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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 202, 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 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/202\hfil Multiple positive solutions]
{Multiple positive solutions for Kirchhoff problems
 with sign-changing potential}

\author[G.-S. Liu,  C.-Y. Lei, L.-T. Guo, H. Rong \hfil EJDE-2015/202\hfilneg]
{Gao-Sheng Liu,  Chun-Yu Lei, Liu-Tao Guo, Hong Rong}

\address{Gao-Sheng Liu \newline
School  of  Science, Guizhou Minzu University,  Guiyang 550025, China}
\email{772936104@qq.com}

\address{Chun-Yu Lei (corresponding author)\newline
School  of  Science, Guizhou Minzu University,  Guiyang 550025, China}
\email{leichygzu@sina.cn, Phone +86 15985163534 }

\address{Liu-Tao Guo \newline
School  of  Science, Guizhou Minzu University,  Guiyang 550025, China}
\email{350630542@qq.com}

\address{Hong Rong \newline
School  of  Science, Guizhou Minzu University,  Guiyang 550025, China}
\email{402453552@qq.com}

\thanks{Submitted June 22, 2015. Published August 4, 2015.}
\subjclass[2010]{35D05, 35J60, 58J32}
\keywords{Kirchhoff type equation;
sign-changing potential; Nehari manifold}

\begin{abstract}
 In this article, we study the existence and multiplicity of positive solutions
 for a class of Kirchhoff type equations with sign-changing potential.
 Using the Nehari manifold, we obtain two positive solutions.
\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 result}

 Consider the  Kirchhoff type problems with Dirichlet boundary value conditions
\begin{equation}\label{1}
\begin{gathered}
-(a+b\int_\Omega(|\nabla u|^2+v(x)u^2)\,dx)(\Delta u-v(x)u)
=h(x)u^{p}+{\lambda}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 $\mathbb{R}^{3}$, $a>0$, $b>0$,
$\lambda>0$, $3<p<5$, $h\in{C(\bar{\Omega})}$, with $h^+=\max\{h,0\}\neq0$,
$v\in{C(\bar{\Omega})}$ is a bounded function with $\|v\|_{\infty}>0$, and
$f(x,u)$ satisfies the following two conditions:
\begin{itemize}
\item[(F1)] $f(x,u)\in{C^1(\Omega\times \mathbb{R})}$ with $f(x,0)\geq0$,
and $f(x,0)\neq0$. There exists a constant
$c_1>0$, such that $f(x,u)\leq c_1(1+u^q)$ for $0<q<1$ and
$(x,u)\in \Omega \times \mathbb{R}^+$.

\item[(F2)] $f_u(x,u)\in{L^{\infty}(\Omega \times \mathbb{R})}$
and for all $u\in H_0^{1}(\Omega)$,
$\int_{\partial \Omega}\frac{\partial}{\partial u}f(x,t|u|)u^2$ has the same
sign for every $t\in(0,+\infty)$.
\end{itemize}

\begin{remark} \label{rmk1.1} \rm
Note that under assumptions (F1) and (F2) hold, we have:
\begin{itemize}
\item[(F3)] there exists a constant $c_2>0$, such that
 $pf(x,u)-uf_{u}(x,u)\leq c_2(1+u)$, for all
$(x,u)\in \Omega \times \mathbb{R}^+$.

\item[(F4)] $F(x,u)-\frac{1}{p+1}f(x,u)u\leq c_2(1+u^2)$, for all
$(x,u)\in \Omega \times \mathbb{R}^+$, where $F(x,u)$ is
defined by $F(x,u)=\int_0^{u}f(x,s)ds$ for $x\in \Omega$, $u\in \mathbb{R}$.
\end{itemize}
\end{remark}

In recent years, the existence and multiplicity of solutions to the nonlocal problem
\begin{equation}\label{1.2}
\begin{gathered}
-\Big(a+b\int_\Omega|\nabla u|^2\,dx\Big)\Delta u=g(x,u) \quad \text{in }\Omega, \\
u=0,    \quad\text{on }    \partial\Omega,
\end{gathered}
\end{equation}
have been studied by various researchers and many interesting and important 
results can be found. For instance, positive solutions could be obtained 
in \cite{CF,CX,TJ}. Especially, Chen et al \cite{CT}  discussed a Kirchhoff 
type problem when $g(x,u)=f(x)u^{p-2}u+\lambda g(x)|u|^{q-2}u$, where 
$1<q<2<p<2^{*}$($2^{*}=\frac{2N}{N-2}$ if $N\geq3$, $2^{*}=\infty$ 
if $N=1,2$), $f(x)$ and $g(x)$ with some proper conditions are sign-changing 
weight functions. And they have obtained the existence of two positive solutions 
if $p>4$, $0<\lambda<\lambda_0(a)$.
Researchers, such as Mao and Zhang \cite{AZ}, Mao and Luan \cite{AS}, found 
sign-changing solutions.
As for infinitely many solutions, we refer readers to \cite{JX,LX}. 
He and Zou \cite{XW} considered the class of Kirchhoff type problem when
$g(x,u)=\lambda f(x,u)$ with some conditions and  proved a sequence of a.e. 
positive weak solutions tending to zero in $L^{\infty}(\Omega)$.
In addition, problems on unbounded domains have been studied by researchers,
such as Figueiredo and Santos Junior \cite{GJ}, Li et al. \cite{YF}, 
Li and Ye \cite{GH}.

Our main result read as follows.

\begin{theorem} \label{thm1.1}
 Assume that conditions {\rm (F1)} and {\rm (F2)} hold. 
Then there exists $\lambda^{*}>0$ such that for any $\lambda\in (0,\lambda^{*})$, 
 problem \eqref{1} has at least two  positive solutions.
\end{theorem}

The article is organized as following: 
Section 2 contains notation and preliminaries. 
Section 3 contains the proof of Theorem \ref{thm1.1}.

