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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 10, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/10\hfil Solutions to Kirchhoff equations]
{Solutions to Kirchhoff equations with combined nonlinearities}

\author[L. Ding, L. Li, J.-L. Zhang \hfil EJDE-2014/10\hfilneg]
{Ling Ding, Lin Li, Jing-Ling Zhang}  % in alphabetical order

\address{Ling Ding \newline
School of Mathematics and Computer Science,
Hubei University of Arts and Science, Hubei 441053, China}
\email{dingling1975@qq.com}

\address{Lin Li \newline 
School of Mathematics and Statistics,
Southwest University, Chongqing 400715, China}
\email{lilin420@gmail.com}

\address{Jing-Ling Zhang \newline
School of Mathematics and Computer Science,
Hubei University of Arts and Science, Hubei 441053, China}
\email{1293503066@qq.com}

\thanks{Submitted July 22, 2013. Published January 7, 2014.}
\subjclass[2000]{35J60, 35J40, 35B38}
\keywords{Kirchhoff equation;  asymptotically linear;
asymptotically 3-linear; \hfill\break\indent positive solution; mountain pass lemma}

\begin{abstract}
 We prove the existence of  multiple positive solutions for the
 Kirchhoff equation
 \begin{gather*}
 -\Big(a+b\int_{\Omega}|\nabla u|^2dx\Big)\Delta u =h(x)u^q+f(x,u), \quad
 x\in \Omega, \\
 u=0, \quad  x\in\partial \Omega,
 \end{gather*}
 Here $\Omega $ is an open bounded domain in $ R^{N}$ ($N=1,2,3$),
 $h(x)\in L^\infty(\Omega)$, $f(x,s)$ is a continuous function which
 is asymptotically linear at zero and is asymptotically 3-linear at infinity.
 Our main tools are the Ekeland's variational principle and  the
  mountain pass lemma.
\end{abstract}

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

\section{Introduction and main results}

In this article, we study the existence of positive solutions for
the  Kirchhoff equation
\begin{equation}
\begin{gathered}
 -\Big(a+b\int_{\Omega}|\nabla u|^2dx\Big)\Delta u
=h(x)u^q+f(x,u), \quad   x\in \Omega, \\
u=0, \quad  x\in\partial \Omega,\label{1}
 \end{gathered}
\end{equation}
where $\Omega $ is a bounded smooth domain in $ R^{N}$ ($N=1,2,3$),
 $a>0$, $b>0$,  $0<q<1$.

To state the assumptions, we recall some results about the
following two eigenvalue problems:
\begin{equation}
-\Delta u=\lambda u \text{ in } \Omega,\quad u=0 \text{ on } \Omega,\label{220}
\end{equation}
and
\begin{equation}
-\Big(\int_{\Omega}|\nabla u|^2dx\Big)\Delta u=\mu u^3 \text{ in } \Omega,\quad
 u=0 \text{ on } \Omega.\label{210}
\end{equation}
Let $\lambda_1$ be the principal eigenvalue of  \eqref{220} and let
$\phi_1>0$ be its associated eigenfunction.
It is known that $\lambda_1$ can be characterized by
$$
\lambda_1=\inf\Big\{\int_\Omega|\nabla u|^2dx: \ u\in H_0^1(\Omega),\;
\int_\Omega|u|^2dx=1\Big\},
$$
where $H_0^1(\Omega)$ is the usual Sobolev space defined as the completion
 of $C_0^\infty(\Omega)$ with respect
to the norm $\|u\|=\big(\int_\Omega|\nabla u|^2dx\big)^{1/2}$.
Moreover, define
$$
\mu_1=\inf\Big\{\|u\|^4: \ u\in H_0^1(\Omega),\ \int_\Omega|u|^4dx=1\Big\}.
$$
As shown in \cite{13}, there exists $\mu_1>0$ which is the principle
eigenvalue of \eqref{210} and there is a corresponding eigenfunction of
$\varphi_1>0$ in $\Omega$.

