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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 301, pp. 1--16.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/301\hfil  Kirchhoff type problems]
{Multiple positive solutions for Kirchhoff type problems involving
 concave and convex nonlinearities in $\mathbb{R}^3$}

\author[X. Cao, J. Xu, J. Wang \hfil EJDE-2016/301\hfilneg]
{Xiaofei Cao, Junxiang Xu, Jun Wang}

\address{Xiaofei Cao \newline
Department of Mathematics,
Southeast University,
Nanjing, Jiangsu 210096, China}
\email{caoxiaofei258@126.com}

\address{Junxiang Xu (corresponding author) \newline
Department of Mathematics,
Southeast University,
 Nanjing, Jiangsu 210096,  China}
\email{xujun@seu.edu.cn}

\address{Jun Wang \newline
Faculty of Science,
Jiangsu University,
Zhenjiang, Jiangsu 212013, China}
\email{wangmath2011@126.com}

\thanks{Submitted July 25, 2016. Published November 25, 2016.}
\subjclass[2010]{35A01, 35A15}
\keywords{Kirchhoff type problem; Ekeland variational principle;
\hfill\break\indent  Nehari manifold}

\begin{abstract}
 In this article, we consider the multiplicity of positive solutions for
 a class of Kirchhoff type problems with concave and convex nonlinearities.
 Under appropriate assumptions, we prove that the problem has at least
 two positive solutions, moreover, one of which is a positive ground
 state solution. Our approach is mainly based on the Nehari manifold,
 Ekeland variational principle and the theory of Lagrange multipliers.
\end{abstract}

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

\section{Introduction and statement of main results}\label{00}

In this article, we consider the multiplicity of positive solutions
for the  Kirchhoff type problem
\begin{equation}\label{a1}
\begin{gathered}
-(a+b\int_{\mathbb{R}^3}|\nabla u|^2\,dx)\Delta u+V(x)u
= f(x)|u|^{q-2}u+g(x)|u|^{p-2}u,\\
u\in H^{1}(\mathbb{R}^3),
\end{gathered}
\end{equation}
where $a$ and $ b$ are positive constants, $1<q<2$,
$4<p<2^{\ast}=6$, $ V(x), f(x), g(x)$ are continuous functions and satisfy
suitable conditions.

Problem \eqref{a1} can be written in the general form
\begin{equation}\label{a2}
\begin{gathered}
-(a+b\int_{\mathbb{R}^{N}}|\nabla u|^2\,dx)\Delta u+V(x)u=h(x,u),\\
u\in H^{1}(\mathbb{R}^{N}),
\end{gathered}
\end{equation}
where $V: \mathbb{R}^{N}\to \mathbb{R}$ is a continuous potential,
$a, b>0$ are constants. For the case of the nonlinearity $h$
is asymptotically linear or superlinear, it has been studied extensively
by many authors, see \cite{BCXW,DGW,GMFG,XHWZ,XHWWZ,JJXW,WLXH,YLFL,JSSL,JWLX,ZTZKP}
 and their references therein.

A special case of \eqref{a2} is the  well-known equation
\begin{equation}\label{a3}
\begin{gathered}
-(a+b\int_{\Omega}|\nabla u|^2\,dx)\Delta u= h(x,u) \quad \text{in }\Omega,\\
u=0 \quad \text{on }\partial\Omega, \\
\end{gathered}
\end{equation}
where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$.
This problem \eqref{a3} is related to the stationary analogue of the equation
\begin{equation}\label{a4}
\begin{gathered}
u_{tt}-(a+b\int_{\Omega}|\nabla u|^2\,dx)\Delta u=h(x,u)
\quad \text{in }\Omega,\\
u=0\quad \text{on }\partial\Omega, \\
\end{gathered}
\end{equation}
proposed by Kirchhoff \cite{GGKKM} in 1883 as an extension of the
classical D'Alembert's wave equation for free vibration of elastic strings.
Kirchhoff's model takes into account the changes in length of the string
produced by transverse vibrations. In \eqref{a4}, $u$ denotes the displacement,
 $h(x,u)$ the external force and $b$ the initial tension while $a$
is related to the intrinsic properties of the string (such as Young's modulus).
Such problems are often viewed as nonlocal because of the presence of
the integral term $\int_{\Omega}|\nabla u|^2\,dx$ which implies
that problem \eqref{a4} is no longer a pointwise identity.
This phenomenon causes some mathematical difficulties which makes the study
of such a class of problem particularly interesting. Besides,
 similar nonlocal problem also appears in other fields such as physical and
 biological systems, where $u$ describes a process that depends on the average
of itself, for example, the population density.

Here we are interested in the nonlinearity  $h$ made up of the combination
of a sublinear term and a superlinear term. This case was considered by
Willem  \cite{MWM} for the  elliptic equation
\begin{equation}\label{a5}
\begin{gathered}
-\Delta u=\lambda|u|^{q-2}u+\mu|u|^{p-2}u,\\
u\in H_0^{1}(\Omega),
\end{gathered}
\end{equation}
where $1<q<2<p<2^{\ast}(2^{\ast}=2N/(N-2)$ if $N\geq 3, 2^{\ast}=\infty$
if $N=1, 2$) and $\Omega$ is a bounded domain in $\mathbb{R}^{N}$.
The author  proved that for every $\mu>0,\lambda\in \mathbb{R}$,
problem \eqref{a5} has a sequence of high energy solutions, and for
every $\lambda>0,\mu \in \mathbb{R}$, problem \eqref{a5} has a sequence
of negative energy solutions.

Ambrosetti, Brezis and Cerami \cite{AAHBGC} studied  equation \eqref{a5}
when $\mu=1$. The authors used sub- and super-solutions to prove that
 when $\mu=1$ there exists $\Lambda>0$ such that  \eqref{a5} admits at
least two positive solutions for $\lambda\in(0, \Lambda)$, one positive solution for
$\lambda=\Lambda$ and no positive solution for $\lambda>\Lambda$.
Note that if $\mu\neq0$ in \eqref{a5}, then  by scaling \eqref{a5} becomes
the situation of $\mu=1$.

Wu \cite{TWU} also studied the  concave-convex elliptic equation
\begin{equation}\label{a6}
\begin{gathered}
-\Delta u+u=f_{\lambda}(x) u^{q-1}+g_{\mu}(x)u^{p-1}\quad \text{in }\mathbb{R}^N,\\
u\geq0 \quad \text{in }\mathbb{R}^N,\\
u\in H^{1}(\mathbb{R}^{N})
\end{gathered}
\end{equation}
where $1<q<2<p<2^{\ast}$ ($2^{\ast}=2N/(N-2)$ if $N\geq 3$,
$2^{\ast}=\infty$ if $N=1, 2$),
\[
f_{\lambda}(x)=\lambda f_{+}(x)+f_{-}(x)(f_{\underline{+}}(x)
:=\underline{+} \max\{0, \underline{+}f(x)\}\neq0)\]
 is sign-changing, $g_{\mu}(x)=a(x)+\mu b(x)$ and the parameters
$\lambda, \mu\geq0$. When the functions $f_{+}(x), f_{-}(x), a(x), b(x)$
satisfy suitable conditions, he proved the multiplicity of positive
solutions for the problem \eqref{a6}.

Recently, Chen, Kuo and Wu \cite{CCYKTW} considered the  Kirchhoff type problem
\begin{equation}\label{a7}
\begin{gathered}
-(a+b\int_{\Omega}|\nabla u|^2\,dx)\Delta u
=\lambda f(x)|u|^{q-2}u+g(x)|u|^{p-2}u,\\
u\in H_0^{1}(\Omega), \\
\end{gathered}
\end{equation}
where $\Omega$ is a smooth bounded domain in $\mathbb{R}^{N}$ with
$1<q<2<p<2^{\ast} (2^{\ast}=2N/(N-2)$ if $N\geq 3, 2^{\ast}=\infty$ if $N=1, 2$),
the parameters $a, b, \lambda>0$ and the weight functions
 $f, g\in C(\overline{\Omega})$. Using Nehari manifold they proved the
existence of solutions for \eqref{a7} with $p>4, p=4$ and $p<4$, respectively.

Motivated by these papers \cite{CCYKTW,MWM,TWU}, we consider the Kirchhoff
problem \eqref{a1} with potential $V(x)$ and concave-convex nonlinearities
on the whole space $\mathbb{R}^3$. To the best of our knowledge,
there are few papers which deal with this type of Kirchhoff problem \eqref{a1}.
The main difficulties lie in  the unboundedness of the domain $\mathbb{R}^3$,
the presence of the nonlocal term and the concave-convex nonlinearities.

Assume that $V(x), f(x), g(x)$ satisfy the  following conditions:
\begin{itemize}
  \item[(A1)] $V(x)\in C(\mathbb{R}^3, \mathbb{R})$,
$V_0:=\inf_{\mathbb{R}^3}V(x)>0$ and for any $M>0$, there exists a
constant $r_0>0$ such that $\operatorname{meas}(\{x \in B_{r_0} (y)
: V(x)\leq M\})\to 0 $  as $|y|\to +\infty$, where $B_{r_0}(y)$
denotes  the ball centered at $y$ with radius $r_0$, meas denotes the
 Lebesgue measure in $\mathbb{R}^3$.

