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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 267, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2013/267\hfil Soliton solutions]
{Soliton solutions for a quasilinear \\ Schr\"{o}dinger equation}

\author[D. Liu\hfil EJDE-2013/267\hfilneg]
{Duchao Liu}  % in alphabetical order

\address{Duchao Liu \newline
School of Mathematics and Statistics, Lanzhou
University, Lanzhou 730000, China} 
\email{liuduchao@gmail.com, Phone +8613893289235, fax +8609318912481}

\thanks{Submitted June 24, 2013. Published December 5, 2013.}
\subjclass[2000]{35B38, 35D05, 35J20}
\keywords{Quasilinear Schr\"{o}dinger equation; soliton solution;
\hfill\break\indent critical point theorem; fountain theorem; dual fountain theorem}

\begin{abstract}
 In this article, critical point theory is used to show the existence
 of nontrivial weak solutions to the quasilinear Schr\"{o}dinger equation
 \[
 -\Delta_p u-\frac{p}{2^{p-1}}u\Delta_p(u^2)=f(x,u)
 \]
 in a bounded smooth domain $\Omega\subset\mathbb{R}^{N}$ with Dirichlet
 boundary conditions.
\end{abstract}

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


\section{Introduction}\label{00}

In this article, we study the soliton solutions for the quasilinear
Schr\"{o}dinger equation 
\begin{equation}\label{(P)}
\begin{gathered}
-\Delta_p u-\frac{p}{2^{p-1}}u\Delta_p(u^2)=f(x,u),  \quad  \text{in } \Omega, \\
         u=0, \quad\text{on } \partial\Omega,
\end{gathered}
\end{equation}
where $\Omega\subset\mathbb{R}^{N}$ is a bounded smooth domain,
$\Delta_{p}u=\operatorname{div}(|\nabla u|^{p-2}\nabla u)$ is the
$p$-Laplacian with $1<p<N$.

When $p=2$, equation \eqref{(P)} is a special case for some physical
phenomena, see e.g. \cite{Liu2,Liu1, Pop}. In fact, solutions for
the problem \eqref{(P)} for $p=2$ are the existence of standing wave
solutions for the following quasilinear Schr\"{o}dinger equations
\begin{equation}\label{(1)}
\begin{aligned}
i\partial_tz=-\Delta z+Wz-f(|z|^2)z-\kappa\Delta h(|z|^2)h'(|z|^2)z,
\end{aligned}
\end{equation}
where $W(x)$, $x\in\mathbb{R}^N$ is a given potential, $\kappa$ is a
real constant and $f$, $h$ are real functions of essentially pure
power forms. The semi linear case corresponding to $\kappa=0$ has
been studied extensively in recent years, see \cite{Bere, Floe,Stra}.
 Quasilinear equations of form \eqref{(1)} appear more
naturally in mathematical physics and have been derived as model of
several physical phenomena corresponding to various types of $h$,
the superfluid film equation in plasma physics by Kurihara in
\cite{Kuri} for $h(s)=s$. In the case $h(s)=(1+s)^{1/2}$,
\eqref{(1)} models the self-channeling of a high-power ultra short
laser in matter, see \cite{Boro, Bran, Chen, Ritc} and the
references in \cite{De}. Equation \eqref{(1)} also appears in plasma
physics and fluid mechanics \cite{Kuri, Laed, Litv, Naka, Pork}, in
the theory of Heisenberg ferromagnets and magnons \cite{Bass, Kose,
Lang, Quis, Take}, in dissipative quantum mechanics \cite{Hass}, and
in condensed matter theory \cite{Makh}. In the mathematical
literature very few results are known about equations of the form
\eqref{(1)} before Liu's research team \cite{Liu1, Liu2}, in which,
the existence of positive solution has been proved in \cite{Liu1} by
using a constrained minimization argument. It is worthy of attention
that another earlier paper \cite{Liu_Wang_Wang} deals with a more
general type equations without using the change of variables
developed in the later literature. The problem \eqref{(1)} was
transformed into a semilinear one by a change of variables and an
Orlicz space framework was used in \cite{Liu2}. Since then several
papers appear in the mathematical literature for the equation
defined in the domain $\mathbb{R}^N$. For example, see \cite{Coli,
Glos, Wang, Liu3, Silv} and a very recent paper \cite{Fang}, in
which the authors established the existence of ground states of
soliton type solutions by a minimization argument. But to our best
knowledge, there is no one considering this problem for the
$p$-Laplacian case in a bounded domain.