In this article, we assume that $h$, $f$ satisfy the following conditions:
\begin{itemize}
\item[(H1)] $h\in L^\infty(\Omega)$ and $h(x)\not\equiv0$;

\item[(F1)] $f\in C(\Omega\times \mathbb{R})$,
 $f(x,0)=0$ for all $x\in \Omega$,  $f(x,s)\geq0$ for all
  $x\in \Omega$ and $s\geq0$;

\item[(F2)]
\[
{\lim_{s\to   0^+}\frac{f(x,s)}{a\lambda_1s+b\mu_1s^3}
=\alpha\in[0,1)},\quad 
{\lim_{s\to +\infty}\frac{f(x,s)}{a\lambda_1s+b\mu_1s^3}
=\beta\in(1,+\infty)}
\]
 uniformly for a.e. $x\in \Omega$.
\end{itemize}
It is obvious that the values of $f(x,s)$ for $s<0$ are irrelevant
for us to seek for positive solutions of \eqref{1}, and we may define
$$
f(x,s)=0\quad \text{for } x\in\Omega,\; s\leq0.
$$
The problem
\begin{equation}
\begin{gathered}
-(a+b\int_{\Omega}|\nabla u|^2dx)\Delta u=g(x,u),\quad   x\in \Omega, \\
 u=0,\quad x\in  \partial\Omega,
\end{gathered}\label{2}
\end{equation}
is related to the stationary analogue of the Kirchhoff equation
which was proposed by Kirchhoff in 1883  \cite{1} as
an generalization of the well-known d'Alembert's  equation
$$
    \rho \frac{\partial ^2u}{\partial t^2}-\Big(\frac{P_0}{h}+
\frac{E}{2L}\int_0^L|\frac{\partial u}{\partial
x}|^2dx\Big)\frac{\partial ^2u}{\partial x^2}=g(x,u)
$$
for free vibrations of elastic strings. Kirchhoff¡¯s model takes into
account the changes in length of the string produced by
transverse vibrations. Here, $L$ is the length of the string, $h$
is the area of the cross section, $E$ is the Young modulus of the
material, $\rho$ is the mass density and $P_0$ is the initial tension.
In \cite{2}, it was pointed out that the problem \eqref{2} models
several physical systems, where $u$ describes a process which
depends on the average of itself. Nonlocal effect also finds its
applications in biological systems.  After \cite{3} and \cite{4},
there are abundant results about Kirchhoff's equations.

Some interesting studies by variational methods can be found in
\cite{9,12,13,14,16,17,18,19,21} references therein and  for Kirchhoff-type
problem \eqref{2}, they consider it
in a bounded domain  $\Omega$. For example, Perera and Zhang \cite{13}
obtain nontrivial solutions of \eqref{2}
with asymptotically 4-linear terms by using Yang index. In \cite{14},
they revisit problem \eqref{2} and establish
the existence of a positive, a negative and a sign-changing solution
by means of invariant sets of descent flow. Similar results can also be
found in Mao and Zhang \cite{12} and in Yang and Zhang  \cite{16}.
Yang and Zhang in \cite{17} obtain the existence of  nontrivial solutions
for \eqref{2} by using the local linking theory.
Sun and Tang \cite{19} prove the existence of a mountain pass type positive
solution for problem \eqref{2} with the  nonlinearity which is asymptotically
linear near zero and superlinear at infinity.
Sun and Liu \cite{21} obtain a nontrivial solution via Morse theory by
computing the relevant critical groups
for problem \eqref{2} with the  nonlinearity which is  superlinear near zero but
asymptotically 4-linear  at infinity and asymptotically near zero but
4-linear  at infinity.
In \cite{44}, the authors obtain the existence of positive solutions
for \eqref{1} with $h\equiv0$ and $f(x,t)=\nu h(x,t)$ by using the
topological degree argument and variational method, where $h$ is a
continuous function which is asymptotically linear at zero and is
asymptotically  3-linear at infinity. Inspired by \cite{44},
we shall study the existence of positive solutions for problem \eqref{1}
with $h\not\equiv0$ and $f$ which is asymptotically   linear  at zero
and  asymptotically 3-linear infinity by using the Ekeland's variational
principle and Mountain Pass Lemma different from \cite{44}.
In \cite{44}, when $N=1,2,3,$ the authors studied equation \eqref{1}
with $h\equiv0$ and obtain the existence results of positive solution for
equation \eqref{1} under the conditions: $a, b>0$,
and $f$ satisfies (F1) and (F2) with $\alpha>1$ and $\beta<1$; $a\geq0$, $b>0$,
and $f$ satisfies (F1) and (F2) with $\alpha<1$ and $\beta>1$, respectively.
But equation \eqref{1} with $h\not\equiv0$
has not been studied. We shall obtain the existence of two positive
solution for equation \eqref{1}  because of the nonlinearity term
 $h(x)t^q$($0<q<1$). By the way, recently, Cheng, Wu and Liu \cite{ChengWuLiu}
apply variant mountain pass theorem and Ekeland variational principle
to study the existence of multiple nontrivial solutions for a class
of Kirchhoff type problems with concave nonlinearity similar
to our problem. But in their article, the nonlinear term is superlinear
at infinity.