  \item[(A2)] $f\in C(\mathbb{R}^3)\cap L^{q^{\ast}}(\mathbb{R}^3)$,
 where $q^{\ast}=p/(p-q)$.

  \item[(A3)] $g\in C(\mathbb{R}^3)\cap L^{\infty}(\mathbb{R}^3)$
and $g(x)>0$, for almost every $x\in\mathbb{R}^3$.

\end{itemize}
Let $\sigma:=(p-2)(2-q)^{(2-q)/(p-2)}(\frac{S_p}{p-q})^{(p-q)/(p-2)}$
and $0<\sigma^{\ast}:=\frac{q}{p-2}\sigma<\sigma$, where $S_p$
is the best Sobolev constant described in the following Lemma \ref{lem2.2}.

\begin{theorem}\label{thm1.1}
Under the assumptions {\rm (A1)--(A3)}, if
$|f|_{q^{\ast}}|g|^{(2-q)/(p-2)}_{\infty}\in(0,\sigma)$,
the problem \eqref{a1} has at least two positive solutions, one
of which has negative energy. In particular, if
$|f|_{q^{\ast}}|g|^{(2-q)/(p-2)}_{\infty}\in(0,\sigma^{\ast})$, the solution
corresponding to the  negative energy is a positive ground state
solution and the other one corresponds to positive energy.
\end{theorem}

In problem \eqref{a1}, because of the unboundedness of the domain $\mathbb{R}^3$
there is no compactness, thus we bring in the hypothesis (A1) to recover the
compactness. Moreover, the presence of the nonlocal term and the
concave-convex nonlinearities prevents us from using the Nehari manifold
method in a standard way as \cite{BCXW,GMFG,XHWZ,XHWWZ,JJXW,WLXH,YLFL,JSSL,JWLX,ZTZKP}. Motivated by papers \cite{{KBYZ,CCYKTW,TWU}}, we connect the Nehari manifold with the fibering map and split the Nehari manifold into three parts which are then considered separately. By putting suitable conditions on continuous functions $f(x)$, $g(x)$ and restricting $|f|_{q^{\ast}}|g|^{(2-q)/(p-2)}_{\infty}$ to a suitable range, we use Ekeland variational principle and the theory of Lagrange multipliers to obtain two positive solutions of the problem \eqref{a1}. In addition, from the condition (A2), we easily see that $f(x)$ is allowed to be sign-changing as \cite{{TWU}}.


\begin{remark}\label{rem1.2} \rm
The condition (A1) was first introduced by Bartsch and Wang in
\cite{BARTS}. Note that it is weaker than both conditions
\begin{itemize}
\item[(1)] $V(x)\in C(\mathbb{R}^3, \mathbb{R}),
\inf_{\mathbb{R}^3}V(x)>0, V (x) \to +\infty $ as $|x| \to
+\infty$ (See \cite{XHWWZ}).
  \item[(2)] $V(x)\in C(\mathbb{R}^3, \mathbb{R}),
\inf_{\mathbb{R}^3}V(x)>0$, for each $M>0$, $\operatorname{meas}(\{x\in
\mathbb{R}^3:V(x)\leq M\})<\infty$ (See \cite{CLLM,XXWW,WZMSC}).
\end{itemize}
These conditions are often used to recover compactness.
\end{remark}

\begin{remark}\label{rem1.3} \rm
We can also consider the  Kirchhoff problem
\begin{equation}\label{a8}
\begin{gathered}
(a+b\int_{\mathbb{R}^{N}}(|\nabla u|^2+V(x)u^2)\,dx)
(-\Delta u+V(x)u)=f(x)|u|^{q-2}u+g(x)|u|^{p-2}u,\\
u\in H^{1}(\mathbb{R}^{N}),
\end{gathered}
\end{equation}
where the parameters $a,b>0$ and $1<q<2<p<2^{\ast}(2^{\ast}=2N/(N-2)$
if $N\geq 3, 2^{\ast}=\infty$ if $N=1, 2$).
Under the assumptions of Theorem \ref{thm1.1}, restricting
$|f|_{q^{\ast}}|g|^{(2-q)/(p-2)}_{\infty}$  to a suitable range as in
Theorem \ref{thm1.1}, we can also obtain the existence of solutions for
\eqref{a8}. It is worth noting that
 \eqref{a8} is similar to  \eqref{a7} in the case of the whole
space $\mathbb{R}^{N}$ and that there is compactness because of  condition (A1).
Hence we can also obtain the results in \cite{CCYKTW}.
However, this type of Kirchhoff \eqref{a8} is different from that of Kirchhoff
 \eqref{a1}.
\end{remark}

This  article is  organized  as follows: Section 2 is dedicated to our
abstract framework and some preliminary results.
Section 3 is concerned with the proof of Theorem \ref{thm1.1}.
Throughout this paper, $C$ and $C_{i}$ are used in various places to denote
distinct constants. $L^{p}(\mathbb{R}^{N})$ is the usual Lebesgue space
endowed with the standard norm $|u|_p=(\int_{\mathbb{R}^{N}}|u|^{p}dx)^{1/p}$
for $1\leq p<\infty$ and $|u|_{\infty}=\sup_{x\in\mathbb{R}^{N}}|u(x)|$
for $p=\infty$. When it causes no confusion, we still denote by $\{u_n\}$
a subsequence of the original sequence $\{u_n\}$.



\section{Preliminary results}\label{1}

In this section, we  recall some  preliminaries and establish the
variational setting for our problem.
Under the assumption (A1), define
\begin{equation*}
E:=\{u\in H^{1}(\mathbb{R}^3): \int_{\mathbb{R}^3}V(x)u^2\,dx<+\infty\},
\end{equation*}
with the associated norm
\begin{equation*}
\|u\|= \Big(\int_{\mathbb{R}^3}(a|\nabla u|^2+V(x)u^2)\,dx\Big)^{1/2},
\end{equation*}
where $H^{1}(\mathbb{R}^3)$ is the well known Sobolev space.

Then the energy functional corresponding to  \eqref{a1} is
\begin{equation}\label{b1}
\begin{split}
I(u)&=\frac{1}{2}\int_{\mathbb{R}^3}(a|\nabla u|^2
 +V(x)u^2)\,dx+\frac{b}{4}\Big(\int_{\mathbb{R}^3}|\nabla u|^2\,dx\Big)^2
-\frac{1}{q}\int_{\mathbb{R}^3}f(x)|u|^q\,dx\\
&\quad -\frac{1}{p}\int_{\mathbb{R}^3}g(x)|u|^{p}\,dx, \quad u\in E.
\end{split}
\end{equation}

\begin{lemma}\label{lem2.1}
If {\rm (A1)--(A3)} hold, then the functional $I\in C^{1}(E,\mathbb{R})$
and for any $u,v\in E$,
\begin{equation}\label{b2}
\begin{split}
\langle I^{'}(u),v\rangle
&=\Big(a+b\int_{\mathbb{R}^3}|\nabla u|^2\,dx\Big)
\int_{\mathbb{R}^3}\nabla u\nabla v\,dx+\int_{\mathbb{R}^3}V(x)uv\,dx\\
&\quad - \int_{\mathbb{R}^3}f(x)|u|^{q-2}uv\,dx
 -\int_{\mathbb{R}^3}g(x)|u|^{p-2}uv\,dx.
\end{split}
\end{equation}
\end{lemma}

The proof of the above lemma is a direct computation, and can be found in
\cite{MWM,XXWW}.
As pointed out previously,  assumption (A1) is used to recover compactness
of embedding theorem, which is given in the next lemma.


\begin{lemma}[\cite{BARTS,WZMSC}] \label{lem2.2}
 Under assumption {\rm (A1)}, the embedding $E\hookrightarrow L^{p}(\mathbb{R}^3)$
is continuous for $p\in[2,2^{\ast}]$ and compact for $p\in[2,2^{\ast})$.
Throughout this paper, we denote by $S_p$ the best Sobolev constant for
the embedding $E\hookrightarrow L^{p}(\mathbb{R}^3)$ which is given by
\begin{equation*}
S_p=\inf_{u\in E\backslash\{0\}}
\frac{\|u\|^2}{(\int_{\mathbb{R}^3}|u|^{p}\,dx)^{2/p}}>0.
\end{equation*}
In particular,
\begin{equation*}
|u|_p\leq S_p^{-1/2}\|u\|,\quad \forall u\in E\backslash\{0\}.
\end{equation*}
\end{lemma}

It is well-known that finding a weak solution of problem \eqref{a1} is
equivalent to finding a critical point of the corresponding functional $I$.
In the following, we are devoted to finding the critical point of the
corresponding functional $I$.

As usual, some energy functional such as $I$ in \eqref{b1} is not bounded
from below on $E$ but, as we will see, is bounded from below on an appropriate
subset of $E$ and a minimizer on this set (if it exists) may give rise
to a solution of corresponding differential equation. A good exemplification
for an appropriate subset of $E$ is the so-called Nehari manifold
\begin{equation*}
\mathcal{N} =\{u\in E:\langle I'(u),u\rangle=0\},
\end{equation*}
where $\langle\cdot, \cdot\rangle$ denotes the usual duality between $E$ and
 $E^{\ast}$.
It is clear to see that $u\in \mathcal{N}$ if and only if
\begin{equation}\label{b3}
\|u\|^2+b|\nabla u|_2^4=\int_{\mathbb{R}^3}f(x)|u|^q\,dx
+\int_{\mathbb{R}^3}g(x)|u|^{p}\,dx.
\end{equation}
Obviously, $\mathcal{N}$ contains all solutions of \eqref{a1}.
 Below, we shall use the Nehari manifold methods to find critical points
for functional $I$.