We consider soliton solutions for the following
quasilinear Schr\"{o}dinger equations of a  more general form than
\eqref{(1)}, in a bounded smooth domain $\Omega\subset\mathbb{R}^N$
with the Dirichlet boundary condition
\[
i\partial_tz=-\Delta_p z+Wz-f(|z|^2)z-\kappa\Delta_p
h(|z|^2)h'(|z|^2)z,
\]
in which $\kappa=\frac{p}{2^{p-1}}>0$, $h(s)=s$ and $f=f(x,s)$ is a
Caratheodory function under some power growth with respect to $s$.
At the same time we assume $W(x)\equiv W$ (a constant) to indicate
that the solution stays at a constant potential level. Putting
$z(x,t)=\exp(-iWt)u(x)$ we obtain the corresponding equation
\eqref{(P)} of elliptic type which has a formal variational
structure, see Section \ref{1}.

For a deep insight into this problem one can find that a major
difficulty of the problem \eqref{(P)} is that the functional
corresponding to the equation is not well defined for all $u\in
W^{1,p}_0(\Omega)$ if $N\geq p$. We generalized the method of a
change of variables developed in \cite{Coli} to overcome this
difficulty, and make a slight different definition of weak
solutions. Then by a standard argument from critical point theory,
we develop the existence of nontrivial solutions to our problem.

This article is organized as follows. In Section \ref{1}, we give the
definition of our weak solutions for our problem; in section
\ref{1.1}, we give some existence theorems of solutions and some
remarks for our theorems.



\section{Definition of weak solution}\label{1}

We assume the perturbation $f(x,t)$ is a Caratheodory function.
Firstly we introduce a variational framework of problem \eqref{(P)}.
Under some increasing conditions on $f$ about the item $u$, we
observe that \eqref{(P)} is the Euler-Lagrange equation associated
with the energy functional
\begin{equation}\label{Ori_Functional}
J(u):=\frac{1}{p}\int_{\Omega}(1+p|u|^p)|\nabla u|^p \,
\mathrm{d}x-\int_{\Omega}F(x,u) \, \mathrm{d}x,
\end{equation}
where $F(x,t)=\int^{t}_{0}f(x,s)\, \mathrm{d}x$.

It is difficult to apply variational methods to the functional $J$
directly. Unless $N=1$, the functional $J$ is not well-defined for
all $u\in W_0^{1,p}(\Omega)$. To overcome this difficulty, we
generalized the method of changing variables developed in
\cite{Coli, Liu2}. That is 
\[
v:=g^{-1}(u), 
\]
where $g$ is defined by
\begin{gather*}
g'(t)=\frac{1}{(1+p|g(t)|^p)^{1/p}}, \quad \forall t\in[0,+\infty];\\
g(t)=-g(-t), \quad  \forall t\in(-\infty,0].
\end{gather*}
We summarize the properties of $g$ as following.