In this article, we denote by $\|\cdot\|_p$ the $L^p(\Omega)$-normal
($1\leq p\leq\infty$). We say that $u\in H_0^1(\Omega)$ is a
positive (nonnegative) weak solution to problem \eqref{1} if $u>0$
($u\geq0$) a.e. $\Omega$ and satisfies
\[
\Big(a+b\int_{\Omega}| \nabla u|^2dx\Big)\int_{\Omega}\nabla u\cdot\nabla vdx
=\int_{\Omega}h(x)u^qvdx+\int_{\Omega}f(x,u)vdx
\]
for all $v\in H_0^1(\Omega)$.
By assumption (F1), we know that to seek a nonnegative weak solution of
\eqref{1} is equivalent to finding a nonzero critical point of the following
functional on $H_0^1(\Omega)$:
\[
I(u)=\frac{1}{2}\int_{\Omega}| \nabla u|^2dx
+\frac{b}{4}\Big(\int_{\Omega}| \nabla u|^2dx\Big)^2
-\frac{1}{q+1}\int_{\Omega}h(x)(u^+)^{q+1}dx
-\int_{\Omega}F(x,u^+)dx,
\]
where $u^+=\max\{0,u\}$, $F(x,s)=\int_0^sf(x,\sigma)d\sigma$.
By (F1) and (F2), $I$ is a $C^1$ functional. By the strong maximum principle,
the nonzero critical points of $I$ are positive solutions to problem \eqref{1}
if $h(x)\geq0$.

Our results are as follows.

\begin{theorem} \label{thm1.1}
  Suppose  that $N=1, 2, 3$, $a>0$, $b>0$, $0<q<1$, $h$ and $f$ satisfy
{\rm (H1), (F1), (F2)}. Assume further that exists $v\in H_0^1(\Omega)$
such that
\begin{itemize}
\item[(H2)] $\int_\Omega h(x)(v^+)^{q+1}dx>0$.
\end{itemize}
Then there exists a constant $m>0$ such that if $\|h\|_\infty<m$,
 problem \eqref{1} has a solution $u_1\in H_0^1(\Omega)$, $u_1\geq0$ and
$I(u_1)<0$. Moreover, if $h(x)\geq0$, then $u_1>0$ a. e. in $\Omega$.
\end{theorem}

 \begin{theorem} \label{thm1.2}
Suppose  that $N=1, 2, 3$, $a>0$, $b>0$, $0<q<1$,  $h$ and $ f$  satisfy
{\rm (H1), (F1), (F2)}.  Assume further $\beta\mu_1$ is not an eigenvalue 
of  \eqref{210}.
 Then there exists a constant $m>0$ such that if $\|h\|_\infty<m$,  
problem \eqref{1} has a nonnegative solution $u_2\in H_0^1(\Omega)$ 
with $u_2>0$ and $I(u_2)>0$ if $h(x)\geq0$.
\end{theorem}

 \begin{remark} \label{rmk1.1}\rm
Theorem 1.1 for problem \eqref{1} with $a, b>0$ generalizes 
\cite[Theorem 1.1]{42} where \eqref{1} with $a=1$ and $b=0$.
\end{remark}

\begin{corollary} \label{coro1.1}
   Suppose  that $N=1, 2, 3$, $a>0$, $b>0$, $0<q<1$, $h$ and 
$ f$ satisfy {\rm (H1), (F1), (F2)}. Assume further that $\beta\mu_1$ 
is not an eigenvalue of  \eqref{210} and $h(x)\geq(\not\equiv)0$.  
Then there exists a constant $m>0$ such that for all 
$h\in L^\infty(\Omega)$ with $\|h\|_\infty<m$,  problem \eqref{1}
 has at least two positive solutions $u_1, u_2\in H_0^1(\Omega)$ 
such that $I(u_1)<0<I(u_2)$.
\end{corollary}

\begin{remark} \label{rmk1.2}\rm
If $h(x)\geq(\not\equiv)0$, it is easy to see that {\rm (H2)} is always 
satisfied. Therefore, Corollary 1.1 is a straightforward conclusion 
of Theorems 1.1 and 1.2 by applying the strong maximum principle \cite{45}.
\end{remark}

This paper is organized as follows.  
In Section 1, we obtain the existence of a local minimum solution 
by the Ekeland's variational principle. 
In Section  2, by using the Mountain Pass Lemma, we obtain the existence
of a mountain pass solution. 
In the following discussion, we denote various positive
  constants as $C$ or $C_i$, $i=1, 2, 3,\dots$.