The Nehari manifold $\mathcal{N}$ is closely linked to the behavior of functions
of the form $K_{u}: t\to I(tu)$ for $t>0$. Such maps are known as fibering maps,
 which were introduced by Dr$\acute{a}$bek and Pohozaev in \cite{PDSP}.
For $u\in E$, let
\begin{align*}
K_{u}(t)&=I(tu)=\frac{1}{2}t^2\|u\|^2+\frac{b}{4}t^4
 |\nabla u|_2^4-\frac{1}{q}t^q\int_{\mathbb{R}^3}f(x)| u|^q\,dx
 -\frac{1}{p}t^{p}\int_{\mathbb{R}^3}g(x)|u|^{p}\,dx;\\
K_{u}'(t)&=t\|u\|^2+t^3b|\nabla u|_2^4
 -t^{q-1}\int_{\mathbb{R}^3}f(x)|u|^q\,dx-t^{p-1}
 \int_{\mathbb{R}^3}g(x)|u|^{p}\,dx;\\
K_{u}''(t)&=\|u\|^2+3t^2b|\nabla u|_2^4-(q-1)t^{q-2}
 \int_{\mathbb{R}^3}f(x)|u|^q\,dx\\
&\quad-(p-1)t^{p-2}\int_{\mathbb{R}^3}g(x)|u|^{p}\,dx.
\end{align*}

\begin{lemma}\label{lem2.3}
 Let $u\in E\backslash\{0\}$ and $t>0$. Then $tu\in \mathcal{N}$
if and only if $K_{u}'(t)=0$, that is, the critical points of $K_{u}(t)$
correspond to the points on the Nehari manifold.
In particular, $u\in \mathcal{N}$ if and only if $K_{u}'(1)=0$.
\end{lemma}

\begin{proof}
 The result is an immediate consequence of the fact that
\begin{equation*}
K_{u}'(t)=\langle I'(tu),u\rangle
=\frac{1}{t}\langle I'(tu),tu\rangle.
\end{equation*}
\end{proof}

Thus, it is natural to split $\mathcal{N}$ into three parts corresponding
to local minima, points of inflection and local maxima.
Accordingly, we define
\begin{gather*}
\mathcal{N}^{+}=\{u\in \mathcal{N}: K_{u}''(1)>0\},\\
\mathcal{N}^{0}=\{u\in \mathcal{N}: K_{u}''(1)=0\},\\
\mathcal{N}^{-}=\{u\in \mathcal{N}: K_{u}''(1)<0\}.
\end{gather*}
It is easy to see that
\begin{equation}\label{b4}
K_{u}''(1)=\|u\|^2+3b|\nabla u|_2^4-(q-1) \int_{\mathbb{R}^3}f(x)|u|^q\,dx
-(p-1)\int_{\mathbb{R}^3}g(x)|u|^{p}\,dx.
\end{equation}
Define
\begin{equation}\label{b5}
\begin{aligned}
\Psi(u)&=K_{u}'(1)=\langle I'(u),u\rangle \\
&=\|u\|^2+b|\nabla u|_2^4-\int_{\mathbb{R}^3}f(x)|u|^q\,dx
-\int_{\mathbb{R}^3}g(x)|u|^{p}\,dx.
\end{aligned}
\end{equation}
Then for $u\in \mathcal{N}$,
\begin{align*}
\frac{d}{dt}\Psi(tu)|_{t=1}
&=\langle \Psi'(u),u\rangle
 =\langle \Psi'(u),u\rangle-\langle I'(u),u\rangle=K_{u}''(1)\\
&=\|u\|^2+3b|\nabla u|_2^4-(q-1) \int_{\mathbb{R}^3}f(x)|u|^q\,dx
-(p-1)\int_{\mathbb{R}^3}g(x)|u|^{p}\,dx.
\end{align*}
For each $u\in \mathcal{N}$, $\Psi(u)=K_{u}'(1)=0$. Thus, we have
\begin{equation}\label{b6}
\begin{split}
K_{u}''(1)&=K_{u}''(1)-(q-1)\Psi(u)\\
&=(2-q)\|u\|^2+(4-q)b|\nabla u|_2^4-(p-q)\int_{\mathbb{R}^3}g(x)|u|^{p}\,dx
\end{split}
\end{equation}
and
\begin{equation}\label{b7}
\begin{split}
K_{u}''(1)&=K_{u}''(1)-(p-1)\Psi(u)\\
&=(2-p)\|u\|^2+(4-p)b|\nabla u|_2^4+(p-q) \int_{\mathbb{R}^3}f(x)|u|^q\,dx.
\end{split}
\end{equation}


To ensure the Nehari manifold $\mathcal{N}$ to be a $C^{1}$-manifold, we need
the following proposition.
Let $\sigma:=(p-2)(2-q)^{(2-q)/(p-2)}(\frac{S_p}{p-q})^{(p-q)/(p-2)}$.

\begin{proposition}\label{prop2.4}
If $|f|_{q^{\ast}}|g|^{(2-q)/(p-2)}_{\infty}\in(0,\sigma)$, then the set
$\mathcal{N}^{0}=\{0\}$.
\end{proposition}

\begin{proof}
Suppose, on the contrary, there exists $u\in \mathcal{N}\backslash \{0\}$
such that $K_{u}''(1)=0$. By Lemma \ref{lem2.2},
\begin{equation}\label{b8}
\int_{\mathbb{R}^3}g(x)|u|^{p}\,dx
\leq|g|_{\infty}S_p^{-p/2}\|u\|^{p}.
\end{equation}
Noting that $1<q<2$ and $4<p<6$, from \eqref{b6} we have
\begin{equation*}
(2-q)\|u\|^2\leq(p-q)|g|_{\infty}S_p^{-p/2}\|u\|^{p},
\end{equation*}
and so
\begin{equation}\label{b9}
\|u\|\geq\Big(\frac{(2-q)S_p^{\frac{p}{2}}}{(p-q)|g|_{\infty}}\Big)^{\frac{1}{p-2}}.
\end{equation}
Moreover, by H\"{o}lder inequality and Lemma \ref{lem2.2}, we have
\begin{equation}\label{b10}
\begin{split}
\int_{\mathbb{R}^3}f(x)|u|^q\,dx
&\leq\Big(\int_{\mathbb{R}^3}|f(x)|^{q\ast}\,dx\Big)^{1/q\ast}
\Big(\int_{\mathbb{R}^3}|u|^{p}\,dx\Big)^{q/p}\\
&=|f|_{q\ast}|u|_p^q\leq|f|_{q\ast}S_p^{-q/2}\|u\|^q.
\end{split}
\end{equation}
From \eqref{b7} we have
\begin{equation*}
(p-2)\|u\|^2\leq(p-q)|f|_{q\ast}S_p^{-q/2}\|u\|^q,
\end{equation*}
which implies that
\begin{equation}\label{b11}
\|u\|\leq\Big(\frac{(p-q)|f|_{q\ast}}{(p-2)S_p^{\frac{q}{2}}}\Big)^{\frac{1}{2-q}}.
\end{equation}
Combining \eqref{b9} and \eqref{b11} we deduce that
\begin{equation*}
|f|_{q^{\ast}}|g|^{\frac{2-q}{p-2}}_{\infty}
\geq\Big(\frac{(2-q)S_p^{\frac{p}{2}}}{p-q}\Big)^{\frac{2-q}{p-2}}
 \frac{p-2}{p-q}S_p^{\frac{q}{2}}
=(p-2)(2-q)^{\frac{2-q}{p-2}}\Big(\frac{S_p}{p-q}\Big)^{\frac{p-q}{p-2}},
\end{equation*}
which contradicts the assumptions. The proof is complete.
\end{proof}

Let
\begin{equation*}
h_{b}(t):=t^{2-q}\|u\|^2+t^{4-q}b|\nabla u|_2^4-t^{p-q}
\int_{\mathbb{R}^3}g(x)|u|^{p}\,dx,
\end{equation*}
then we have
\begin{equation}\label{b12}
K_{u}'(t)=t^{q-1}\Big(h_{b}(t)-\int_{\mathbb{R}^3}f(x)|u|^q\,dx\Big).
\end{equation}
Clearly, $h_{b}(0)=0, h_{b}(t)>0$ for $t$ is small enough and
$h_{b}(t)\to-\infty$ as $t\to\infty$. From $1<q<2,~4<p<2^{\ast}=6$ and
\begin{align*}
h_{b}'(t)&=t^{p-q-1}\Big((2-q)t^{2-p}\|u\|^2+(4-q)t^{4-p}b|\nabla u|_2^4
-(p-q)\int_{\mathbb{R}^3}g(x)|u|^{p}\,dx\Big)\\
&=0,
\end{align*}
we can infer that there is a unique $t_{b,\max}>0$ such that $h_{b}(t)$
achieves its maximum at $t_{b,\max}$, increasing for $t\in[0, t_{b,\max})$
and decreasing for $t\in( t_{b,\max}, \infty)$ with
$\lim_{t\to\infty}h_{b}(t)=-\infty$.