\begin{lemma}\label{g}
The function $g$ defined above satisfies the following conditions:
\begin{enumerate}
\item $g(0)=0$;
\item $g$ is uniquely defined, $C^\infty$ and invertible;
\item $0<g'(t)\leq1$ for all $t\in\mathbb{R}$;
\item $\frac{1}{2}g(t)\leq tg'(t)\leq g(t)$ for all $t>0$;
\item $g(t)/t\nearrow1$, as $t\to 0+$;
\item $|g(t)|\leq|t|$ for all $t\in\mathbb{R}$;
\item $g(t)/\sqrt{t}\nearrow K_0:=\sqrt{2}p^{-1/(2p)}$, as $t\to  +\infty$;
\item $|g(t)|\leq K_0|t|^{1/2}$ for all $t\in\mathbb{R}$;
\item $g^2(t)-g(t)g'(t)t\geq0$ for all $t\in\mathbb{R}$;
\item There exists a positive constant $C$ such that $|g(t)|\geq C|t|$ 
 for $|t|\leq1$ and $|g(t)|\geq C|t|^{1/2}$ for $|t|\geq1$;
\item $|g(t)g'(t)|<K_0^{2}$ for all $t\in \mathbb{R}$;
\item $g''(t)<0$ when $t>0$ and $g''(t)>0$ when $t<0$.
\end{enumerate}
\end{lemma}

\begin{proof}
The conclusions (1), (2) and (3) are trivial.
 To establish the left hand side of inequality (4), we need to
show that, for all $t\geq0$,
\[
(1+p|g(t)|^p)^{1/p}g(t)\leq2t.
\]
To prove this we study the function
$h:\mathbb{R}^+\to  \mathbb{R}$, defined by
\[
h(t):=2t-(1+p|g(t)|^p)^{1/p}g(t).
\]
We have $h(0)=0$, and since
$g'(t)(1+p|g(t)|^p)^{1/p}=1$ for all $t\in\mathbb{R}$, we have
\[
h'(t)=|g'(t)|^p\geq0.
\]
Hence the left hand side inequality is proved. The right hand side
inequality can be proved in a similar way.

It is easy to get (5) and (6) by (4).
We give the proof of (7) by the Principle of L'Hospital. In
fact, since $g(t)\to +\infty$ as $t\to +\infty$, we
have
\begin{align*}
\lim_{t\to +\infty}\frac{g(t)}{t^{1/2}}
&=\lim_{t\to +\infty}\Big(\frac{(g(t))^{2}}{t}\Big)^{1/2}
=\Big(\lim_{t\to +\infty}\frac{(g(t))^{2}}{t}\Big)^{1/2}\\
&=\Big(\lim_{t\to +\infty}\frac{2g(t)g'(t)}{t'}\Big)^{1/2}\\
&=\Big(\lim_{t\to +\infty}\frac{2g(t)}{(1+p|g(t)|^p)^{1/p}}\Big)^{1/2}\\
&=\Big(\frac{2}{p^{1/p}}\Big)^{1/2}=K_0.
\end{align*}
Then (7) is proved by (4).
It is easy to get (8) by (4).

We can get (9) from (4). Inequalities in (10) are trivial and (11)
is from (4) and (8).\\
For (12), it is easy to see
\[
g''(t)=-p\big(1+p|g(t)|^p\big)^{-\frac{1}{p}-1}|g(t)|^{p-2}g(t)g'(t).
\]
So the conclusion of (12) is true.
\end{proof}

We assume the following conditions on $f$:
\begin{itemize}
\item[(F1)] $|f(x,t)|\leq C(1+|t|^{2q-1})$ holds for some positive 
constant $C$, all $x\in\Omega$ and $t\in\mathbb{R}$, where 
$1\leq q < p^{*}:=\frac{Np}{N-p}$.
\end{itemize}
Under condition (F1), consider the  functional 
\begin{equation}
\Phi(v):=\frac{1}{p}\int_{\Omega}|\nabla v|^p\,
\mathrm{d}x-\int_{\Omega}F(x,g(v))\, \mathrm{d}x.
\end{equation}
Then $\Phi$ is well defined on the space $W_0^{1,p}(\Omega)$ 
(equipped with the norm $\|v\|:=(\int_\Omega|\nabla v|^p\, \mathrm{d}x)^{1/p}$), 
and $\Phi\in C^1(W_0^{1,p}(\Omega);\mathbb{R})$ by assumption (F1) and
Lemma \ref{g}. Thus for all $w\in W_0^{1,p}(\Omega)$, we have
\[
\langle\Phi'(v),w\rangle=\int_{\Omega}|\nabla v|^{p-2}\nabla v\nabla
w\, \mathrm{d}x-\int_\Omega f(x,g(v))g'(v)w\, \mathrm{d}x,
\]
where $\langle\cdot,\cdot\rangle$ is the duality pairing between
$W_0^{1,p}(\Omega)$ and $(W_0^{1,p}(\Omega))^*$. Then the critical
points of $\Phi$ are weak solutions (in the usual sense) for the
problem
\begin{equation}\label{Real}
\begin{gathered}
-\Delta_p v=f(x,g(v))g'(v), \quad \text{ in }\Omega,\\
v=0, \quad\text{on }\partial\Omega.
\end{gathered}
\end{equation}
By setting $v=g^{-1}(u)$, it is easy to see that equation \eqref{Real}
is equivalent to our problem \eqref{(P)}, which takes $u=g(v)$ as its solution.