\section{Existence of a local minimum}

In this section, we prove Theorem 1.1 by Ekeland's variational principle. 
We need the following Lemmas.

 \begin{lemma} \label{lem2.1}
 Suppose  that $N=1,2,3$, $a>0$, $b>0$, $0<q<1$, $h$ and $ f$ satisfy
{\rm (H1), (F1), (F2)}.  Then there exists a constant $m>0$ such that 
if $\|h\|_\infty<m$, we have
\begin{itemize}
\item[(a)] There exist $\rho, \gamma>0$ such that
 $I(u)|_{\|u\|=\rho}\geq \gamma>0$.

\item[(b)] There exists an $e\in \mathbb{R}\setminus B_\rho(0)$ such that 
$I(e)<0$.
\end{itemize}
\end{lemma}

\begin{proof} (a) By (F2), $\beta\in(1,+\infty)$ and noticing that 
$f(x,s)/s^{p-1} \to 0$ as $s \to +\infty$ uniformly in $x \in \Omega$ 
for any fixed $ p\in(4,6)$ if $ N=3$; $p\in(4,+\infty)$ if $ N=1,2$. 
Given $\varepsilon\in(0,1)$, there exist $\delta, M_\varepsilon>0$ 
satisfying $0<\delta<+\infty$ such that
$$
f(x,s)<\Big(\alpha+\varepsilon\Big)(a\lambda_1 s+b\mu_1 s^3),\quad 0<s<\delta,
$$
and
$$
f(x,s)< M_\varepsilon s^{p-1},\quad \delta<s,
$$
where $ p\in(4,6)$ if $ N=3$; $p\in(4,+\infty)$ if $ N=1,2$.
Together with (F1) and  $f(x,s)=0$ for $x\in\Omega$, $s\leq0$, we obtain
$$
f(x,s)<a\lambda_1(\alpha+\varepsilon)|s|
+b\mu_1(\alpha+\varepsilon)|s|^3+M_\varepsilon s^{p-1},\quad s\in R.
$$
This yields
\begin{equation}
F(x,s)\leq \frac{a\lambda_1}{2}(\alpha+\varepsilon)|s|^2
+\frac{b\mu_1}{4}(\alpha+\varepsilon)|s|^4+ A|s|^p,\quad s\in R,\label{63}
\end{equation}
where  $A=M_\varepsilon/p$.
Furthermore, by (F2), for the above $\varepsilon$, we have
$$
f(x,s)>(\beta-\varepsilon)(a\lambda_1 s+b\mu_1 s^3),\quad s>\delta_\infty.
$$
Thus, we obtain
$$
F(x,s)>(\beta-\varepsilon)
\Big(\frac{a\lambda_1}{2} s^2+\frac{b\mu_1 }{4}s^4\Big),\quad
s>\delta_\infty.
$$
Together with (F1) and  $f(x,s)=0$ for $x\in\Omega$, $s\leq0$, 
there exists a constant $B>0$ such that
\begin{equation}
F(x,s)\geq \frac{a}{2}(\beta-\varepsilon)\lambda_1|s|^2
+\frac{b}{4}(\beta-\varepsilon)\mu_1|s|^4-B,\quad s\in R. \label{3}
\end{equation}
Since $\alpha<1$, we can choose $\varepsilon>0$ such that 
$\varepsilon<1-\alpha$. By  (H1), \eqref{63},
$\lambda_1\|u\|_2^2\leq\|u\|^2$, $\mu_1\|u\|_4^4\leq\|u\|^2$, 
the Sobolev's embedding theorem: 
$\|u\|_{q+1}^{q+1}\leq K\|u\|^{q+1}$, $\|u\|_{p+1}^{p+1}\leq M\|u\|^{p+1}$ 
and the Young inequality, we have
\begin{equation}
\begin{aligned}
&I(u)\\
&= \frac{a}{2}\int_{\Omega}| \nabla u|^2dx+\frac{b}{4}
\Big(\int_{\Omega}| \nabla u|^2dx\Big)^2
 -\frac{1}{q+1}\int_{\Omega}h(x)(u^+)^{q+1}dx-\int_{\Omega}F(x,u^+)dx\\
&\geq \frac{a}{2}\|u\|^2+\frac{b}{4}\|u\|^4-\frac{\|h\|_\infty}{q+1}
 \|u^+\|_{q+1}^{q+1}-
 \frac{a}{2}(\alpha+\varepsilon)\lambda_1\|u^+\|_2^2\\
&\quad -\frac{b}{4}(\alpha+\varepsilon)\mu_1\|u^+\|_4^4
 -A\|u^+\|_p^{p}\\
&\geq \frac{a}{2}\|u\|^2+\frac{b}{4}\|u\|^4-\frac{\|h\|_\infty}{q+1}
 \|u\|_{q+1}^{q+1}-
\frac{a}{2}(\alpha+\varepsilon)\|u\|^2-\frac{b}{4}(\alpha+\varepsilon)
 \|u\|^4-A\|u\|_p^{p}\\
&\geq \frac{a[1-(\alpha+\varepsilon)]}{2}\|u\|^2
 +\frac{b[1-(\alpha+\varepsilon)]}{4}\|u\|^4
 -\frac{\|h\|_\infty K}{q+1}\|u\|^{q+1}-AM\|u\|^{p}\\
&\geq \|u\|^2\big(C_1-C_2\|h\|_\infty\|u\|^{q-1}-C_3\|u\|^{p-2}\big),
\end{aligned} \label{17}
\end{equation}
where $C_1=\frac{a[1-(\alpha+\varepsilon)]}{2}$,
 $C_2=\frac{ K}{q+1}$ and $C_3=AM$.
 Let 
$$
g(t)=C_2\|h\|_\infty t^{q-1}+C_3t^{p-2}\quad \text{for } t\geq0.
$$ 
Clearly, 
$$
g'(t)=C_2(q-1)\|h\|_\infty  t^{q-2}+(p-2) C_3t^{p-3}.
$$
From $g'(t_0)=0$, we have
$$
t_0=(C_4\|h\|_\infty)^{\frac{1}{p-q-1}},\quad 0<q<1<4<p,
$$
where $C_4=\frac{C_2(1-q)}{(p-2)C_3}$. Then
 $$
g(t_0)=C_2\|h\|_\infty (C_4\|h\|_\infty)^{\frac{q-1}{p-q-1}}
+C_3(C_4\|h\|_\infty)^{\frac{p-2}{p-q-1}}=C_5\|h\|_\infty^{\frac{p-2}{p-q-1}},
$$
where $C_5=C_2 C_4^{\frac{q-1}{p-q-1}}+C_3C_4^{\frac{p-2}{p-q-1}}$ and 
$\frac{p-2}{p-q-1}>0$ because $0<q<1<4<p$. Thus, for any $p>4$, 
there exists $m>0$ such that $g(t_0)<C_1$ if $\|h\|_\infty<m$. 
Then, if $\|h\|_\infty<m$ and taking $\rho=t_0$, from \eqref{17},  
(a) is proved.

