ satisfies the $(Ce)_c$ for all $c\in \mathbb {R}$.
\end{definition}

The following lemma, which can be found in \cite{DG}, is our main tool in this
article.

\begin{lemma}[Mountain Pass Theorem] \label{lem2.1}
Let $H$ be a real Banach space and $I\in C^1(H, \mathbb R)$ satisfying
the $(Ce)$ condition. Suppose $I(0)=0$,
\begin{itemize}
\item[(i)] there are constants $\rho, \beta> 0$ such that
$I|_{\partial B_\rho}\geq\beta$ where
$$
B_\rho=\{u\in H : \|u\|\leq \rho\};
$$

\item[(ii)] there is $u_1\in H$ and $\|u_1\|>\rho$ such that $I(u_1)<0$.
\end{itemize}
Then $I$ possesses a critical value $c\geq \beta$.  Moreover $c$ can be
characterized as
$$
c =\inf_{g\in\Gamma}\max_{u\in g([0,1])} I(u), \quad
\Gamma= \{g\in  C([0, 1],H) : g(0) = 0, g(1) = u_1\}.
$$
\end{lemma}

We give a lemma about the $(Ce)$ condition which will play an important role
in the proof of our theorems.

\begin{lemma} \label{lem2.2}
Assume that $f(x,t)$ satisfies \eqref{8.1} and (F4), then $I$ satisfies
the $(Ce)$ condition.
\end{lemma}

\begin{proof}
Suppose that $\{u_n\} $ is a $(Ce)_c$ sequence for $c\in \mathbb {R}$
\begin{equation}\label{eq6.4}
I(u_n)\to c, \quad   (1+\|u_n\|)I'(u_n)\to 0 \quad\text{as } n\to \infty.
\end{equation}
Now firstly, we prove that $\{u_n\} $ is a bounded sequence.
From \eqref{eq6.2}, \eqref{eq6.3} and \eqref{eq6.4}, we obtain
\begin{equation}\label{eq6.5}
1+c \geq I(u_n)-\frac{1}{4} I'(u_n)u_n
=\frac{a}{4}\|u_n\|^2+\int_\Omega \Big(\frac{1}{4}f(x,u_n)u_n-F(x,u_n)\Big)dx.
\end{equation}
By (F4), there exists $M>0$ such that
\begin{equation}\label{eq6.6}
\frac{1}{4}f(x,t)t-F(x,t)+\frac{a\lambda_1}{4}|t|^2\geq -M
\end{equation}
for all $x\in\Omega$ and $t\in\mathbb {R}$.
And let $u_n=\phi_n+w_n$, where $\phi_n\in E_1$ and $w_n\in E^{\perp}_1$.
From \eqref{eq6.5} and \eqref{eq6.6}, one obtains
\begin{equation} \label{eq6.7}
\begin{aligned}
1+c
&\geq I(u_n)-\frac{1}{4} I'(u_n)u_n  \\
&=\frac{a}{4}\|u_n\|^2-\frac{a\lambda_1}{4}|u_n|^2_2
+\int_\Omega \Big(\frac{1}{4}f(x,u_n)u_n-F(x,u_n)
 +\frac{a\lambda_1}{4}|u_n|^2\Big)dx \\
&\geq \frac{a}{4}\big(1-\frac{\lambda_1}{\lambda_2}\big)\|w_n\|^2-M|\Omega|
\end{aligned}
\end{equation}
which implies that $\|w_n\|$ is bounded. We claim that $\{u_n\}$ is a bounded
sequence. Otherwise, there is a subsequence of $\{u_n\}$ satisfying
$\|u_n\| \to+\infty$ as $n\to+\infty$.
Then we obtain
 $$
\frac{w_n}{\|u_n\|} \to 0 \in H.
$$
Since $\phi_n/\|u_n\|$ is bounded in $E_1$ ($E_1$ has finite dimension),
we have
$\phi_n/\|u_n\|\to v$ in $E_1$.
By
$$
v_n:=\frac{u_n}{\|u_n\|}=\frac{\phi_n+w_n}{\|u_n\|}
=\frac{\phi_n}{\|u_n\|}+\frac{w_n}{\|u_n\|}\to v\in  E_1,
$$
one has
\begin{equation}\label{eq6.8}
\frac{u_n(x)}{\|u_n\|}\to v(x)\quad\text{a.e. in }\Omega.
\end{equation}
 From $\|v_n\|=1$, we obtain that $\|v\|=1$. And by $v\in E_1$, one has
 that $v(x)>0$ or $v(x)<0$, which implies that
\begin{equation}\label{eq6.9}
|u_n(x)|\to +\infty\quad\text{as }n\to+\infty
\end{equation}
for all $x\in \Omega$ by \eqref{eq6.8}.
It follows from \eqref{eq6.7}, \eqref{eq6.9} and Fatou's lemma that
\begin{align*}
1+c
&\geq I(u_n)-\frac{1}{4} I'(u_n)u_n  \\
&=\frac{a}{4}\|u_n\|^2+\int_\Omega \Big(\frac{1}{4}f(x,u_n)u_n-F(x,u_n)\Big)dx \\
&\geq \int_\Omega \Big(\frac{1}{4}f(x,u_n)u_n-F(x,u_n)+\frac{a\lambda_1}{4}|u_n|^2
\Big)dx \\
&\to+\infty\quad\text{as }n \to +\infty,
\end{align*}
which is a contradiction. Then we get that $\{u_n\}$ is bounded in $H$.
Since $f(x,t)$ is subcritical growth, we can easily obtain that
$\{u_n\}$ has a convergence subsequence. Hence, $I$ satisfies the $(Ce)$
condition.
\end{proof}