\section{Preliminaries}

Throughout this article, we use the following notation:
The space $H_0^{1}(\Omega)$ is equipped with the norm
$\|u\|^2=\int_{\Omega}(|\nabla u|^2+v(x)|u|^2)\,dx$.
Let $S_r$ be the best Sobolev constant for the embedding of 
$H_0^{1}(\Omega)$ into $L^{r}(\Omega)$, where $1\leq r < 6$, then
\begin{equation}\label{2.1}
\frac{1}{S_{p+1}^{2(p+1)}}
\leq \frac{\|u\|^{2(p+1)}}{(\int_{\Omega}|u|^{p+1})^2} .
\end{equation}
We define a functional $I_{\lambda}(u)$: $H_0^{1}(\Omega)\to \mathbb{R}$ by
\begin{equation}\label{2.2}
I_{\lambda}(u)=\frac{a}{2}\|u\|^2+\frac{b}{4}\|u\|^4
-\frac{1}{p+1}H(u)-\lambda\int_{\Omega}F(x,|u|)\,dx
\quad\text{for } u\in H_0^{1}(\Omega) ,
\end{equation}
where
\[
H(u)=\int_{\Omega}h(x)|u|^{p+1}\,dx.
\]
The weak solutions of \eqref{1} is the critical points of the functional 
$I_{\lambda}$. Generally speaking, a function $u$ is called a solution 
of \eqref{1} if
$u\in H_0^{1}(\Omega)$ and for all $\varphi\in H_0^{1}(\Omega)$ it
holds
\[
(a+b\|u\|^2)\int_{\Omega}(\nabla u \cdot \nabla \varphi+v(x)u\varphi)\,dx
=\int_{\Omega}h(x)|u|^{p-1}|u|\varphi \,dx+\lambda\int_{\Omega}f(x,|u|)\varphi \,dx.
\]
As $I_{\lambda}(u)$ is unbounded below on $H_0^{1}(\Omega)$,
it is useful to consider the functional on the Nehari manifold:
\[
\mathcal{N}_\lambda(\Omega)
=\{u\in H_0^{1}(\Omega)\backslash{\{0\}}: \langle I'_{\lambda}(u),u\rangle=0 \}.
\]
It is obvious that the Nehari manifold contains all the nontrivial critical 
points of $I_{\lambda}$, thus, for $u\in \mathcal{N}_\lambda(\Omega)$, 
if and only if
\begin{equation}\label{2.3}
(a+b\|u\|^2)\|u\|^2-\int_{\Omega}h(x)|u|^{p+1} \,dx
-\lambda\int_{\Omega}f(x,|u|)|u|\,dx=0.
\end{equation}
Define
\[
  \psi_{\lambda}(u)=\langle I'_{\lambda}(u),u \rangle,
\]
then it follows that
\begin{gather}\label{2.4}
  I_{\lambda}(tu)=\frac{a}{2}t^{2}\|u\|^{2}+\frac{b}{4}t^{4}\|u\|^{4}
-\frac{t^{p+1}}{p+1}\int_{\Omega}h(x)|u|^{p+1}\,dx-\lambda\int_{\Omega} F(x,|tu|)\,dx,\\
\label{2.5}
   \psi_{\lambda}(tu)=at^{2}\|u\|^{2}+bt^{4}\|u\|^{4}
-t^{p+1}\int_{\Omega}h(x)|u|^{p+1}\,dx-\lambda\int_{\Omega} f(x,|tu|)|tu|\,dx, \\
\label{2.6}
\begin{aligned}
\langle \psi'_{\lambda}(tu),tu \rangle
&=2at^{2}\|u\|^{2}+4bt^{4}\|u\|^{4}-(p+1)t^{p+1}\int_{\Omega}h(x)|u|^{p+1}\,dx\\
&\quad -\lambda\int_{\Omega} f_{u}(x,|tu|)|tu|^{2}\,dx
 -\lambda\int_{\Omega} f(x,|tu|)|tu|\,dx.
\end{aligned}
\end{gather} 
Notice that $\psi_{\lambda}(tu)=0$ if and only if 
$tu\in \mathcal{N}_{\lambda}(\Omega)$. And we divide 
$\mathcal{N}_{\lambda}(\Omega)$ into three parts:
\begin{gather*}
\mathcal{N}^{-}_{\lambda}(\Omega)=\{u\in \mathcal{N}_{\lambda}(\Omega):
\langle\psi'_{\lambda}(u),u \rangle<0\},\\
\mathcal{N}^{+}_{\lambda}(\Omega)=\{u\in \mathcal{N}_{\lambda}(\Omega):
\langle\psi'_{\lambda}(u),u \rangle>0\},\\
\mathcal{N}^{0}_{\lambda}(\Omega)=\{u\in \mathcal{N}_{\lambda}(\Omega):
\langle\psi'_{\lambda}(u),u \rangle=0\}.
\end{gather*}
Then we have the following results.

\begin{lemma} \label{lem2.1} 
There exists a constant $\lambda_1>0$, for $0<\lambda<\lambda_1$, such that 
$\mathcal{N}^{0}_{\lambda}(\Omega)=\emptyset$.
\end{lemma}