(b) For $t>0$ large enough, by \eqref{3} and $0<q<1$, taking
$\varepsilon>0$ such that $\varepsilon<\min\{\beta-1,1-\alpha\}$, we have
\begin{align*}
I(t\varphi_1)
&= \frac{a t^2}{2}\int_{\Omega}| \nabla \varphi_1|^2dx
 +\frac{b t^4}{4}\left(\int_{\Omega}| \nabla \varphi_1|^2dx\right)^2
 -\frac{t^{q+1}}{q+1}\int_{\Omega}h(x)\varphi_1^{q+1}dx\\
&\quad -\int_{\Omega}F(x,t\varphi_1)dx\\
&\leq \frac{a t^2}{2}\| \varphi_1\|^2+\frac{b t^4}{4}\|\varphi_1\|^4
 -\frac{t^{q+1}}{q+1}\int_{\Omega}h(x)\varphi_1^{q+1}dx
 -\frac{at^2}{2}(\beta-\varepsilon)\lambda_1\|\varphi_1\|_2^2\\
&\quad -\frac{bt^4}{4}(\beta-\varepsilon)\mu_1\|\varphi_1\|_4^4 +B|\Omega|\\
&\leq \frac{a t^2}{2}\| \varphi_1\|^2+\frac{b t^4}{4}\|\varphi_1\|^4
 -\frac{t^{q+1}}{q+1}\int_{\Omega}h(x)\varphi_1^{q+1}dx
 -\frac{bt^4}{4}(\beta-\varepsilon)\|\varphi_1\|^4+B|\Omega|\\
&= \frac{a t^2}{2}\| \varphi_1\|^2-\frac{b t^4}{4}(\beta-\varepsilon-1)
 \|\varphi_1\|^4-\frac{t^{q+1}}{q+1}\int_{\Omega}h(x)\varphi_1^{q+1}dx+B|\Omega|\\
&\to -\infty %\label{18}
\end{align*}
as $t\to\infty$. So we can choose $t^0>0$ large enough  
and $e=t\varphi_1$ so that $I(e)<0$ and $\|e\|>\rho$.
\end{proof}