\begin{proposition}\label{prop2.5}
Suppose that $|f|_{q^{\ast}}|g|^{(2-q)/(p-2)}_{\infty}\in(0,\sigma)$ and
$u\in E\backslash \{0\}$. Then

(i) if $\int_{\mathbb{R}^3}f(x)|u|^q\,dx\leq0$, then there is a unique
$t^{-}>t_{b,\max}$ such that $t^{-}u\in \mathcal{N}^{-}$ and
\begin{equation*}
I(t^{-}u)=\sup_{t\geq0}I(tu);
\end{equation*}

(ii) if $\int_{\mathbb{R}^3}f(x)|u|^q\,dx>0$, then there are unique $t^{+}$
and $t^{-}$ with $0<t^{+}<t_{b,\max}<t^{-}$ such that $t^{+}u\in \mathcal{N}^{+}$,
$t^{-}u\in \mathcal{N}^{-}$ and
\begin{equation*}
I(t^{+}u)=\inf_{0\leq t\leq t_{b,\max}}I(tu),\quad
I(t^{-}u)=\sup_{t\geq t_{b,\max}}I(tu).
\end{equation*}
\end{proposition}

\begin{proof}
Since $b>0$, we have
\begin{equation*}
h_{b}(t)>h_0(t)=t^{2-q}\|u\|^2-t^{p-q}\int_{\mathbb{R}^3}g(x)|u|^{p}\,dx,
\end{equation*}
where $h_0(t)=h_{b}(t)|_{b=0}$.
It is clear that $h_0(t)$ has a unique critical point at $t_{0,\max}=t_{0,\max}(u)$,
where
\begin{equation*}
t_{0,\max}=\Big(\frac{(2-q)\|u\|^2}{(p-q)\int_{\mathbb{R}^3}g(x)|u|^{p}\,dx}
\Big)^{\frac{1}{p-2}}.
\end{equation*}
It follows that
\begin{equation}\label{b13}
\begin{split}
h_0(t_{0,\max})
&=\|u\|^q\Big(\frac{\|u\|^{p}}{\int_{\mathbb{R}^3}g(x)| u|^{p}\,dx}
 \Big)^{\frac{2-q}{p-2}}\Big(\frac{2-q}{p-q}\Big)^{\frac{2-q}{p-2}}\frac{p-2}{p-q}\\
&\geq\|u\|^q\Big(\frac{\|u\|^{p}}{|g|_{\infty}S_p^{-p/2}\|u\|^{p}}
 \Big)^{\frac{2-q}{p-2}}\Big(\frac{2-q}{p-q}\Big)^{\frac{2-q}{p-2}}\frac{p-2}{p-q}\\
&=\|u\|^q\Big(\frac{(2-q)S_p^{\frac{p}{2}}}{|g|_{\infty}(p-q)}
 \Big)^{\frac{2-q}{p-2}}\frac{p-2}{p-q}>0.
\end{split}
\end{equation}
Thus, $h_{b}(t_{b,\max})>h_0(t_{0,\max})>0$.

From $|f|_{q^{\ast}}|g|^{(2-q)/(p-2)}_{\infty}\in(0,\sigma)$, \eqref{b10}
and \eqref{b13} we also have
\begin{equation}\label{b14}
\begin{split}
\int_{\mathbb{R}^3}f(x)|u|^q\,dx
&<\|u\|^q\Big(\frac{(2-q)S_p^{\frac{p}{2}}}{|g|_{\infty}(p-q)}\Big)^{\frac{2-q}{p-2}}
 \frac{p-2}{p-q}\\
&\leq h_0(t_{0,\rm max})<h_{b}(t_{b,\rm max}).
\end{split}
\end{equation}

(i) If $\int_{\mathbb{R}^3}f(x)|u|^q\,dx\leq0$, noting that $h_{b}(0)=0$
and $h_{b}(t)\to-\infty$ as $t\to\infty$, by \eqref{b12} there is a
unique $t^{-}>t_{b,\max}$ such that $K_{u}'(t^{-})=0$, that is
$t^{-}u\in \mathcal{N}$. Moreover, for $tu\in \mathcal{N}$, $K_{u}'(t)=0$.
By \eqref{b12} we obtain that
\begin{equation*}
K_{u}''(t)=t^{q-1}h_{b}'(t)<0.
\end{equation*}
Thus, if $\int_{\mathbb{R}^3}f(x)|u|^q\,dx\leq0$, there is a unique
$t^{-}>t_{b,\max}$ such that $t^{-}u\in \mathcal{N}^{-}$ and
\begin{equation*}
I(t^{-}u)=\sup_{t\geq0}I(tu).
\end{equation*}

(ii) If $\int_{\mathbb{R}^3}f(x)|u|^q\,dx>0$, by \eqref{b12} and \eqref{b14}
we know there are unique $t^{+}$ and $t^{-}$ with $0<t^{+}<t_{b,\max}<t^{-}$
such that  $K_{t^{+}u}'(1)=0$, $K_{t^{-}u}'(1)=0$, that is
$t^{+}u,t^{-}u\in \mathcal{N}$.

From $K_{u}''(t)=t^{q-1}h_{b}'(t)$ and $h_{b}'(t^{+})>0>h_{b}'(t^{-})$, we have
$t^{+}u\in \mathcal{N}^{+}$, $t^{-}u\in \mathcal{N}^{-}$ and
\begin{equation*}
I(t^{+}u)=\inf_{0\leq t\leq t_{b,\rm max}}I(tu), \quad
I(t^{-}u)=\sup_{t\geq t_{b,\rm max}}I(tu).
\end{equation*}
\end{proof}


The forthcoming lemma obtains the minimizing sequence of the energy
functional $I$ on Nehari manifold $\mathcal{N}$.

\begin{lemma}\label{lem2.6}
The energy functional $I$ is coercive and bounded from below on
$\mathcal{N}$.
\end{lemma}

\begin{proof}
For $u\in \mathcal{N}$, then, by H\"older inequality
and  Lemma \ref{lem2.2},
\begin{align*}
I(u)
&=I(u)-\frac{1}{4}\langle I'(u),u\rangle\\
&=\frac{1}{4}\|u\|^2-\Big(\frac{1}{q}-\frac{1}{4}\Big)
 \int_{\mathbb{R}^3}f(x)|u|^q\,dx
 +\Big(\frac{1}{4}-\frac{1}{p}\Big)\int_{\mathbb{R}^3}g(x)|u|^{p}\,dx\\
&\geq\frac{1}{4}\|u\|^2-\Big(\frac{1}{q}-\frac{1}{4}\Big)
 |f|_{q^{\ast}}S_p^{-q/2}\|u\|^q.
\end{align*}
This completes the proof.
\end{proof}


\begin{lemma}\label{lem2.7}
If $|f|_{q^{\ast}}|g|^{(2-q)/(p-2)}_{\infty}\in(0,\sigma)$,
the set $\mathcal{N}^{-}$ is closed in $E$.
\end{lemma}

\begin{proof}
Let $\{u_n\}\subset \mathcal{N}^{-}$ such that $u_n\to u$ in $E$.
In the following we prove $u\in \mathcal{N}^{-}$. Indeed, by
$\langle I'(u_n), u_n\rangle=0$ and
\begin{equation*}
\langle I'(u_n), u_n\rangle-\langle I'(u), u \rangle
=\langle I'(u_n)-I'(u), u\rangle + \langle I'(u_n), u_n-u\rangle\to 0,\quad
\text{as } n\to\infty,
\end{equation*}
we have $\langle I'(u), u\rangle=0$. So $u\in \mathcal{N}$.

For any $u\in \mathcal{N}^{-}$, from \eqref{b6} we have
\begin{equation*}
(2-q)\|u\|^2+(4-q)b|\nabla u|_2^4<(p-q)\int_{\mathbb{R}^3}g(x)|u|^{p}dx.
\end{equation*}
Similar to the proof of \eqref{b9}, we have
\begin{equation}\label{b15}
\|u\|\geq\Big(\frac{(2-q)S_p^{\frac{p}{2}}}{(p-q)|g|_{\infty}}\Big)^{\frac{1}{p-2}}.
\end{equation}
Hence $\mathcal{N}^{-}$ is bounded away from 0.