\begin{proof} 
By contradiction, suppose $u\in \mathcal{N}^{0}_{\lambda}(\Omega)$, we obtain
\begin{align*}
\langle \psi'_{\lambda}(u),u \rangle
&= 2a\|u\|^{2}+4b\|u\|^{4}-(p+1)\int_{\Omega}h(x)|u|^{p+1}\,dx\\
&\quad -\lambda\int_{\Omega} f_{u}(x,|u|)|u|^{2}\,dx
 -\lambda\int_{\Omega} f(x,|u|)|u|\,dx=0.
\end{align*} 
On one hand, from \eqref{2.1}, \eqref{2.3}, \eqref{2.6} and (F2), one deduces that
\begin{align*}
a\|u\|^{2}+3b\|u\|^{4}
&= p\int_{\Omega}h(x)|u|^{p+1}\,dx+\lambda\int_{\Omega} f_{u}(x,|u|)u^{2}\,dx\\
&\leq  L\|u\|^{p+1}+\lambda L'\|u\|^{2},
\end{align*} 
where $L=p\|h\|_{\infty}S_{p+1}^{p+1}$, 
$L'=\|f_{u}(x,|u|)\|_{L^{\infty}}S_2^{2}$,
then
\[
  L\|u\|^{p+1}\geq (a-\lambda L')\|u\|^{2}+3b\|u\|^{4}\geq (a-\lambda L')\|u\|^{2},
\]
consequently,
\begin{equation}\label{2.7}
  \|u\|^{2}\geq \Big(\frac{a-\lambda L'}{L}\Big)^{\frac{2}{p-1}}.
\end{equation}
On the other hand, by \eqref{2.1}, \eqref{2.3}, \eqref{2.6} and (F3), we obtain
\begin{align*}
a(p-1)\|u\|^2+(bp-3)\|u\|^4
&\leq \lambda \Big(\int_{\Omega}(pf(x,|u|)-f_{u}(x,|u|)|u|)|u|\,dx\Big)\\
&\leq c_2\lambda  \int_{\Omega}(|u|+|u|^2)\,dx\\
&\leq \lambda c_2|\Omega|^{\frac{1}{2}}S_1 \|u\| +\lambda c_2 S_2^{2}\|u\|^2,
\end{align*}
then
\[
\lambda c_2|\Omega|^{\frac{1}{2}}S_1 \|u\| +\lambda c_2 S_2^{2}\|u\|^2
\geq a(p-1)\|u\|^2,
\]
thus one has
\begin{equation}\label{2.8}
\|u\|^2\leq \Big(\frac{\lambda c_2 S_1|\Omega|^{1/2}}{a(p-1)
-c_2\lambda S_2^{2}}\Big)^2.
\end{equation}
It follows from \eqref{2.7} and \eqref{2.8} that
\[
\Big(\frac{a-\lambda L'}{L}\Big)^\frac{2}{p-1}
\leq \|u\|^2\leq \Big(\frac{\lambda c_2 S_1|\Omega|^{1/2}}{a(p-1)
-c_2\lambda S_2^{2}}\Big)^2,
\]
 which is a contradiction when $\lambda$ is small enough. 
So there exists a constant $\lambda_1>0$ such that 
$\mathcal{N}^{0}_{\lambda}(\Omega)=\emptyset$. The proof is complete.
\end{proof}

\begin{lemma} \label{lem2.2} 
There exists a constant $\lambda_2>0$, for $0<\lambda<\lambda_2$, such 
that $\mathcal{N}^{\pm}_{\lambda}(\Omega)\neq \emptyset$.
\end{lemma}

\begin{proof}
 For $u\in H^{1}_0(\Omega)$, $u\neq0$, let
\begin{gather*}
  A_{u}(t)=\frac{a}{2}t^{2}\|u\|^2+\frac{b}{4}t^{4}\|u\|^4
-\frac{t^{p+1}}{p+1}\int_{\Omega}h(x)|u|^{p+1}\,dx, \\
 K_{u}(t)=\int_{\Omega}F(x,|tu|)\,dx,
\end{gather*}
then $I_{\lambda}(tu)= A_{u}(t)-\lambda K_{u}(t)$, 
hence if $\psi_{\lambda}(tu)=\langle I'_{\lambda}(tu),tu\rangle = 0$,
then $A'_{u}(t)-\lambda K'_{u}(t)=0$,
where
\begin{gather*}
  A'_{u}(t)=at^{2}\|u\|^2+bt^{3}\|u\|^4-t^{p}\int_{\Omega}h(x)|u|^{p+1}\,dx, \\
 K'_{u}(t)=\int_{\Omega}f(x,|tu|)|u|\,dx.
\end{gather*}
By (F1), one obtains
\begin{equation}\label{2.9}
K'_{u}(t)=\int_{\Omega}f(x,|tu|)|u|\,dx
\leq \int_{\Omega}c_2(1+|tu|^{q})|u|\,dx.
\end{equation}
We consider the following two cases:
\smallskip

\noindent\textbf{Case 1.}
When $H(u)\leq 0$ and $\int_{\Omega}f(x,t|u|)u^2\,dx>0$, we have 
$A'_{u}(t)>0$, $A_{u}(0)=0$ and $A_{u}(t)$ increases sharply when 
$t\to\infty$. At the same time, $K'_{u}(t)>0$, $K_{u}(0)$ is a positive constant 
and $K_{u}(t)$ increases relatively slowly when $t\to\infty$ since \eqref{2.9}.
When $H(u)\leq 0$ and $\int_{\Omega}f(x,t|u|)u^2\,dx\leq0$, we have
$K'_{u}(t)\leq0$, $K_{u}(0)$ is a positive constant and $K_{u}(t)$ 
decreases slowly when $t\to\infty$ since \eqref{2.9}.