\begin{proof}[Proof of Theorem 1.1]
 Set $\rho$ as in Lemma 2.1(a), define
$$
\overline{B}_\rho=\{u\in H_0^1(\Omega): \|u\|\leq\rho\},\quad
\partial B_\rho=\{u\in H_0^1(\Omega):\ \|u\|=\rho\}
$$
and $\overline{B}_\rho$ is a complete metric space with the distance
$$
\text{dist}(u,v)=\|u-v\|\ \text{for}\ u,\ v\in \overline{B}_\rho.
$$
By Lemma 2.1, 
\begin{equation}
 I(u)|_{\partial B_\rho}\geq \gamma>0. \label{200}
\end{equation}
Clearly, $I\in C^1(\overline{B}_\rho, \mathbb{R})$, hence $I$ is lower 
semicontinuous and bounded from below on
$\overline{B}_\rho$. Let
\begin{equation} c_1=\inf\{I(u):\ u\in \overline{B}_\rho\}. \label{201}
\end{equation}
We claim that \begin{equation} c_1<0.\label{202} 
\end{equation}
Indeed, let $v\in H_0^1(\Omega)$ be given by (H2), that is, 
$\int_\Omega h(x)(v^+)^{q+1}dx>0,$ then for $t>0$ small enough such 
that for any $\varepsilon>0$, we have $|tv|<\varepsilon$. 
Therefore, together (F2) and $\alpha>1$ imply
\begin{align*}
I(tv)&= \frac{a t^2}{2}\int_{\Omega}| \nabla v|^2dx
 +\frac{b t^4}{4}\Big(\int_{\Omega}| \nabla v|^2dx\Big)^2
 -\frac{t^{q+1}}{q+1}\int_{\Omega}h(x)(v^+)^{q+1}dx\\
&\quad -\int_{\Omega}F(x,tv^+)dx\\
&\leq \frac{a t^2}{2}\| v\|^2+\frac{b t^4}{4}\|v\|^4
 -\frac{t^{q+1}}{q+1}\int_{\Omega}h(x)(v^+)^{q+1}dx\\
&\quad - \frac{at^2}{2}(\alpha+\varepsilon)\lambda_1\|v\|_2^2
 -\frac{bt^4}{4}(\alpha+\varepsilon)\mu_1\|v\|_4^4
<0,
\end{align*}
if $t>0$ small enough, because $0<q<1$. 
So \eqref{202} is proved.

By the Ekeland's variational principle \cite[Theorem 1.1]{33}
 in $\overline{B}_\rho$ and \eqref{201}, there is a minimizing
sequence $\{u_n\}\subset \overline{B}_\rho$  such that
\begin{itemize}
\item[(i)] $c_1<I(u_n)<c_1+\frac{1}{n}$,
\item[(ii)] $I(w)\geq I(u_n)-\frac{1}{n}\|w-u_n\|$
 for all $w\in \overline{B}_\rho$. %\label{203}
\end{itemize}
So, $I'(u_n)\to0$ in $H_0^{-1}(\Omega)$ as $n\to\infty$.
 Moreover, by (i) and (ii), we obtain $I(u_n)\to c_1<0$ as $n\to\infty$.

 From the above discussion, we know that  $\{u_n\}$ is a bounded $(PS)$ 
sequence, there exist a subsequence (still denoted by $\{u_n\}$) and  
$u_1\in H_0^1(\Omega)$ such that 
\begin{equation} \label{iiii}
\begin{gathered}
   u_n\rightharpoonup u_1 \quad \text{weakly  in } H_0^1(\Omega), \\
  u_n\to u_1 \quad \text{a.e. in } \Omega,\\
u_n\to u_1 \quad  \text{strongly in }L^r(\Omega)
 \end{gathered}
\end{equation}
as $n\to\infty$, where $r\in[1,6]$ if $N=3$ and $r\in(1,+\infty)$ if $N=1,2$.
 Thus, we have
$\lim_{n\to\infty}\langle I'(u_n),v\rangle=\langle I'(u_1),v\rangle=0$ 
for all $v\in H_0^1(\Omega)$ and
$\lim_{n\to\infty} I(u_n)=c_1<0$.  Moreover, it follows 
from $\langle I'(u_1), u_1^-\rangle=(a+b\|u_1\|^2)\|u_1^-\|^2 =0$ that 
$u_1=u_1^+\geq0$, where $u_1^-=\max\{-u_1,0\}$. Therefore, 
$u_1$ is a nonnegative critical point of $I$. Furthermore, if $h(x)\geq0$, 
the strong maximum principle \cite{45} implies that $u_1$ is a positive 
solution of problem \eqref{1}.
\end{proof}