By \eqref{b6}, it follows that $K_{u_n}''(1)\to K_{u}''(1)$.
From $K_{u_n}''(1)<0$, we have $K_{u}''(1)\leq0$.
By Proposition \ref{prop2.4}, for
$|f|_{q^{\ast}}|g|^{(2-q)/(p-2)}_{\infty}\in(0,\sigma), K_{u}''(1)<0$.
Thus we deduce $u\in \mathcal{N}^{-}$.
\end{proof}

 The following lemma yields a $(PS)_{c}$ sequence from the minimizing sequence
of the energy functional $I$ on Nehari manifold $\mathcal{N}$.

\begin{lemma}\label{lem2.8}
If $|f|_{q^{\ast}}|g|^{(2-q)/(p-2)}_{\infty}\in(0,\sigma)$,
then for every $u\in \mathcal{N}^{+}$, there exist $\epsilon>0$
and a differentiable function
$\varphi^{+}: B_{\epsilon}(0)\to \mathbb{R_{+}}:=(0,+\infty)$ such that
\begin{equation*}
\varphi^{+}(0)=1,\quad \varphi^{+}(w)(u-w)\in \mathcal{ N}^{+}, \quad
\forall w\in B_{\epsilon}(0)
\end{equation*}
and
\begin{equation}\label{b16}
 \langle (\varphi^{+})'(0),w\rangle= L(u,w)/K''_{u}(1),
\end{equation}
where
\begin{align*}
L(u, w)&=2\langle u,w\rangle+4b\int_{\mathbb{R}^3}| \nabla u|^2
 \nabla u \nabla w\,dx -q \int_{\mathbb{R}^3}f(x)|u|^{q-2}uw\,dx\\
&\quad -p\int_{\mathbb{R}^3}g(x)|u|^{p-2}uw\,dx.
\end{align*}
Moreover, for any $C_1, C_2>0$, there exists $C>0$ such that if
$C_1\leq\|u\|\leq C _2$,
\[
|\langle (\varphi^{+})'(0),w\rangle|\leq C\|w\|.
\]
\end{lemma}

\begin{proof}
We define $F:\mathbb{R}\times E\to \mathbb{R}$ by
$F(t,w)=K'_{u-w}(t)$,
it is easy to see $F$ is differentiable. Since $F(1,0)=0$ and
\begin{equation*}
F_{t}(1,0)=K_{u}''(1)>0,
\end{equation*}
we apply the implicit function theorem at point $(1,0)$ to obtain the
 existence of $\epsilon>0$ and differentiable function
$\varphi^{+}: B_{\epsilon}(0)\to \mathbb{R_{+}}:=(0,+\infty)$ such that
\begin{equation*}
\varphi^{+}(0)=1,\quad F(\varphi^{+}(w),w)= 0, \quad \forall w\in B_{\epsilon}(0).
\end{equation*}
Thus,
\begin{equation*}
\varphi^{+}(w)(u-w)\in \mathcal{N},\quad \forall w\in B_{\epsilon}(0).
\end{equation*}

Next, we prove $\varphi^{+}(u-w)\in \mathcal{N}^{+},\quad
\forall w\in B_{\epsilon}(0)$.
Indeed, by $u\in \mathcal{N}^{+}$ and the set $\mathcal{N}^{-}\cup \mathcal{N}^{0}$
is closed, we know $\operatorname{dist}(u, \mathcal{N}^{-}\cup \mathcal{N}^{0})>0$.
Since $\varphi^{+}(w)(u-w)$ is continuous with respect to $w$, when $\epsilon$
is small enough, we know for $w\in B_{\epsilon}(0)$
\begin{equation*}
\|\varphi^{+}(w)(u-w)-u\|<\frac{1}{2}\operatorname{dist}
(u, \mathcal{N}^{-}\cup \mathcal{N}^{0}),
\end{equation*}
so
\begin{align*}
\|\varphi^{+}(w)(u-w)-\mathcal{N}^{-}\cup \mathcal{N}^{0}\|
&\geq \operatorname{dist}(u, \mathcal{N}^{-}\cup \mathcal{N}^{0})
 -\operatorname{dist}(\varphi^{+}(w)(u-w),u)\\
&>\frac{1}{2}\operatorname{dist}(u, \mathcal{N}^{-}\cup \mathcal{N}^{0})>0.
\end{align*}
Thus, $\varphi^{+}(w)(u-w)\in \mathcal{N}^{+}$ for all $w\in B_{\epsilon}(0)$.

Also by the differentiability of the implicit function theorem, we have
\begin{equation*}
\langle (\varphi^{+})'(0),w\rangle
= -\frac{\langle F_{w}(1,0),w\rangle}{F_{t}(1,0)}.
\end{equation*}
Note that $L(u,w)=-\langle F_{w}(1,0),w\rangle$ and
$K''_{u}(1)=F_{t}(1,0)$. So we prove \eqref{b16}.

Next we prove that for any $C_1, C_2>0$, if
$C_1\leq\|u\|\leq C _2$, $u\in \mathcal{N}^{+}$, there exists $\delta>0$
such that $K''_{u}(1)\geq\delta>0$.

On the contrary. If there exists a sequence $\{u_n\}\in \mathcal{N}^{+}$,
$C_1\leq\|u_n\|\leq C _2$, such that for any $\delta_n$ sufficiently small,
$K''_{u_n}(1)\leq\delta_n$, $\delta_n\to0$ as $n\to\infty$.
From \eqref{b6} we have
\begin{equation*}
(2-q)\|u_n\|^2+(4-q)b|\nabla u_n|_2^4
=(p-q)\int_{\mathbb{R}^3}g(x)|u_n|^{p}\,dx+O(\delta_n),
\end{equation*}
where $O(\delta_n)\to0$ as $n\to\infty$.

Noting that $1<q<2$, $4<p<6$, $C_1\leq\|u_n\|\leq C _2$ and \eqref{b8},
we have
\begin{equation*}
(2-q)\|u_n\|^2\leq(p-q)|g|_{\infty}S_p^{-p/2}\|u_n\|^{p}+O(\delta_n),
\end{equation*}
and so
\begin{equation}\label{b17}
\|u_n\|\geq\Big(\frac{(2-q)S_p^{\frac{p}{2}}}{(p-q)|g|_{\infty}}
\Big)^{\frac{1}{p-2}}+O(\delta_n).
\end{equation}
From \eqref{b7} we also have
\begin{equation*}
(p-2)\|u_n\|^2+(p-4)b|\nabla u_n|_2^4
=(p-q)\int_{\mathbb{R}^3}f(x)|u_n|^q\,dx+O(\delta_n).
\end{equation*}
In view of \eqref{b10}, we have
\begin{equation*}
(p-2)\|u_n\|^2\leq(p-q)|f|_{q\ast}S_p^{-q/2}\|u_n\|^q+O(\delta_n),
\end{equation*}
which implies
\begin{equation}\label{b18}
\|u_n\|\leq\Big(\frac{(p-q)|f|_{q\ast}}{(p-2)S_p^{\frac{q}{2}}}\Big)^{\frac{1}{2-q}}
+O(\delta_n).
\end{equation}
Combining  \eqref{b17} and \eqref{b18} as $n\to\infty$, we deduce a contradiction.

Thus if $C_1\leq\|u\|\leq C _2$, there exist $C>0$ such that
\begin{equation*}
|\langle (\varphi^{+})'(0),w\rangle|\leq C\|w\|.
\end{equation*}
This completes the proof.
\end{proof}

Similarly, we establish the following lemma.

\begin{lemma}\label{lem2.9}
Assume $|f|_{q^{\ast}}|g|^{(2-q)/(p-2)}_{\infty}\in(0,\sigma)$, then for
 every $u\in \mathcal{N}^{-}$, there exist  $\epsilon>0$ and a differentiable
function $\varphi^{-}: B_{\epsilon}(0)\to \mathbb{R_{+}}:=(0,+\infty)$ such that
\begin{gather*}
\varphi^{-}(0)=1,\quad \varphi^{-}(w)(u-w)\in \mathcal{N}^{-},\quad
\forall w\in B_{\epsilon}(0), \\
\langle (\varphi^{-})'(0),w\rangle=L(u,w)/K''_{u}(1),
\end{gather*}
where $L(u,w)$ is defined in Lemma \ref{lem2.8}.
Moreover, for any $C_1, C_2>0$, there exists $C>0$ such that if
 $C_1\leq\|u\|\leq C _2$,
\begin{equation*}
|\langle (\varphi^{-})'(0),w\rangle|\leq C\|w\|.
\end{equation*}
\end{lemma}

 The following lemma aims at obtaining the critical point of $I$ on
the whole space from the local minimizer for $I$ on Nehari manifold .

\begin{lemma}\label{lem2.10}
Suppose that $u$ is a local minimizer for $I$ on $\mathcal{N}^{+}$
(or $\mathcal{N}^{-}$). Then $I'(u)=0$.
\end{lemma}

\begin{proof}
If $u\neq0$, $u$ is a local minimizer for $I$ on $\mathcal{N}^{+}$
(or $\mathcal{N}^{-}$), then $u$ is a nontrivial solution of the optimization
problem
\begin{equation*}
 \text{minimize $I$ subject to } \Psi(u)=0,
\end{equation*}
where $\Psi(u)$ is described in  \eqref{b5}. Note that $\Psi'(u)\neq 0$,
$\mathcal{N}^{+}$ (or $\mathcal{N}^{-}$) is a local differential manifold.
So by the theory of Lagrange multipliers, there exists $\mu\in\mathbb{R}$
such that $I'(u)=\mu\Psi'(u)$. Thus
\begin{equation*}
\langle I'(u),u\rangle=\mu\langle\Psi'(u),u\rangle.
\end{equation*}
Since $u\in  \mathcal{N}^{+}$ (or $\mathcal{N}^{-}$),
 $\langle I'(u),u\rangle=0$ and $\langle\Psi'(u),u\rangle=K_{u}''(1)\neq 0$.
Hence, $\mu=0$. Thus the proof is complete.
\end{proof}