Through the above discussion, we obtain there exists $t_1$ such that 
$t_1 u\in \mathcal{N}_{\lambda}(\Omega)$  to every situation. 
When $0<t<t_1$, one gets $\psi_{\lambda}(tu)<0$ and when $t>t_1$,
 we have $\psi_{\lambda}(tu)>0$, then $t_1 u$ is the local minimizer 
of $I_{\lambda}(u)$, so $t_1 u\in \mathcal{N}^{+}_{\lambda}(\Omega)$. 
In conclusion, when $H(u)\leq 0$, one has 
$\mathcal{N}^{+}_{\lambda}(\Omega)\neq \emptyset$.
\smallskip

\noindent\textbf{Case 2.}
When $H(u)>0$ and $\int_{\Omega}f(x,t|u|)u^2\,dx>0$, we have
 $A'_{u}(t)>0$ as $t\to 0$ and $A'_{u}(t)<0$ for $t\to \infty$, 
so $A_{u}(t)$ increases as $t\to 0$ and then decreases as $t\to \infty$. 
At the same time, $K'_{u}(t)>0$, $K_{u}(0)$ is a positive constant and 
$K_{u}(t)$ increases relatively slowly when $t\to\infty$ since \eqref{2.9}.
When $H(u)>0$ and $\int_{\Omega}f(x,t|u|)u^2\,dx<0$, we have
 $A'_{u}(t)>0$ as $t\to 0$ and $A'_{u}(t)<0$ for $t\to \infty$,
 so $A_{u}(t)$ increases as $t\to 0$ and then decreases as $t\to \infty$. 
At the same time, $K'_{u}(t)<0$, $K_{u}(0)$ is a positive constant and 
$K_{u}(t)$ decreases slowly when $t\to\infty$ since \eqref{2.9}.

Through the above discussion, if $\lambda$ is small enough, there exists 
$t_1<t_2$, such that $\psi_{\lambda}(tu)=0$, for $0<t<t_1$, 
$\psi_{\lambda}(tu)<0$, for $t_1<t<t_2$, $\psi_{\lambda}(tu)>0$, and for
 $t>t_2$, $\psi_{\lambda}(tu)<0$. Thus $t_1 u$ is the local minimizer 
of $I_{\lambda}(u)$ and $t_2 u$ is the local maximizer of $I_{\lambda}(u)$. 
So there exists $\lambda_2>0$, when $\lambda< \lambda_2$,
one gets $t_1 u\in \mathcal{N}^{+}_{\lambda}(\Omega)$ and 
$t_2 u\in \mathcal{N}^{-}_{\lambda}(\Omega)$. Therefore one concludes that when 
$H(u)>0$ and $\lambda$ is small enough, 
$\mathcal{N}^{\pm}_{\lambda}(\Omega)\neq \emptyset$. This completes the proof.
\end{proof}

\begin{lemma} \label{lem2.3} 
Operator $I_{\lambda}$ is coercive and bounded below on
 $\mathcal{N}_\lambda(\Omega)$.
\end{lemma}

\begin{proof}
From \eqref{2.1}, \eqref{2.2}, \eqref{2.3} and (F4), one has
\begin{align*}
I_{\lambda}(u)
&= a\Big(\frac{1}{2}-\frac{1}{p+1}\Big)\|u\|^2
 +b \Big(\frac{1}{4}-\frac{1}{p+1}\Big)\|u\|^4\\
&\quad -\lambda\int_{\Omega}(F(x,|u|-\frac{1}{p+1}f(x,|u|)|u|)\,dx\\
&\geq  a \Big(\frac{1}{2}-\frac{1}{p+1}\Big)\|u\|^2
 +b \Big(\frac{1}{4}-\frac{1}{p+1}\Big)\|u\|^4
 -\lambda c_3 \int_{\Omega}(1+|u|^2)\,dx\\
&\geq  a \Big(\frac{1}{2}-\frac{1}{p+1}\Big)\|u\|^2
 +b \Big(\frac{1}{4}-\frac{1}{p+1}\Big)\|u\|^4
 -\lambda c_3\Big(|\Omega|+S_2^{2}\|u\|^2\Big)\\
&\geq  \Big(\frac{a(p-1)}{2(p+1)}  -\lambda c_3 S_2^{2}\Big)\|u\|^2
 +b \Big(\frac{1}{4}-\frac{1}{p+1}\Big)\|u\|^4-\lambda c_3|\Omega|.
\end{align*} 
By $3<p<5$, it follows that $I_{\lambda}(u)$ is coercive and bounded below 
on $\mathcal{N}_\lambda(\Omega)$. The proof is complete.
\end{proof}
 
\begin{remark} \label{rmk2.3}\rm
 From Lemmas \ref{lem2.1} and \ref{lem2.2}, one has 
$\mathcal{N}_\lambda(\Omega)=\mathcal{N}^{+}_\lambda(\Omega)\cup
 \mathcal{N}^{-}_\lambda(\Omega)$ for all
 $0<\lambda<\min\{\lambda_1,\lambda_2\}$. Furthermore, we obtain 
$\mathcal{N}^{+}_\lambda(\Omega)$ and $\mathcal{N}^{-}_\lambda(\Omega)$
 are non-empty, thus, we may define
\[
\alpha^{+}_\lambda=\inf_{u\in \mathcal{N}^{+}_\lambda(\Omega)}I_{\lambda}(u), 
\quad \alpha^{-}_\lambda=\inf_{u\in \mathcal{N}^{-}_\lambda(\Omega)}I_{\lambda}(u).
\]
\end{remark}

 \begin{lemma} \label{lem2.4} 
If $u\in H_0^{1}(\Omega)\backslash \{0\}$, there exists a constant
$\lambda_3>0$, such that $I_{\lambda}(tu)>0$, for $\lambda<\lambda_3$.
\end{lemma}