Motivated by the above, we give the following definition of the weak
solution for problem \eqref{(P)}.

\begin{definition} \label{weak_solution} \rm
We say $u$ is a weak solution for \eqref{(P)}, 
if $v=g^{-1}(u)\in W_0^{1,p}(\Omega)$ is a critical
point of the following functional corresponding to problem
\eqref{Real}:
\[
\Phi(v):=\frac{1}{p}\int_{\Omega}|\nabla v|^p\,
\mathrm{d}x-\int_{\Omega}F(x,g(v))\, \mathrm{d}x.
\]
\end{definition}

\section{Existence of weak solutions}\label{1.1}

For simplicity, we make a use of the following notation. $X$
denotes Sobolev space $W_{0}^{1,p}(\Omega)$ with the norm
$\|\cdot\|:=(\int_\Omega|\nabla\cdot|^p\,
\mathrm{d}x)^{1/p}$; $X^{*}$ denotes the conjugate space for
$X$; $L^{p}(\Omega)$ denotes Lebesgue space with the norm
$|\cdot|_{p}$; $\langle\cdot,\cdot\rangle$ is the dual pairing on
the space $X^{*}$ and $X$; by $\to $ (resp.
$\rightharpoonup$) we mean strong (resp. weak) convergence.
$|\Omega|$ denotes the Lebesgue measure of the set
$\Omega\subset\mathbb{R}^{N}$; $C,C_1,C_{2},\dots$ denote
(possibly different) positive constants.

Let $\varphi(v):=\frac{1}{p}\int_{\Omega}|\nabla v|^p\,\mathrm{d} x$ for all
$v\in X$. It is obvious that the functional $\varphi$ is a
continuously G\^{a}teaux differentiable whose G\^{a}teaux derivative
at the point $v\in X$ is the functional $\varphi'(u)\in X^*$, given
by
\[
\langle\varphi'(v),u\rangle=\int_\Omega |\nabla v|^{p-2}\nabla
v\nabla u \,\mathrm{d} x;
\]
Let $\mathcal{F}(v)=\int_\Omega F(x,g(v))\,\mathrm{d} x$. Then by the notation
of section \ref{1}, $\Phi(v)=\varphi(v)-\mathcal{F}(v)$.
It is well known that the following lemma holds for the functional
$\varphi$.

\begin{lemma}\label{(S+)}
\begin{itemize}
\item[(i)] $\varphi':X\to  X^*$ is a continuous and strictly
monotone operator;
\item[(ii)] $\varphi'$ is a mapping of type $(S_+)$, i.e. if $v_n\rightharpoonup
v$ in $X$ and
$\limsup_{n\to +\infty}\langle\varphi'(v_n)-\varphi'(v),v_n-v\rangle\leq0$,
then $v_n\to  v$ in $X$;
\item[(iii)] $\varphi'(v):X\to  X^*$ is a homeomorphism;
\item[(iv)] $\varphi$ is weakly lower semicontinuous.
\end{itemize}
\end{lemma}

If $f$ is independent of $u$, we have the following result.