\section{Proof of Theorem \ref{thm1.1}}\label{2}

For proving Theorem \ref{thm1.1},
we first show that any Palais-Smale sequence of $I$ has a strongly
convergent subsequence in $E$.


\begin{lemma}\label{lem3.1}
Each Palais-Smale sequence $\{u_n\}\subset \mathcal{N}^{+}$
(or $\mathcal{N}^{-}$) for $I$ on $E$ has a strongly convergent subsequence.
\end{lemma}

\begin{proof}
 Assume that $\{u_n\}\subset \mathcal{N}^{+}$(or $\mathcal{N}^{-}$) such that
$I(u_n)\to c$ and $I'(u_n)\to0$ as $n\to\infty$. As in  Lemma \ref{lem2.6},
we know the Palais-Smale sequence $\{u_n\}\subset \mathcal{N}^{+}$
(or $\mathcal{N}^{-}$) for $I$ on $E$ is bounded. And by  Lemma \ref{lem2.2},
going if necessary to a subsequence,  we have
\begin{gather*}
u_n\rightharpoonup u\quad \text{in } E, \\
u_n\to u\quad \text{in } L^{r}(\mathbb{R}^3), ~r\in [2,2^{\ast}).
\end{gather*}
Note that
\begin{align*}
&(I'(u_n)-I'(u),u_n-u) \\
&=( I'(u_n),u_n-u)-( I'(u),u_n-u)\\
&=\Big(a+b\int_{\mathbb{R}^3}|\nabla u_n|^2\,dx\Big)
 \int_{\mathbb{R}^3}|\nabla(u_n-u)|^2\,dx+\int_{\mathbb{R}^3}V(x)| u_n-u|^2\,dx\\
&\quad -b\Big(\int_{\mathbb{R}^3}|\nabla u|^2\,dx
 -\int_{\mathbb{R}^3}|\nabla u_n|^2\,dx\Big)
 \int_{\mathbb{R}^3}\nabla u\nabla (u_n-u)\,dx\\
&\quad -\int_{\mathbb{R}^3}f(x)(|u_n|^{q-2}u_n-|u|^{q-2}u)(u_n-u)\,dx\\
&\quad -\int_{\mathbb{R}^3}g(x)(|u_n|^{p-2}u_n-|u|^{p-2}u)(u_n-u)\,dx\\
&\geq  \|u_n-u\|^2-b\Big(\int_{\mathbb{R}^3}|\nabla u|^2\,dx
 -\int_{\mathbb{R}^3}|\nabla u_n|^2\,dx\Big)
 \int_{\mathbb{R}^3}\nabla u\nabla(u_n-u)\,dx\\
&\quad -\int_{\mathbb{R}^3}f(x)(|u_n|^{q-2}u_n-|u|^{q-2}u)(u_n-u)\,dx\\
&\quad -\int_{\mathbb{R}^3}g(x)(|u_n|^{p-2}u_n-|u|^{p-2}u)(u_n-u)\,dx,\\
\end{align*}
then we can deduce that $\|u_n-u\|\to0$ as $n\to\infty$.
Indeed, from the boundedness of $\{ u_n\}$ in $E$ and  Lemma \ref{lem2.2},
$\{ u_n\}$ is bounded in $L^{r}(\mathbb{R}^3)$, $r\in[2, 6)$.
By  using twice H\"{o}lder inequality we obtain
\begin{align*}
&\big|\int_{\mathbb{R}^3}f(x)(|u_n|^{q-2}u_n-|u|^{q-2}u)(u_n-u)\,dx\big|\\
&\leq \Big(\int_{\mathbb{R}^3}|f|^{q^{\ast}}\,dx\Big)^{1/q^{\ast}}
 \Big(\int_{\mathbb{R}^3}||u_n|^{q-2}u_n-|u|^{q-2}u|^{p/q}|
 u_n-u|^{p/q}\,dx\Big)^{q/p}\\
&\leq C|f|_{q^{\ast}}(|u_n|_p^{q-1}+|u|_p^{q-1})|u_n-u|_p\to 0,\quad
 \text{as } n\to\infty,
\end{align*}
where $C$ is a positive constant. Similarly, we have
\begin{equation*}
\big|\int_{\mathbb{R}^3}g(x)(|u_n|^{p-2}u_n-|u|^{p-2}u)(u_n-u)\,dx\big|
\to 0,\quad \text{as } n\to\infty\,.
\end{equation*}
From 
\begin{equation*}
b\Big(\int_{\mathbb{R}^3}|\nabla u|^2\,dx-\int_{\mathbb{R}^3}|\nabla u_n|^2\,dx\Big)
\int_{\mathbb{R}^3}\nabla u\nabla (u_n-u)\,dx\to0,\quad \text{as } n\to\infty,
\end{equation*}
and
\begin{equation*}
( I'(u_n)-I'(u),u_n-u)\to0,\quad \text{as } n\to\infty,
\end{equation*}
we have $\|u_n-u\|\to0$ as $n\to\infty$. This completes the proof.
\end{proof}

\begin{lemma}\label{lem3.2}
If $|f|_{q^{\ast}}|g|^{(2-q)/(p-2)}_{\infty}\in(0,\sigma)$, then the
minimization problem
\begin{equation*}
c_1=inf_{\mathcal{N}^{+}}I
\end{equation*}
is solved at a point $u_1\in \mathcal{N}^{+}$, moreover this value is a critical
point of $I$.
\end{lemma}

\begin{proof}
First we prove the minimizing sequence $\{u_n\}\subset \mathcal{N}^{+}$ 
is a $(PS)_{c_1}$ sequence on $E$. Indeed, by  Lemma \ref{lem2.6} and
Ekeland Variational Principle \cite{MWM} on $\mathcal{N}^{+}\cup \mathcal{N}^{0}$, 
there exists a minimizing sequence $\{u_n\}\subset \mathcal{N}^{+}\cup N^{0}$ 
such that
\begin{gather}\label{c1}
\inf_{u\in \mathcal{N}^{+}\cup \mathcal{N}^{0}}I(u)
\leq I(u_n)<\inf_{u\in \mathcal{N}^{+}\cup \mathcal{N}^{0}}I(u)+\frac{1}{n}, \\
\label{c2}
I(u_n)-\frac{1}{n}\|v-u_n\|\leq I(v),\quad 
\forall v\in \mathcal{N}^{+}\cup \mathcal{N}^{0}.
\end{gather}
From  Proposition \ref{prop2.5}, we know for each 
$u\in E\backslash \{0\}$, there is a unique $t^{+}$ such that 
$t^{+}u\in \mathcal{N}^{+}$, then $inf_{u\in \mathcal{N}^{+}}I\leq I(t^{+}u)$. 
Now we  prove that for each $u\in \mathcal{N}^{+}, I(u)<0$. 
Indeed, for each $u\in \mathcal{N}^{+}$, $K_{u}''(1)>0$. From  \eqref{b7}, we have
\begin{equation*}
(p-q)\int_{\mathbb{R}^3}f(x)|u|^q\,dx>(p-2)\|u\|^2+(p-4)b|\nabla u|_2^4.
\end{equation*}
Then for each $u\in \mathcal{N}^{+}$,
\begin{equation}\label{c3}
\begin{split}
I(u)&=I(u)-\frac{1}{p}\langle I'(u),u\rangle\\
&=\frac{p-2}{2p}\|u\|^2+\frac{p-4}{4p}b|\nabla u|_2^4
 -\frac{p-q}{pq}\int_{\mathbb{R}^3}f(x)|u|^q\,dx\\
&<\frac{p-2}{2p}\|u\|^2+\frac{p-4}{4p}b|\nabla u|_2^4
 -\frac{1}{pq}\left((p-2)\|u\|^2+(p-4)b|\nabla u|_2^4\right)\\
&=\frac{(p-2)(q-2)}{2pq}\|u\|^2+\frac{(p-4)(q-4)}{4pq}b|\nabla u|_2^4<0.
\end{split}
\end{equation}
From the above, we know that $\inf_{u\in \mathcal{N}^{+}}I(u)<0$.