\begin{proof}  
For every $u\in H^{1}_0(\Omega)$, $u\neq0$, if $H(u)\leq 0$, by \eqref{2.4},
we obtain $I_{\lambda}(tu)>0$ when $t$ is large enough. Assume $H(u)>0$, and
let
\[
\phi_1(t)=\frac{a}{2}t^{2}\|u\|^{2}-\frac{t^{p+1}}{p+1}H(u).
\]
Through calculations, one obtains that $\phi_1(t)$ takes on a maximum at
\[
  t_{\rm max}=\Big(\frac{a\|u\|^2}{H(u)}\Big)^{\frac{1}{p-1}}.
\]
It follows that
\begin{align*}
\phi_1(t_{\rm max})
&=\frac{p-1}{2(p+1)}\Big( \frac{(a\|u\|^2)^{p+1}}
{(\int_{\Omega}h(x)|u|^{p+1}\,dx)^2}\Big)^{\frac{1}{p-1}}\\
&\geq \frac{p-1}{2(p+1)}\Big( \frac{a^{p+1}}{\|h^{+}\|_{\infty}^{2}S_{p+1}^{2(p+1)}}
\Big)^{\frac{1}{p-1}}:=\delta_1.
\end{align*}
When $1\leq r<6$, one has
\begin{equation}\label{2.10}
\begin{aligned}
(t_{\rm max})^{r}\int_{\Omega}|u|^{r}\,dx
&\leq S_{r}^{r}\Big(\frac{a\|u\|^2}{H(u)}\Big)^{\frac{r}{p-1}}
 (\|u\|^{2})^{r/2}\\
&=  S_{r}^{r}a^{-\frac{r}{2}} \Big(\frac{(a\|u\|^2)^{p+1}}{(H(u))^{2}}
 \Big)^{\frac{r}{2(p-1)}}\\
&=  S_{r}^{r}a^{-\frac{r}{2}} \Big(\frac{2(p+1)}{p-1}\Big)^{r/2}
\Big(\phi_1(t_{\rm max})\Big)^{r/2}\\
&=  c\big(\phi_1(t_{\rm max})\big)^{r/2}.
\end{aligned}
\end{equation}
Then by (F1) and (F4), we deduce that
\begin{equation}\label{2.11}
\begin{aligned}
&\int_{\Omega}F(x,t_{\rm max}|u|)\,dx\\
&\leq \frac{1}{p+1}\int_{\Omega}c_4 (2+|t_{\rm max}u|^2)\,dx
  +\int_{\Omega}c_1(|t_{\rm max}u|+|t_{\rm max}u|^{q+1})\\
&\leq  B_0+B_1 \phi_1(t_{\rm max})
 +B_2 (\phi_1(t_{\rm max}))^{1/2}
 +B_3 \phi_1(t_{\rm max})^\frac{q+1}{2}.
\end{aligned}
\end{equation}
Since
\[
I_{\lambda}(t_{\rm max}u)=A_{u}(t_{\rm max})
-\lambda K_{u}(t_{\rm max})\geq  \phi_1(t_{\rm max})
-\lambda \int_{\Omega}F(x,t_{\rm max}|u|)\,dx,
\]
according to \eqref{2.4}, \eqref{2.10} and \eqref{2.11}, one obtains
\begin{align*}
I_{\lambda}(t_{\rm max}u)
&\geq   \phi_1(t_{\rm max})-\lambda \int_{\Omega}F(x,t_{\rm max}|u|)\,dx\\
&\geq   \phi_1(t_{\rm max})-\lambda\Big[ B_0+B_1 \phi_1(t_{\rm max})
 +B_2 (\phi_1(t_{\rm max}))^{1/2}
 +B_3 \phi_1(t_{\rm max})^\frac{q+1}{2}\Big]\\
&\geq   \delta_1\Big[1-\lambda\Big(B_0\delta^{-1}+B_1
 +B_2\delta^{-\frac{1}{2}}+B_3\delta^{\frac{q-1}{2}}\Big)\Big].
\end{align*}
So, if $\lambda < \lambda_3=(2(B_0\delta^{-1}+B_1
+B_2\delta^{-\frac{1}{2}}+B_3\delta^{\frac{q-1}{2}}))^{-1}$,
we obtain $I_{\lambda}(t_{\rm max}u)>0$.
\end{proof}

\begin{remark} \label{rmk2.4} \rm
If $\lambda < \lambda_3$ and $u\in \mathcal{N}_{\lambda}^{-}(\Omega)$, by (F2), 
we conclude that there is a global maximum on $u$ for $I_{\lambda}(u)$, 
then $I_{\lambda}(u)>I_{\lambda}(t_{\rm max}u)>0$.
\end{remark}

\begin{lemma} \label{lem2.5} 
If $u\in H_0^{1}(\Omega)\backslash \{0\}$, there exists a constant $\lambda_4>0$ 
such that $\psi_{\lambda}(tu)=\langle I'_{\lambda}(tu),tu \rangle>0$ when 
$\lambda<\lambda_4$.
\end{lemma}