\begin{theorem} \label{thm3.1}
If $f(x,u)=f(x), f\in L^{r}(\Omega)$ in which
$\frac{1}{r}+\frac{1}{p^*}<1$, then \eqref{(P)} has a unique weak
solution.
\end{theorem}

\begin{proof} 
It is clear that $(f,u):=\int_\Omega f(x)u\,\mathrm{d} x,\quad\forall u\in X$
defines a continuous linear functional on X. By Lemma \ref{(S+)}
(iii), \eqref{(P)} has a unique weak solution. 
\end{proof} 

Next we assume the following conditions on $f$,
\begin{itemize}
\item[(F2)] There exists $p^{*}>\theta>p$, $M>0$ such that $|t|\geq M$ implies
\[
0<\theta F(x,t)\leq \frac{1}{2}tf(x,t). 
\]
\item[(F3)] $f(x,t)=o(|t|^{p-1}), t\to 0$, for $x\in\Omega$
uniformly.
\item[(F4)] $f(x,-t)=-f(x,t), x\in\Omega,t\in\mathbb{R}$.
\end{itemize}

\begin{lemma}\label{weak_strong_contionuous}
Under assumption {\rm (F1)},
\begin{itemize}
\item[(i)]the functional
$\mathcal{F}$ is sequentially weak-strong continuous, i.e.,
$v_n\rightharpoonup v$ in $X$ implies
$\mathcal{F}(v_n)\to \mathcal{F}(v)$;
\item[(ii)]
$\mathcal{F}'(v_n)\to \mathcal{F}(v)$ in $X^*$ as
$v_n\rightharpoonup v$ in $X$.
\end{itemize}
\end{lemma}

\begin{proof} (i) By (F1) and Lemma \ref{g}, we have
\[
|F(x,g(t))|=\int_{0}^{g(t)}|f(x,s)|\,\mathrm{d} s
\leq C(1+|g(t)|^{2q})
\leq C(1+|t|^q).
\]
Then the Caratheodory mapping
$F(x,g(\cdot)):L^{q}(\Omega)\to  L^1(\Omega)$ is continuous.
Since $v_n\rightharpoonup v$ in $X$, by the Sobolev compact
imbedding, it is east to see $v_n\to  v$ in $L^q(\Omega)$.
Then $F(x,v_n(x))\to F(x,v(x))$ in $L^1(\Omega)$, which means
$\mathcal{F}(v_n)\to \mathcal{F}(v)$.

(ii) By (F1) and Lemma \ref{g}, we have
\[
 |f(x,g(t))g'(t)|\leq C(1+|g(t)|^{2q-1}|g'(t)|)
 \leq C(1+|g(t)|^{2q-2})
 \leq C(1+|t|^{q-1}).
\]
Hence, the mapping $L^q(\Omega)\to  L^{q'}(\Omega)$:
$v\mapsto f(x,g(v))g'(v)$ is continuous. Then it is easy to see that
$\mathcal{F}\in C^1(X)$ and $\mathcal{F}':X\to  X^*$ defined
by
\[
\langle\mathcal{F}'(v),u\rangle=\langle\mathcal{F}'(v),u\rangle_{L^{q'},L^{q}}
=\int_{\Omega}f(x,g(v(x)))g'(v(x))u(x)\,\mathrm{d} x,
\]
for all $v,u\in X\subset L^{q}(\Omega)$,
is completely continuous. In fact, we have the following
decomposition for the operator
\[
\mathcal{F}':X\xrightarrow{i}
L^{\Psi}(\Omega)\xrightarrow[]{f(x,g(\cdot))g'(\cdot)}
L^{\Psi'}(\Omega)\xrightarrow{j}(L^\Psi(\Omega))^*\xrightarrow{k}
X^*,
\]
i.e.,
\[
\mathcal{F}'(u)=k\circ j\circ f\circ i(u),\quad\forall u\in X,
\]
in which, $i$ is compact, $j$ is homeomorphic, and $k$ means
restriction on $X^*$ of functionals in $(L^\Psi(\Omega))^*$. Then it
is clear that $\mathcal{F}'$ is completely continuous.
\end{proof}