Since $I(0)=0$, we have 
$\inf_{u\in \mathcal{N}^{+}\cup \mathcal{N}^{0}}I(u)
=\inf_{u\in \mathcal{N}^{+}}I(u)=c_1$.
Thus we may assume $u_n\in \mathcal{N}^{+}$, $I(u_n)\to c_1<0$.
By Lemma \ref{lem2.8} , since 
$|f|_{q^{\ast}}|g|^{(2-q)/(p-2)}_{\infty}\in(0,\sigma)$, 
we can find $\epsilon_n>0$ and differentiable function 
$\varphi^{+}_n=\varphi^{+}_n(w)>0 $ such that
\begin{equation*}
\varphi_n^{+}(w)(u_n-w)\in \mathcal{N}^{+},\quad
\forall w\in B_{\epsilon_n}(0).
\end{equation*}
By the continuity of $\varphi_n^{+}(w)$ and $\varphi_n^{+}(0)=1$,
 without loss of generality, we can assume $\epsilon_n$ is sufficiently 
small such that $\frac{1}{2}\leq \varphi^{+}_n(w)\leq\frac{3}{2}$ for
 $\|w\|<\epsilon_n$.
From $\varphi_n^{+}(w)(u_n-w)\in \mathcal{N}^{+}$ and \eqref{c2}, 
we have
\begin{equation*}
I(\varphi^{+}_n(w)(u_n-w))\geq I(u_n)-\frac{1}{n}\|\varphi^{+}_n(w)(u_n-w)-u_n\|,
\end{equation*}
which implies 
\begin{align*}
&\langle I'(u_n), \varphi^{+}_n(w)(u_n-w)-u_n\rangle
+o\big(\|\varphi^{+}_n(w)(u_n-w)-u_n\|\big)\\
&\geq-\frac{1}{n}\| \varphi^{+}_n(w)(u_n-w)-u_n\|.
\end{align*}
Consequently,
\begin{align*}
&\varphi^{+}_n(w)\langle I'(u_n),w\rangle+(1-\varphi^{+}_n(w))
 \langle I'(u_n),u_n\rangle\\
&\leq \frac{1}{n}\|\left(\varphi^{+}_n(w)-1\right)u_n
-\varphi^{+}_n(w)w\|+o\left(\|\varphi^{+}_n(w)(u_n-w)-u_n\|\right).
\end{align*}
By the choice of $\epsilon_n$ and  $1/2\leq \varphi^{+}_n(w)\leq3/2$, 
we infer that there exists $C_3>0$ such that
\begin{equation*}
|\langle I'(u_n),w\rangle|\leq\frac{1}{n}\|\langle (\varphi^{+}_n)'(0),w\rangle u_n\|
+\frac{C_3}{n}\|w\|+o\left(|\langle (\varphi^{+}_n)'(0),w\rangle|(\|u_n\|
+\|w\|)\right).
\end{equation*}
Next we prove for $\{u_n\}\subset \mathcal{N}^{+}$, $\inf_n\|u_n\|\geq C_1>0$,
 where $C_1$ is a constant. Indeed, if not, then $I(u_n)$ would converge to zero,
 which contradict with $I(u_n)\to c_1<0$. Moreover, by Lemma \ref{lem2.6}
we know that $I$ is coercive on $\mathcal{N}^{+}$, $\{ u_n\}$ is bounded in $E$. 
Thus, there exists $C_2>0$ such that $0<C_1\leq\|u_n\|\leq C_2$.
From Lemma \ref{lem2.8}, $|\langle (\varphi^{+}_n)'(0),w\rangle|\leq C\|w\|$.
So
\begin{equation*}
|\langle I'(u_n),w\rangle|\leq\frac{C}{n}\|w\|+\frac{C}{n}\|w\|+o(\|w\|)
\end{equation*}
and
\begin{equation}\label{c4}
\begin{gathered}
\|I'(u_n)\|=\sup_{w\in E\backslash \{0\}}\frac{|\langle I'(u_n),w\rangle|}{\|w\|}\leq\frac{C}{n}+o(1),\\
\|I'(u_n)\|\to 0,\quad\text{as } n\to\infty.
\end{gathered}
\end{equation}
Thus, $\{u_n\}\subset \mathcal{N}^{+}$ is $(PS)_{c_1}$ for $ I$ on
$E$. From Lemma \ref{lem3.1}, there is a strongly convergent subsequence 
$\{u_n\}$, we still denote by $\{u_n\}$, $u_n\to u_1$ in $E$.
From the above we know that there exist $C_1, C_2>0$ such that
$0<C_1\leq\|u_n\|\leq C_2$, then $0<C_1\leq\|u_1\|\leq C_2$. 
Thus $u_1\neq 0$.

Next we prove $ u_1\in \mathcal{N}^{+}$. Indeed, by \eqref{b6}, it follows 
that $K_{u_n}''(1)\to K_{u_1}''(1)$. From $K_{u_n}''(1)>0$, we have 
$K_{u_1}''(1)\geq0$. By Proposition \ref{prop2.4}, we know 
$K_{u_1}''(1)>0$. Thus we deduce
\begin{equation*}
u_1\in \mathcal{N}^{+},\quad 
 I(u_1)=\lim_{n\to\infty} I(u_n)=\inf_{u\in \mathcal{N}^{+}}I(u).
\end{equation*}
We recall  \cite{LCE} that 
$\int_{\mathbb{R}^3}|\nabla|u||^2\,dx=\int_{\mathbb{R}^3}|\nabla u|^2\,dx$, 
therefore  $I(u_1)=I(|u_1|)$ and $| u_1|\in \mathcal{N}^{+}$, then without 
loss of generality we may assume that $u_1$ is positive. 
Combining this with Lemma \ref{lem2.10}, we obtain the results.
\end{proof}

\begin{lemma}\label{lem3.3}
If  $|f|_{q^{\ast}}|g|^{(2-q)/(p-2)}_{\infty}\in(0,\sigma)$, then the minimization 
problem
\begin{equation*}
c_2=\inf_{\mathcal{N}^{-}}I
\end{equation*}
is solved at a point $u_2\in \mathcal{N}^{-}$ which is a critical point for $I$.
\end{lemma}

\begin{proof}
From Lemma \ref{lem2.7}, $\mathcal{N}^{-}$ is closed in $E$. 
By Lemma \ref{lem2.6}, we know $I$ is coercive on $\mathcal{N}^{-}$. 
So we use Ekeland Variational Principle \cite{MWM} on $\mathcal{N}^{-}$ 
to obtain a minimizing sequence $\{u_n\}\subset \mathcal{N}^{-}$ such that
\begin{gather*}
\inf_{u\in \mathcal{N}^{-}}I(u)\leq I(u_n)<\inf_{u\in \mathcal{N}^{-}}I(u)
 +\frac{1}{n},\\
I(u_n)-\frac{1}{n}\|v-u_n\|\leq I(v),\quad \forall v\in \mathcal{N}^{-}.
\end{gather*}
In view of \eqref{b15} and  Lemma \ref{lem2.6} we know the there exist 
$C_1, C_2>0$ such that
\begin{equation*}
0<C_1\leq\|u_n\|\leq C_2.
\end{equation*}
Hence by Lemma \ref{lem2.9}, in the same way as Lemma \ref{lem3.2}, 
there exists a minimizing sequence $\{u_n\}\subset \mathcal{N}^{-}$ 
is the $(PS)_{c_2}$ sequence on E. From Lemmas \ref{lem3.1}, we know there 
is a strongly convergent subsequence $\{u_n\}$, we still denote by 
$\{u_n\}$, $u_n\to u_2$ in $E$. By Lemma \ref{lem2.7} the set 
$\mathcal{N}^{-}$ is closed, we know $u_2\in \mathcal{N}^{-}$. 
Thus, $I(u_2)=\lim_{n\to\infty} I(u_n)=\inf_{u\in \mathcal{ N}^{-}}I(u)$. 
Since $I(u_2)=I(|u_2|)$ and $|u_2|\in \mathcal{N}^{-}$, then without 
loss of generality we may assume that $u_2$ is positive. 
Combining with Lemma \ref{lem2.10}, we prove the claim.
\end{proof}

Now we are in a position to give the proof of the main results.

\begin{proof}[Proof of theorem  \ref{thm1.1}]
From Lemmas \ref{lem3.2} and  \ref{lem3.3}, we know if
$|f|_{q^{\ast}}|g|^{(2-q)/(p-2)}_{\infty}\in(0,\sigma)$, then problem \eqref{a1} 
has at least two positive solutions $u_1$ and $u_2$.
 From the proof of Lemma \ref{lem3.2} we know that 
if $|f|_{q^{\ast}}|g|^{(2-q)/(p-2)}_{\infty}\in(0,\sigma)$, then the 
 positive solution of  \eqref{a1} $u_1$ belongs to $\mathcal{N}^{+}$ and 
$I(u_1)<0$. If 
$0<|f|_{q^{\ast}}| g|^{(2-q)/(p-2)}_{\infty}<\sigma^{\ast}
:=\frac{q}{p-2}\sigma<\sigma$,
where $\sigma$ is described in Proposition \ref{prop2.4},
then by \eqref{b15} we can infer that
\begin{align*}
I(u)&=I(u)-\frac{1}{4}\langle I'(u),u\rangle\\
&=\frac{1}{4}\|u\|^2-\big(\frac{1}{q}-\frac{1}{4}\big)
 \int_{\mathbb{R}^3}f(x)|u|^q\,dx+\big(\frac{1}{4}-\frac{1}{p}\big)
 \int_{\mathbb{R}^3}g(x)|u|^{p}\,dx\\
&\geq\frac{1}{4}\|u\|^2-\big(\frac{1}{q}-\frac{1}{4}\big)
 |f|_{q_{\ast}}S_p^{-q/2}\|u\|^q\\
&=\|u\|^q\Big(\frac{1}{4}\|u\|^{2-q}-(\frac{1}{q}
 -\frac{1}{4})|f|_{q_{\ast}}S_p^{-q/2}\Big)\\
&\geq\Big(\frac{(2-q)S_p^{\frac{p}{2}}}{(p-q)|g|_{\infty}}\Big)^{\frac{q}{p-2}}
 \Big(\frac{1}{4}\Big(\frac{(2-q)S_p^{\frac{p}{2}}}{(p-q)| g|_{\infty}}
 \Big)^{\frac{2-q}{p-2}}-\big(\frac{1}{q}-\frac{1}{4}\big)|f|_{q^{\ast}}S_p^{-q/2}
 \Big)\\
&\geq\Big(\frac{(2-q)S_p^{\frac{p}{2}}}{(p-q)|g|_{\infty}}\Big)^{\frac{q}{p-2}}
 \Big(\frac{1}{4}\Big(\frac{(2-q)S_p^{\frac{p}{2}}}{(p-q)| g|_{\infty}}
 \Big)^{\frac{2-q}{p-2}}-\frac{p-q}{4q}|f|_{q^{\ast}}S_p^{-q/2}\Big)
>0.
\end{align*}
In fact, if $|f|_{q^{\ast}}|g|^{(2-q)/(p-2)}_{\infty}\in(0,\sigma^{\ast})$, 
for any $u\in \mathcal{N}^{-}$, $I(u)>0$, where $\sigma^{\ast}=q/(p-2)\sigma$. 
From Lemma \ref{lem3.3} the positive solution of problem \eqref{a1}
 $ u_2\in \mathcal{N}^{-}$, then for 
$|f|_{q^{\ast}}|g|^{(2-q)/(p-2)}_{\infty}\in (0,\sigma^{\ast})$, $I(u_2)>0$. 
From the above, we know that 
if $|f|_{q^{\ast}}|g|^{(2-q)/(p-2)}_{\infty}\in (0,\sigma^{\ast})$, then
$I(u_1)=\inf_{u\in \mathcal{N}}I(u)$, and  $u_1$ is a positive ground state 
solution of  \eqref{a1}. This completes the proof.
\end{proof}