\begin{proof}  
For every $u\in H^{1}_0(\Omega)$, $u\neq0$, if $H(u)\leq 0$, by \eqref{2.5}, 
we get $\psi_{\lambda}(tu)>0$ when $t$ is large enough. 
Assume $H(u)>0$, and let
\[
\psi_1(t)=at^{2}\|u\|^{2}-t^{p+1}H(u).
\]
Through calculations, we obtain that $\psi_1(t)$ takes on a maximum at
\[
 \tilde{t}_{\rm max}=\Big(\frac{2a\|u\|^2}{(p+1)H(u)}\Big)^{\frac{1}{p-1}}.
\]
It follows that
\begin{align*}
\psi_1(\tilde{t}_{\rm max})
&= \Big(\frac{2a}{p+1}\Big)^{\frac{2}{p-1}}\Big(\frac{p-1}{p+1}\Big)
\Big( \frac{(\|u\|^2)^{p+1}}{(\int_{\Omega}h(x)|u|^{p+1}\,dx)^2}\Big)^{\frac{1}{p-1}}\\
&\geq \Big(\frac{2a}{p+1}\Big)^{\frac{2}{p-1}}\Big(\frac{p-1}{p+1}\Big)
\Big( \frac{1}{\|h^{+}\|_{\infty}^{2}S_{p+1}^{2(p+1)}}\Big)^{\frac{1}{p-1}}
:=\delta_2.
\end{align*}
Similar to the proof of Lemma \ref{lem2.4}, when $1\leq r<6$, one obtains
\begin{equation}\label{2.12}
(\tilde{t}_{\rm max})^{r}\int_{\Omega}|u|^{r}\,dx
\leq \tilde{c}\left(\psi_1(\tilde{t}_{\rm max})\right)^{r/2}.\\
\end{equation}
According to (F1), we deduce that
\begin{equation}\label{2.13}
\begin{aligned}
\int_{\Omega}f(x,\tilde{t}_{\rm max}|u|)|\tilde{t}_{\rm max}u|\,dx
&\leq c_1 \int_{\Omega}\left(|\tilde{t}_{\rm max}u|
 +|\tilde{t}_{\rm max}u|^{q+1}\right)\,dx\\
&\leq b_0\left(\psi_1(\tilde{t}_{\rm max})\right)^{1/2}
+b_1\left(\psi_1(\tilde{t}_{\rm max})\right)^\frac{q+1}{2},
\end{aligned}
\end{equation}
then, by \eqref{2.5}, \eqref{2.12} and \eqref{2.13}, it follows that
\begin{align*}
\psi_{\lambda}(\tilde{t}_{\rm max}u)
&\geq   \psi_1(\tilde{t}_{\rm max})
 - \lambda \int_{\Omega}f(x,\tilde{t}_{\rm max}|u|)|\tilde{t}_{\rm max}u|\,dx\\
&\geq  (\psi_1(\tilde{t}_{\rm max}))^{\frac{1+q}{2}}
 \left(\psi_1(\tilde{t}_{\rm max}))^{\frac{1-q}{2}}
 -\lambda (b_0(\psi_1(\tilde{t}_{\rm max}))^{-\frac{q}{2}}+b_1)\right)\\
&\geq   \delta_2^\frac{1+q}{2}\left( \delta_2^{\frac{1-q}{2}}
 -\lambda (b_0\delta_2^{-\frac{q}{2}}+b_1) \right),
\end{align*}
consequently, when $\lambda < \lambda_{4}
= \delta_2^{\frac{1-q}{2}}/2(b_0\delta_2^{-\frac{q}{2}}+b_1)$,
 we obtain $\psi_{\lambda}(\tilde{t}_{\rm max}u)>0$.
\end{proof}

\begin{remark} \label{rmk2.5} \rm
We claim that:
(1) If $H(u)\leq 0$ for every $u \in H_0^{1}(\Omega)\backslash \{0\}$, 
there exists $t_1$ such that $I_{\lambda}(t_1u)<0$ for 
$t_1u \in \mathcal{N}_{\lambda}^{+}(\Omega)$. Indeed, obviously, 
in this condition, $\psi_{\lambda}(0)<0$  and 
$\lim_{t\to\infty}\psi_{\lambda}(tu)=+\infty $, therefore, there exists 
$t_1>0$ such that $\psi_{\lambda}(tu)=0$. Because of $\psi_{\lambda}(tu)<0$ 
for $0<t<t_1$ and $\psi_{\lambda}(tu)>0$ for $t>t_1$, we obtain that 
$t_1u \in \mathcal{N}_{\lambda}^{+}(\Omega)$ and 
$I_{\lambda}(t_1u)<I_{\lambda}(0)=0$.

(2) If $H(u)>0$ for $0<\lambda< \lambda_1$, there exists $t_1<t_2$, 
such that $t_1u \in \mathcal{N}_{\lambda}^{+}(\Omega)$,
$t_2u \in \mathcal{N}_{\lambda}^{-}(\Omega)$ and $I_{\lambda}(t_1u)<0$. 
Indeed, in this condition, one gets $\psi_{\lambda}(0)<0$
and $\lim_{t\to\infty}\psi_{\lambda}(tu)=-\infty $. 
By Lemma \ref{lem2.5}, there exists $T>0$ such that $\psi_{\lambda}(Tu)>0$, 
therefore, we could obtain there exists $0<t_1<T<t_2$, such that  
$\psi_{\lambda}(t_1u)=\psi_{\lambda}(t_2u)=0$, 
$t_1u \in \mathcal{N}_{\lambda}^{+}(\Omega)$,
$t_2u \in \mathcal{N}_{\lambda}^{-}(\Omega)$ and 
$I_{\lambda}(t_1u)<I_{\lambda}(0)=0$.
\end{remark}

\begin{lemma} \label{lem2.6} 
Suppose $\{u_{n}\} \subset H_0^{1}(\Omega)$ is a $(PS)_{c}$ sequence for 
$I_{\lambda}(u)$,  then $\{u_{n}\}$ is bounded in $H_0^{1}(\Omega)$.
\end{lemma}

\begin{proof}
 Let $\{u_{n}\} \subset H_0^{1}(\Omega)$ be such that
\[\label{1.24}
I_{\lambda}(u_n)\to c,\quad I_{\lambda}'(u_n)\to0 \quad\text{as }n\to\infty.
\]
We claim that $\{u_n\}$ is bounded in $H_0^1(\Omega)$.
Otherwise, we can suppose that $\|u_n\|\to\infty$ as $n\to\infty$.
It follows from \eqref{2.1}, \eqref{2.4}, \eqref{2.5} and (F4) that
\begin{align*}
&1+c+o(1)\|u_n\| \\
&\geq  I_{\lambda}(u_n)-\frac{1}{p+1}\langle I'_{\lambda}(u_n),u_n\rangle\\
&\geq  a\Big(\frac{1}{2}-\frac{1}{p+1}\Big)\|u_n\|^2
 +b\Big(\frac{1}{4}-\frac{1}{p+1}\Big)\|u_n\|^4\\
&\quad  -\lambda \int_{\Omega}[F(x,|u_n|)-\frac{1}{p+1}f(x,|u_n|)|u_n|]\,dx\\
&\geq  a\Big(\frac{1}{2}-\frac{1}{p+1}\Big)\|u_n\|^2
 +b\Big(\frac{1}{4}-\frac{1}{p+1}\Big)\|u_n\|^4 
 -\lambda c_3\int_{\Omega}(1+|u_n|^2)\,dx\\
&\geq  a\Big(\frac{1}{2}-\frac{1}{p+1}\Big)\|u_n\|^2
 +b\Big(\frac{1}{4}-\frac{1}{p+1}\Big)\|u_n\|^4
  -\lambda c_3\left(|\Omega|+S_2^{2}\|u_n\|^2\right)\\
&\geq  \Big(\frac{a(p-1)}{2(p+1)}-\lambda c_3 S_2^{2}\Big)\|u_n\|^2
 +b\Big(\frac{1}{4}-\frac{1}{p+1}\Big)\|u_n\|^4
  -\lambda c_3 |\Omega|.
\end{align*}
Since $3<p<5$, it follows that the last inequality is an
absurd. Therefore, $\{u_n\}$ is bounded in $H_0^{1}(\Omega)$.
So Lemma \ref{lem2.6} holds.
\end{proof}