\section{Proof of main results}

\begin{proof}[Proof of Theorem \ref{thm1.1}]
 Let
$$
\overline{u}=\Big(\int_\Omega \nabla u \cdot \nabla \phi_1 dx \Big)\phi_1, \quad
\widetilde{u}=u-\overline{u},
$$
where the $\phi_1$ is the first eigenfunction corresponding to $\lambda_1$.

The following statements come from \cite{SQ}.
First, there exist a real function $g\in L^1(\Omega)$, and
$G\in C(\mathbb{R},\ \mathbb{R})$ which is subadditive; that is,
$$
G(s+t)\leq G(s)+G(t)
$$
for all $s,\ t\in \mathbb{R}$, and coercive; that is,
$G(t)\to+\infty$ as $|t|\to\infty$, and satisfies
$$
G(t)\leq|t|+4
$$
for all $t\in \mathbb{R}$, such that
$$
F(x, t)-\frac{a\lambda_1}{2}t^2-\frac{b\mu_1}{4}t^4\leq-G(t)+g(x)
$$
for all $t\in \mathbb{R}$ and $x\in \Omega$.

Second, the functional $\int_\Omega G(v)dx$ is coercive on $E_1$
(this result also can be seen in \cite{SQ}). We claim that $I(u)$ is coercive.
\begin{align*}
&\int_\Omega \Big( F(x, u)-\frac{a\lambda_1}{2}u^2-\frac{b\mu_1}{4}u^4 \Big)dx\\
&\leq -\int_\Omega G(u)dx+\int_\Omega g(x)dx \\
& \leq -\int_\Omega\left( G(\overline{u})-G(-\widetilde{u})\right) dx
 +\int_\Omega g(x)dx \\
& \leq -\int_\Omega G(\overline{u})dx +|\widetilde{u}|_{1}+4 |\Omega|
 +\int_\Omega g(x)dx \\
& \leq -\int_\Omega G(\overline{u})dx +C_1(\|\widetilde{u}\|+1)
\end{align*}
for all $u\in H$ and some
$$
C_1=C+4  |\Omega|+\int_\Omega g(x)dx,
$$
where $C$ is a positive constant in Sobolev's inequality,
$$
|u|_{1}\leq C\|u\|,\quad |u|_{2}\leq C\|u\|
$$
for all $u\in H$. Hence we have
\begin{align*}
I(u_n)
&=\frac{a}{2}\|u\|^2+\frac{b}{4}\|u\|^4-\int_\Omega F(x,u)dx \\
&=\frac{a}{2}\|u_n\|^2-\frac{a\lambda_1}{2}|u_n|^2_2
 +\frac{b}{4}\|u_n\|^4-\frac{b\mu_1}{4}|u_n|^4_4 \\
&\quad+\int_\Omega\Big(\frac{a\lambda_1}{2}|u_n|^2+\frac{b\mu_1}{4}|u_n|^4
 -F(x,u_n)\Big)dx \\
&\geq\frac{a}{2}\|u_n\|^2-\frac{a\lambda_1}{2}|u_n|^2_2
 +\int_\Omega\Big(\frac{a\lambda_1}{2}|u_n|^2+\frac{b\mu_1}{4}|u_n|^4-F(x,u_n)\Big)dx \\
&\geq \frac{a}{2}\big(1-\frac{\lambda_1}{\lambda_2}\big)
 \|\widetilde{u}\|^2+\int_\Omega G(\overline{u})dx-C_1(\|\widetilde{u}\|+1)
\end{align*}
for all $u\in H$. By the coercivity of the functional
$\int_\Omega G(v)dx$ on $E_1$ and that fact
$$
\|u\|^2=\|\overline{u}\|^2+\|\widetilde{u}\|^2,
$$
which implies that the functional $I(u)$ is coercive. $I$ satisfies the $(Ce)$
condition and is bounded from below. By (F2), we have
$$
F(x,t)\geq \frac{a\lambda}{2}t^2-C|t|^p
$$
for all $x\in \Omega$ and $t\in\mathbb{R}$, which implies that
\begin{align*}
I(u)
&\leq\frac{a}{2}\|u\|^2+\frac{b}{4}\|u\|^4-\frac{a\lambda}{2}|u|_2^2+C|u|_p^p \\
&=\frac{a}{2}\big(1-\frac{\lambda}{\lambda_1}\big)\|u\|^2
+\frac{b}{4}\|u\|^4+C\|u\|^p
<0
\end{align*}
for $u\in E_1\cap B_\delta$,  $\lambda>\lambda_1$, where $\delta>0$
small enough and $E_1$ is the subspace of $H$ spanned by $\phi_1$ the
eigenfunctions of $\lambda_1$. Then $I(u)$ achieves the negative infimum.
 This completes the proof
\end{proof}