\subsection*{Acknowledgments}
This work is supported by the Innovation Project for college postgraduates 
in Jiangsu Province(KYLX15-0108), the Natural Science Foundation of China
(Nos: 11371090, 11571140, 11601204, 11671077), the  Natural Science Foundation
 of Jiangsu Province (Nos: BK 20140525, BK 20150478, BK 20160063) 
and the Natural Science Foundation of Outstanding Young Scholars of Jiangsu 
Province (Nos: BK 20160063).

\begin{thebibliography}{00}

\bibitem{AAHBGC} A. Ambrosetti, H. Brezis, G. Cerami; 
\emph{Combined effects of concave and convex nonlinearities in some elliptic problems},
 J. Funct. Anal., 122 (1994), 519-543.

\bibitem{AAAZZ}A. Azzollini, P. d'Avenia, A. Pomponio; 
\emph{Multiple critical points for a class of nonlinear functionals}, 
Ann. Mat. Pura Appl., 190 (2011), 507-523.

\bibitem{TBTBT} T. Bartsch; 
\emph{Infinitely many solutions of a symmetric Dirichlet problem}, 
Nonlinear Anal. TMA, 20 (1993), 1205-1206.

\bibitem{HBPLL} H. Berestycki, P. L. Lions; 
\emph{Nonlinear scalar field equations I}, Arch. Rational Mech. Anal.,
 82(4) (1983), 313-345.

 \bibitem{BARTS}T. Bartsch, Z. Q. Wang; 
\emph{Existence and multiple results for some superlinear elliptic problems on 
$\mathbb{R}^N$}, Commun. Partial Differ. Equ., 20 (1995), 1725-1741.

\bibitem{KBYZ} K. J. Brown, Y. P. Zhang; 
\emph{The Nehari manifold for a semilinear elliptic equation with a sign-changing 
weight function}, J. Differential Equations, 193 (2003), 481-499.

\bibitem{KBTW} K. J. Brown, T. F. Wu; 
\emph{A fibering map approach to a semilinear elliptic booundary value problem},
 Electron. J. Differential Equations, 69 (2007), 1-9.

\bibitem{KCC} K. C. Chang; 
\emph{Methods in Nonlinear Analysis}, Springer, 2005.

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

\bibitem{CLLM} S. J. Chen, L. Li; 
\emph{Multiple solutions for the nonhomogeneous Kirchhoff equation on $R^{N}$}, 
Nonlinear Anal. Real World Appl., 14 (2013), 1477-1486.

\bibitem{BCBC} B. T. Cheng; 
\emph{New existence and multiplicity of nontrivial solution for nonlocal 
elliptic Kirchhoff type problems}, J. Math. Anal. Appl., 394 (2012), 488-495.

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

\bibitem{DGW} G. W. Dai, R. F. Hao; 
\emph{Existence of solutions for a {$p(x)$}-{K}irchhoff-type equation}, 
J. Math. Anal. Appl., 1 (2009), 275--284.

\bibitem{YDYD} Y. H. Ding; 
\emph{Variational Methods for Strongly Indefinite Problems}, 
World Scientific Press, 2007.

\bibitem{PDSP} P. Dr\'abek, S. Pohozaev; 
\emph{Positive solutions for the p-Laplacian: application of the fibering method}, 
Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 703-726.

\bibitem{LCE} L. C. Evans; 
\emph{Partial Differential Equations}, Graduate Studies in Mathematics, 
vol. 19 (American Mathematical Society, Providence, 1998).

\bibitem{GMFG} Giovany M. Figueiredo; 
\emph{Existence of a positive solution for a Kirchhoff problem type with 
critical growth via truncation argument}, J. Math. Anal. Appl., 401 (2013), 706-713.

\bibitem{XHWZ} X. M. He, W. M. Zou; 
\emph{Multiplicity of solutions for a class of Kirchhoff type problems}, 
Acta Math. Appl. Sin. Engl. Ser., 26 (2010), 387-394.

\bibitem{XHWWZ} X. M. He, W. M. Zou; 
\emph{Existence and concentration behavior of positive solutions for a 
Kirchhoff equation in $R^3$}, J. Differential Equations, 2 (2012), 1813-1834.

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

\bibitem{GGKKM} G. Kirchhoff; 
\emph{Mechanik}, Teubner, Leipzig, 1883.

\bibitem{JLLLL} J. L. Lions; 
\emph{On some questions in boundary value problems of mathematical physics}, 
North-Holland Math. Stud., 30 (1978), 284-346.

\bibitem{ZSLSJG} Z. S. Liu, S. J. Guo; 
\emph{On ground states for the Kirchhoff-type problem with a general critical
 nonlinearity}, J. Math. Anal. Appl., 426 (2015), 267-287.

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

\bibitem{LFYL} 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.

\bibitem{YLFL} Y. H. Li, F. Y. Li, J. P. Shi; 
\emph{Existence of positive solutions to Kirchhoff type problems with zero mass}, 
J. Math. Anal. Appl., 410 (2014), 361-374.

\bibitem{ZLFL} Z. P. Liang, F. Y. Li, J. P. Shi; 
\emph{Positive solutions to Kirchhoff type equations with nonlinearity 
having prescribed asymptotic behavior},  Ann. Inst. H. Poincar\'e Anal. 
Non Lin\'eaire, 31 (2014), 155-167.

\bibitem{ZLLZQW} Z. L. Liu, Z. Q. Wang; 
\emph{Multiple bound states of nonlinear Schr\"odinger systems}, 
Comm. Math. Phys., 282 (2008), 721-731.

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

\bibitem{JSSL} J. Sun, S. B. Liu; 
\emph{Nontrival solution of Kirchhoff type problems},
 Appl. Math. Lett., 25 (2012), 500-504.

\bibitem{YZXL} Y. J. Sun, X. Liu;
 \emph{Existence of Positive Solutions for Kirchhoff Type Problems with
 Critical Exponent}, J. Part. Diff. Eq., 25 (2012), 85-96.

\bibitem{HSCT}J. J. Sun, C. L. Tang; 
\emph{Existence and multiplicity of solutions for Kirchhoff type equations}, 
Nonlinear Anal., 74 (2011), 1212-1222.

\bibitem{ASTW} A. Szulkin, T. Weth; 
\emph{The method of Nahari manifold}, Boston, 2010, 597-632.

\bibitem{MWM} M. Willem;
\emph{Minimax Theorems}, Progr. Nonlinear Differential Equations Appl., 
vol. 24, Birkhauser, Basel, 1996.

\bibitem{TWU} T. F. Wu; 
\emph{Multiple positive solutions for a class of concave-convex ellipic 
problems in $\mathbb{R}^{N}$ involving sign-changing weight}, 
J. Funct. Anal., 258 (2010), 99-131.

\bibitem{XXWW} X. Wu; 
\emph{Existence of nontrivial solutions and high energy solutions for 
Schroinger-Kirchhoff type equations in $\mathbb{R}^N$},
 Nonlinear Anal. Real World Appl., 12 (2011), 1278-1287.

\bibitem{JWLX} J. Wang, L. X. Tian, J. X. Xu, F. B. Zhang; 
\emph{Multiplicity and concentration of positive solutions for a Kirchhoff 
type problem with critical growth}, J. Differential Equations, 253 (2012), 2314-2351.

\bibitem{ZTZKP} Z. T. Zhang, K. Perera; 
\emph{Sign changing solutions of Kirchhoff type problems via invariant 
sets of descent flow}, J. Math. Anal. Appl., 317 (2)(2006), 456-463.

\bibitem{WZMSC} W. M. Zou, M. Schechter; 
\emph{Critical Point Theory and Its Applications}, Springer, New York 2006.

\end{thebibliography}

\end{document}