\section{Proof of Theorem \ref{thm1.1}}

Let $ \lambda^{*}=\min\{\lambda_1,\lambda_2,\lambda_3,\lambda_{4}\}$, 
then Lemmas \ref{lem2.1}--\ref{lem2.6} hold for every $\lambda \in (0, \lambda^{*})$. 
We prove Theorem \ref{thm1.1} by three steps.
\smallskip

\noindent\textbf{Step 1.} 
We claim that $I_{\lambda}(u)$ has a minimizer on 
$\mathcal{N}_{\lambda}^{+}(\Omega)$.
Indeed, from Remark \ref{rmk2.5}, there exists 
$u\in \mathcal{N}_{\lambda}^{+}(\Omega)$ such that $I_{\lambda}(u)<0$, 
so it follows that
$\inf_{u\in \mathcal{N}_{\lambda}^{+}(\Omega)}I_{\lambda}(u)<0$.
By Lemma \ref{lem2.3},  let $\{u_{n}\}$ be a sequence
minimizing for $I_{\lambda}(u)$ on $\mathcal{N}_{\lambda}^{+}(\Omega)$.
Clearly, this minimizing sequence is of course bounded, up to a 
subsequence (still denoted $\{u_{n}\}$), there exists $u_1\in H_0^{1}(\Omega)$ 
such that
\begin{gather*}
u_n\rightharpoonup u_1,\quad\text{weakly in }H_0^{1}(\Omega),\\
u_n\to u_1,\quad\text{strongly in } L^p(\Omega)\; (1\leq p<6),\\
u_n(x)\to u_1,\quad\text{a.e. in }\Omega.
\end{gather*}
Now we claim that $u_{n}\to u_1$ in $H_0^{1}(\Omega)$. In fact, 
set $\lim_{n\to \infty}\|u_{n}\|^2=l^2$.
By the Ekeland's variational principle \cite{EI}, it follows that
\begin{align*}
 o(1)&=\langle I'_{\lambda}(u_{n}),u_1 \rangle\\
&= \left(a+bl^2\right)\int_{\Omega}(\nabla u_{n}\cdot\nabla u_1 + v(x)u_{n}u_1)\,dx\\
 &\quad  - \int_{\Omega} h(x)|u_{n}|^{p}u_1\,dx
 -\lambda \int_{\Omega} f(x,|u_{n}|)|u_1|\,dx,
\end{align*} 
thus one obtains
\begin{equation}\label{3.1}
0= (a+bl^2)\|u_1\|^{2}- \int_{\Omega} h(x)|u_1|^{p+1}\,dx
-\lambda \int_{\Omega} f(x,|u_1|)|u_1|\,dx.
\end{equation}
Replacing $u_1$ with $u_{n}$, we obtain
\begin{align*}
o(1)&=\langle I'_{\lambda}(u_{n}),u_{n} \rangle \\
&= \left(a+bl^2\right)l^2-\int_{\Omega} h(x)|u_{n}|^{p+1}\,dx
 -\lambda \int_{\Omega} f(x,|u_{n}|)|u_{n}|\,dx,
\end{align*} 
consequently, one obtains
\begin{equation}\label{3.2}
0= (a+bl^2)l^2-\int_{\Omega} h(x)|u_1|^{p+1}\,dx
-\lambda \int_{\Omega} f(x,|u_1|)|u_1|\,dx.
\end{equation}
According to \eqref{3.1} and \eqref{3.2}, we obtain 
$\|u_1\|^{2}=l^2=\lim_{n\to \infty}\|u_{n}\|^2$, which suggests that 
$u_{n}\to u_1$ in $H_0^{1}(\Omega)$. Therefore, by Remark \ref{rmk2.5}, one obtains
\[
 I_{\lambda}(u_1)=\alpha^{+}_\lambda
=\lim_{n\to \infty}I_{\lambda}(u_{n})
=\inf_{u\in \mathcal{N}_{\lambda}^{+}(\Omega)}I_{\lambda}(u)<0.
\]
So we proved the claim.
\smallskip