\begin{remark} \rm
Under assumption (F1), by lemma \ref{weak_strong_contionuous} and
Lemma \ref{(S+)}, we know that $\Phi'=\varphi'-\mathcal{F}'$ is of
type $(S_+)$.
\end{remark}


\begin{theorem} \label{CE} 
If {\rm (F1)} holds and $q<p$, then  \eqref{(P)} has
a weak solution. 
\end{theorem}

\begin{proof} By (F1), we have the estimate
\[
|F(x,t)|\leq C(1+|t|^{2q}).
\]
Then $\Phi$ is coercive because of the  inequality
\[
\begin{aligned}
\Phi(v)&=\frac{1}{p}\int_{\Omega}|\nabla v|^p\,
\mathrm{d}x-\int_{\Omega}F(x,g(v))\, \mathrm{d}x\\
&\geq \frac{1}{p}\|v\|^p-C\int_{\Omega}|g(v)|^{2q}\, \mathrm{d}x-C\\
&\geq\frac{1}{p}\|v\|^p-C\int_{\Omega}|v|^{q}\, \mathrm{d}x-C\\
&\geq\frac{1}{p}\|v\|^p-C|v|_{q}^{q}-C\\
&\geq\frac{1}{p}\|v\|^p-C\|v\|^{q}-C\to +\infty, \quad
\text{as }\|v\|\to +\infty.
\end{aligned}
\]
By Lemma \ref{(S+)} and Lemma \ref{weak_strong_contionuous}, it is
easy to verify that $\Phi$ is weakly lower semicontinuous. Then
$\Phi$ has a minimum point $v$ in $X$ and $v$ is a weak solution of
\eqref{(P)}, which completes the proof.
\end{proof}

\begin{lemma} \label{PS} 
Under assumptions {\rm (F1)} and {\rm (F2)}, the functional $\Phi$
satisfies the (PS) condition. 
\end{lemma}

\begin{proof} 
Suppose that $\{v_{n}\}\subset X$, $|\Phi(v_{n})|\leq B$ for
some $B\in\mathbb{R}$, and $\Phi'(v_{n})\to  0$  in
$X^{*}$  as $n\to \infty$.

After integrating, we obtain from the assumption (F2) that there
exists $C_1$ such that 
\begin{equation} \label{F31}
 C_1(|t|^{2\theta}-1)\leq F(x,t)\quad \forall x\in \Omega,t\in\mathbb{R}.
\end{equation}
Let $c:=\sup_{n}\Phi(v_{n})$ and
$\beta\in(\frac{1}{\theta},\frac{1}{p})$ for large $n$. From Lemma
\ref{g} (4) and (10), (F6) and the inequality \eqref{F31} we have
\begin{align*}
&c+1+\|v_{n}\|\\
&\geq
\Phi(v_{n})-\beta\langle\Phi'(v_{n}),v_{n}\rangle\\
&=\frac{1}{p}\|v_{n}\|^{p}-\beta\|v_{n}\|^{p}+\int_{\Omega}\big(\beta
f(x,g(v_{n}))g'(v_{n})v_n-F(x,g(v_{n}))\big)\, \mathrm{d}x\\
&\geq
\frac{1}{p}\|v_{n}\|^{p}-\beta\|v_{n}\|^{p}+\int_{\Omega}\Big(
\frac{1}{2}\beta f(x,g(v_{n}))g(v_{n})-F(x,g(v_{n}))\Big)\, \mathrm{d}x\\
&\geq\Big(\frac{1}{p}-\beta
\Big)\|v_{n}\|^{p}+(\theta\beta-1)\int_{\Omega}F(x,g(v_{n}))\, \mathrm{d}x\\
&\geq\Big(\frac{1}{p}-\beta
\Big)\|v_{n}\|^{p}+C_1(\theta\beta-1)\int_{\Omega}|g(v_{n})|^{2\theta}\, \mathrm{d}x-C_{3}\\
&\geq
\Big(\frac{1}{p}-\beta\Big)\|v_{n}\|^{p}+C_2(\theta\beta-1)|v_{n}|_{\theta}^{\theta}-C_{3},\\
&\geq\Big(\frac{1}{p}-\beta\Big)\|v_{n}\|^{p}-C_{3},
\end{align*}
noticing that $\frac{1}{p}-\beta>0$, and $\theta\beta-1>0$, we
obtain the boundedness of $\{v_{n}\}$ in $X$. Without of loss of
generality, we assume $v_n\rightharpoonup v$, then
$\langle\Phi'(v_n)-\Phi'(v),v_n-v\rangle\to 0$. Since $\Phi'$
is of type $(S_+)$, we have $v_n\to  v$ in $X$. \end{proof}