\section{Existence of a mountain pass solution}

In this section, we use a variant version of mountain pass theorem to obtain
a nonzero critical point of functional I; this theorem is used 
also in \cite{42} and its proof can be found in \cite{41}, let us recall 
first this theorem.

 \begin{lemma}[Mountain Pass Theorem] \label{lem3.1}
Let $E$ be a real Banach space with its dual
space $E^*$ and suppose that $I\in C^1(E,R)$ satisfy the condition
$$
\max\{I(0),I(e)\}\leq\kappa<\gamma\leq\inf_{\|u\|=\rho}\{I(u)\}
$$
for some $\kappa<\gamma$, $\rho>0$ and $e\in E$ with $\|e\|>\rho$.
 Let $c\geq\gamma$ be characterized by
$$
c=\inf_{h\in \Gamma}\max_{t\in[0,1]}I(h(t)),
$$
where $\Gamma=\{h\in([0,1],E)| h(0)=0, h(1)=e\}$ is the set of continuous 
paths joining $0$ and $e$. Then, there exists a sequence $\{u_n\}\subset E$ 
such that
$$
I(u_n)\to c\geq\gamma\quad \text{and}\quad (1+\|u_n\|)\|I'(u_n)\|_{E^{-1}}\to0
$$
as $n\to\infty$.
\end{lemma}

\begin{proof}[Proof of Theorem 1.2]
Let $\rho$, $\gamma$ and $e$ be given in Lemma 2.1, applying
Lemma 3.1 with $\kappa=0$, $E=H_0^1(\Omega)$, and for $c$ defined as in
Lemma 3.1, then there exists a sequence
 $\{u_n\}\subset H_0^1(\Omega)$ such that
 $$
I(u_n)\to c\geq\gamma\quad \text{and}\quad (1+\|u_n\|)\|I'(u_n)\|_{E^{-1}}\to0
$$
as $n\to\infty$. This implies that 
\begin{gather}\label{204}
\begin{aligned}
&\frac{a}{2}\int_{\Omega}| \nabla u_n|^2 dx+\frac{b}{4}
 \Big(\int_{\Omega}| \nabla u_n|^2dx\Big)^2
 -\frac{1}{q+1}\int_{\Omega}h(x)(u_n^+)^{q+1}dx\\
&-\int_{\Omega}F(x,u_n^+)dx=c+o(1),
\end{aligned}\\
\begin{aligned}
&a\int_{\Omega}\nabla u_n\cdot\nabla \varphi dx+b\int_{\Omega}| \nabla u_n|^2dx
\int_{\Omega}\nabla u_n\cdot\nabla \varphi dx
 -\frac{1}{q+1}\int_{\Omega}h(x)(u_n^+)^{q}\varphi\\
& -\int_{\Omega}f(x,u_n^+)\varphi dx=o(1),\quad \text{for }
  \varphi\in H_0^1(\Omega),
\end{aligned}\label{205}
\\
\label{206}
a\int_{\Omega}| \nabla u_n|^2dx+b\Big(\int_{\Omega}| \nabla u_n|^2dx\Big)^2
-\int_{\Omega}h(x)(u_n^+)^{q+1}dx-\int_{\Omega}f(x,u_n^+)u_n^+dx=o(1).
\end{gather}
By the compactness of Sobolev embedding and the standard procedures, 
we know that, if $\{u_n\}$ is bounded in $H_0^1(\Omega)$, there exists 
$u_2\in H_0^1(\Omega)$ such that $I'(u_2)=0$ and  $I(u_2)=c>0$ and $u_2$ 
is a nonnegative weak solution of problem \eqref{1}, which is positive if 
$h(x)\geq0$  by the strong maximum principle. Moreover, $u_2$ is 
different from the solution $u_1$ obtained in Theorem 1.1 since $I(u_1)=c_1<0$.
 So, to prove Theorem 1.2, we only need to prove that $\{u_n\}$ given 
by \eqref{204}$-\eqref{206}$ is bounded in $H_0^1(\Omega)$.

Next, we shall show that $\{u_n\}$  is bounded in $H_0^1(\Omega)$. 
By contradiction, we suppose that
$\|u_n\|\to\infty$ as $n\to\infty$, and set $w_n=\frac{u_n}{\|u_n\|}$. 
Clearly, $\{w_n\}$ is bounded in $H_0^1(\Omega)$. Thus, there exist 
a subsequence, still denoted by $\{w_n\}$, and $w\in H_0^1(\Omega)$, such that
%   \label{iiii}
\begin{gather*}
   w_n\rightharpoonup w\quad \text{weakly  in } H_0^1(\Omega), \\
  w_n\to w\quad \text{a.e. in }\Omega,\\
w_n\to w \quad \text{strongly in }L^r(\Omega)
 \end{gather*}
as $n\to\infty$, where $r\in[1,6]$ if $N=3$ and $r\in(1,+\infty)$ if $N=1, 2$.