\noindent\textbf{Step 2.} 
$I_{\lambda}(u)$ has a minimizer on $\mathcal{N}_{\lambda}^{-}(\Omega)$.
As a matter of fact, from Remark \ref{rmk2.4}, we have $I_{\lambda}(u)>0$ for 
$u\in \mathcal{N}_{\lambda}^{-}(\Omega)$, so it follows that
$\inf_{u\in \mathcal{N}_{\lambda}^{-}(\Omega)}I_{\lambda}(u)>0$. 
Similarly to step 1,
we define a sequence $\{u_{n}\}$ as a minimizing for 
$I_{\lambda}(u)$ on $\mathcal{N}_{\lambda}^{-}(\Omega)$, and there 
exists $u_2\in H_0^{1}(\Omega)$ such that
\begin{gather*}
u_n\rightharpoonup u_2,\quad \text{weakly in } H_0^{1}(\Omega),\\
u_n\to u_2,\quad \text{strongly in } L^p(\Omega)\; (1\leq p<6),\\
u_n(x)\to u_2,\quad\text{a.e. in }\Omega.
\end{gather*}
We claim that $H(u_{n})>0$. By contradiction, assume $H(u_{n})\leq0$, 
then $-pH(u_{n})\geq0 $,
from $u_{n}\in \mathcal{N}_{\lambda}^{-}(\Omega)$, by \eqref{2.1}, \eqref{2.4}, 
\eqref{2.5} and (F2), it follows that
\begin{align*}
a\|u_{n}\|^2&<a\|u_{n}\|^2+3b\|u_{n}\|^4-pH(u_{n})\\
&< \lambda \int_{\Omega}f_{u}(x,|u_{n}|)|u_{n}|^2\,dx\\
&\leq\lambda \|f_{u}(x,|u_{n}|)\|_{L^{\infty}}S_2^{2}\|u_{n}\|^2,
\end{align*} 
which is a contradiction when $\lambda$ is small enough. 
 We get $H(u_{n})>0$. Therefore $H(u_2)>0$ as $n\to \infty$. 
Similar to the proof of step 1, one can get $u_{n}\to u_2$ in $H_0^{1}(\Omega)$.
Therefore,
\[
 I_{\lambda}(u_2)=\alpha^{-}_\lambda
=\lim_{n\to \infty}I_{\lambda}(u_{n})
=\inf_{u\in \mathcal{N}_{\lambda}^{-}(\Omega)}I_{\lambda}(u)>0.
\]
From above discussion, we obtain that $I_{\lambda}(u)$ has a minimizer
on $\mathcal{N}_{\lambda}^{-}(\Omega)$.

By Step 1 and Step 2, there exist $u_1 \in \mathcal{N}_{\lambda}^{+}(\Omega)$ and
$u_2 \in \mathcal{N}_{\lambda}^{-}(\Omega)$
such that $I_{\lambda}(u_1)= \alpha^{+}_\lambda<0$ and 
$ I_{\lambda}(u_2)= \alpha^{-}_\lambda>0$.
It follows that $u_1$ and $u_2$  are nonzero solutions of (1.1).
Because of  $I_{\lambda}(u)=I_{\lambda}(|u|)$, one gets $u_1, u_2\geq0$. 
Therefore, by the Harnack inequality (see \cite[Theorem 8.20]{DN}), 
we have $u_1, u_2>0$ a.e. in $\Omega$. 
Consequently the proof of Theorem \ref{thm1.1} is complete.

\subsection*{Acknowledgments}
This research was supported by the Research Foundation of Guizhou Minzu 
University (No. 15XJS013; No. 15XJS012).


\begin{thebibliography}{99}

\bibitem{AS} A. M. Mao, S. X. Luan;
\emph{Sign-changing solutions of a class of nonlocal quasilinear elliptic boundary 
value problems}, J. Math. Anal. Appl. 383 (2011), 239--243.

\bibitem{AZ} A. M. Mao, Z. T. Zhang;
\emph{Sign-changing and multiple solutions of Kirchhoff type problems without 
the P.S. condition}, Nonlinear Anal. 70 (2009), 1275--1287.

\bibitem{CF} C. O. Alves, F. J. S. A. Corr\'ea, T. F. Ma;
\emph{Positive solutions for a quasilinear elliptic equation of Kirchhoff type}, 
Comput. Math. Appl. 49 (2005), 85--93.

\bibitem{CT} C. Chen, Y. Kuo, T. Wu;
\emph{The Nehari manifold for a Kirchhoff type problem involving sign-changing 
weight functions}, J. Differential Equations. 250 (2011), 1876--1908.

\bibitem{CX} C. T. Cheng, X. Wu;
\emph{Existence results of positive solutions of Kirchhoff type problems},
 Nonlinear Anal. 71 (2009), 4883--4892.

\bibitem{DN} D. Gilbarg, N. S. Trudinger;
\emph{Elliptic Partial Differential Equations of Second Order}, 
Springer-Verlag, Berlin, 2001.

\bibitem{EI} I. Ekeland;
\emph{On the variational principle}, J. Math. Anal. Appl. 47 (1974), 324--353.

\bibitem{GH} G. B. Li, H. Y. Ye;
\emph{Existence of positive ground state solutions for the nonlinear 
Kirchhoff type equations in $\mathbb{R}^{3}$}, 
J. Differential Equations. 257 (2014), 566--600.

\bibitem{GJ} G. M. Figueiredo, J. R. Santos Junior;
\emph{Multiplicity of solutions for a Kirchhoff equation with subcritical 
or critical growth}, Diff. Integral Equations. 25 (2012), 853--868.

\bibitem{JW} J. J. Nie, X. Wu;
\emph{Existence and multiplicity of non-trivial solutions for 
Schr\"odinger-Kirchhoff-type equations with radial potential}, 
Nonlinear Anal. 75 (2012), 3470--3479.

\bibitem{JX} J. H. Jin, X. Wu;
\emph{Infinitely many radial solutions for Kirchhoff-type problems in 
$\mathbb{R}^N$}, J. Math. Annal. Appl. 369 (2010), 564--574.

\bibitem{LX} L. Wei, X. M. He;
\emph{Multiplicity of high energy solutions for superlinear Kirchhoff equations}, 
J. Appl. Math. Comput. 39 (2012), 473--487.

\bibitem{TJ} T. F. Ma, J. E. M. Rivera;
\emph{Positive solutions for a nonlinear nonlocal elliptic transmission problem}, 
Appl. Math. Lett. 16 (2003), 243--248.

\bibitem{XW} X. M. He, W. M. Zou;
\emph{Infinitely many positive solutions for Kirchhoff-type problems}, 
Nonlinear Analysis. 70 (2009), 1407--1414.

\bibitem{YF} Y. H. Li, F. Y. Li, J. P. Shi;
\emph{Existence of a positive solution to Kirchhoff type problems without 
compactness conditions}, J. Differential Equations. 253 (2012), 2285--2294.

\end{thebibliography}

\end{document}