\begin{proof}[Proof of Theorem \ref{thm1.2}]
By Lemmas \ref{lem2.1} and \ref{lem2.2}, it is sufficient to show that $I$ satisfies
(i) and (ii).
\smallskip

\noindent\textbf{Step 1.} There are constants $\rho,\ \beta> 0$ such
that $I(u)\geq\beta$ for all $\|u\|=\rho$.
In fact, by (F5), it is easy to see that
$$
F(x,t)\leq\frac{a\lambda_1}{2}t^2+\frac{b(\mu_1-\varepsilon)}{4}t^4+C|t|^p
$$
for all $t\in \mathbb{R}$ and $x\in \Omega$,
\begin{align*}
I(u)&\geq\frac{a}{2}\|u\|^2+\frac{b}{4}\|u\|^4
 -\frac{a\lambda_1}{2}|u|_2^2-\frac{b(\mu_1-\varepsilon)}{4}|u|_4^4
 -C\int_\Omega |u|^pdx \\
&\geq\frac{b}{4}\big(1-\frac{\mu_1-\varepsilon}{\mu_1}\big)\|u\|^4-C\gamma_p\|u\|^p.
\end{align*}
Note that $4< p < 2^*$, then for $\varepsilon$ small enough.
So there exists $\beta>0$ such that $I(u)\geq\beta$ for all $\|u\|=\rho$,
where $\rho>0$ small enough.
\smallskip


\noindent\textbf{Step 2.} There exists $u_1\in H$ and $\|u_1\|>\rho$ such that
 $I(u_1)<0$.
Indeed, for small $\varepsilon > 0$, by the definition of $\mu_1$, we can
choose $u\in S$ satisfying
 \begin{equation}\label{eq7.6}
\mu_1+\frac{\varepsilon}{2}\geq\|u\|^4.
\end{equation}
It follows from (F3) that
\begin{equation}\label{eq7.7}
F(x,t)\geq\frac{b(\mu_1+\varepsilon)}{4}t^4-C.
\end{equation}
Hence, by \eqref{eq7.6} and \eqref{eq7.7}, we have
\begin{equation} \label{eq7.8}
\begin{aligned}
I(tu)
&\leq\frac{a}{2}t^2\|u\|^2+\frac{b}{4}t^4\|u\|^4
 -\frac{b}{4}t^4(\mu_1+\varepsilon)+C|\Omega| \\
&\leq\frac{a}{2}t^2\|u\|^2+\frac{b}{4}t^4\mu_1
 +\frac{b\varepsilon}{8}t^4-\frac{b}{4}t^4(\mu_1+\varepsilon)+C|\Omega| \\
&=-\frac{b\varepsilon}{8}t^4+\frac{a}{2}t^2\|u\|^2+C|\Omega|.
\end{aligned}
\end{equation}
Thus, $I(tu)\to-\infty$ as $t\to\infty$.
Therefore, there is $u_1 \in H$ with $\|u_1\|>\rho$ such that $I(u_1)< 0$.
 This completes the proof.
\end{proof}


\subsection*{Acknowledgements} The authors would like to thank the
anonymous referees and the editors for their valuable suggestions.

This research was supported by National Natural Science
Foundation of China (No.11471267).



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\end{thebibliography}

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