Similarly, $w_n^+=\frac{u_n^+}{\|u_n\|}$ also satisfies
% \label{iiii}
\begin{gather*}
   w_n^+\rightharpoonup w^+\quad \text{weakly  in } H_0^1(\Omega), \\
  w_n^+\to w^+\quad \text{a.e.  in}\ \Omega,\\
w_n^+\to w^+ \quad \text{strongly in } L^r(\Omega)
 \end{gather*}
as $n\to\infty$.
We first claim that $w\not\equiv0$. Indeed, if $w\equiv0$, then by (H1), 
we have
 \begin{equation}
\lim_{n\to\infty}\int_{\Omega}h(x)(w_n^+)^{q+1}dx=0.\label{208}
\end{equation}
Moreover, by  (F1)-(F2), for any $\varepsilon>0$, if $s>0$ large enough, 
we obtain
$$
(\beta-\varepsilon)a\lambda_1s+(\beta-\varepsilon)b\mu_1s^3<
f(x,s)<(\beta+\varepsilon)a\lambda_1s+(\beta+\varepsilon)b\mu_1s^3.
$$
Therefore, we deduce
$$
(\beta-\varepsilon)a\lambda_1s-\varepsilon b\mu_1s^3<
f(x,s)-\beta b\mu_1s^3<(\beta+\varepsilon)a\lambda_1s+\varepsilon b\mu_1s^3.
$$
It implies that
\begin{align*}
&\frac{(\beta-\varepsilon)\lambda_1}{\|u_n\|^2}\int_{\Omega}w_n^+\varphi dx
-\varepsilon b\mu_1\int_{\Omega}(w_n^+)^{3}\varphi dx\\
&< \int_{\Omega}\frac{f(x,u_n^+)-b\beta\mu_1(u_n^+)^3}{\|u_n\|^3}\varphi dx\\
&< \frac{(\beta+\varepsilon)\lambda_1}{\|u_n\|^2}\int_{\Omega}w_n^+\varphi dx
 -\varepsilon b\mu_1\int_{\Omega}(w_n^+)^{3}\varphi dx
\end{align*}
for any $\varphi\in H_0^1(\Omega)$.
By the arbitrariness of $\varepsilon$, we obtain
\begin{equation}
\lim_{n\to+\infty}\int_{\Omega}\frac{f(x,u_n^+)
-b\beta\mu_1(u_n^+)^3}{\|u_n\|^3}\varphi dx=0.\label{209}
\end{equation}
Multiplying \eqref{205} by $\frac{1}{\|u_n\|^3}$, we have
 \begin{equation}
\begin{aligned}
&\frac{a}{\|u_n\|^2}\int_{\Omega}\nabla w_n\cdot\nabla\varphi dx
 +b\int_{\Omega}\nabla w_n\cdot\nabla\varphi dx
 -\frac{1}{\|u_n\|^{3-q}}\int_{\Omega}h(x)(w_n^+)^{q}\varphi dx\\
& -b\beta\mu_1\int_{\Omega}(w_n^+)^{3}\varphi dx
 -\int_{\Omega}\frac{f(x,u_n^+)-b\beta\mu_1(u_n^+)^3}{\|u_n\|^3}\varphi dx=o(1).
\end{aligned} \label{207}
\end{equation}
Letting $n\to\infty$ in \eqref{207}, according to  $\|u_n\|\to\infty$ 
as $n\to\infty$, \eqref{208}, \eqref{209} and $b\neq0$, we have
\begin{align*}
\int_{\Omega}\nabla w\cdot\nabla\varphi dx
=\beta\mu_1\int_{\Omega}(w^+)^{3}\varphi dx
\end{align*}
and $w\neq0$. Hence, $\beta\mu_1$ is  an eigenvalue of \eqref{210}, which
 contradicts with the assumption. The proof is complete.
\end{proof}

\subsection*{Acknowledgments}
The authors express their gratitude to the anonymous referees for the
useful comments and remarks.

This research was supported by National Natural Science Foundation 
of China (No. 11101347), the Key Project in Science and Technology
 Research Plan of the Education
Department of Hubei Province (No. D20112605, No. D20122501),
Fundamental Funds for the Central Universities (No. XDJK2013D007)
and the Science and Technology Funds of Chongqing Educational
Commission (No. KJ130703).

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\end{document}