\begin{theorem} 
Under assumption {\rm (F1), (F2), (F3)} and $q>p$, problem \eqref{(P)}
has a nontrivial solution. 
\end{theorem}

\begin{proof}
We will show that the functional $\Phi$ satisfies the Mountain
Pass Theorem. By Lemma \ref{PS}, $\Phi$ satisfies (PS) condition
in $X$. Since $p<q<p^*$, $X\subset L^p(\Omega)$; i.e., there exists
a $C>0$ such that
\[
|v|_p\leq C\|v\|, \quad\forall v\in X.
\]
By assumption (F3) and Lemma \ref{g}, for small $\epsilon>0$,  we have
\[
F(x,g(t))\leq\epsilon|g(t)|^p+C|g(t)|^{2q-1}\leq\epsilon|t|^p+C|t|^q,
\quad\forall (x,t)\in\Omega\times\mathbb{R}.
\]
So we have
\begin{align*}
\Phi(v)&\geq\frac{1}{p}\int_\Omega|\nabla v|^p\,\mathrm{d}
x-\epsilon\int_\Omega|v|^p\,\mathrm{d}
x-C\int_\Omega|v|^q\,\mathrm{d} x\\
&\geq\frac{1}{p}\|v\|^p-C\|v\|^p-C\|v\|^q\\
&\geq\frac{1}{2p}\|v\|^p-C\|v\|^q, \text{ when }\|v\|\leq 1.
\end{align*}
So there exist $r>0$ and $\delta>0$ such that $\Phi(v)\geq\delta>0$
for every $\|v\|=r$.

From the assumption (F2) and Lemma \ref{g}, there exists a constant
$C_1>0$ such that
\[
F(x,g(t))\geq
C_1|g(t)|^{2\theta}\geq C_2|t|^\theta,\quad \text{for }|t|\geq M.
\]
For $w\in X\backslash\{0\}$ and $t>1$, in view of the above in
equality, we have
\begin{align*}
\Phi(tw)&=\frac{1}{p}\int_\Omega|t\nabla w|^p\,\mathrm{d} x-\int_\Omega
F(x,tw)\,\mathrm{d} x\\
&\leq Ct^p\|w\|^p-C\int_\Omega|tw|^\theta\,\mathrm{d} x-C\\
&\leq Ct^p\|w\|^p-Ct^\theta|w|_{\theta}^\theta-C
\to -\infty, \quad \text{as }t\to +\infty.
\end{align*}
Obviously we have $\Phi(0)=0$, so $\Phi$ satisfies the geometry
conditions of the Mountain Pass Theorem in \cite{Willem}. Then
$\Phi$ admits at least one nontrivial critical which corresponds to
the weak solution of \eqref{(P)}.
\end{proof}

Thanks to  Lemma \ref{g}, the translation $g$ is strictly
increasing and $g$ is odd, which means that the functional
$\mathcal{F}$ is even. This allows us to make an application of
Fountain theorem and Dual Fountain theorem to obtain infinitely many
solutions to \eqref{(P)}.

\begin{theorem} \label{Fountain} 
Let {\rm (F1), (F2), (F4)} hold and $p^*>q>p$, then
\eqref{(P)} has a sequence of weak solutions 
$\{\pm u_k\}_{k=1}^{\infty}$ such that $\Phi(\pm u_k)\to +\infty$ as
$k\to +\infty$. 
\end{theorem}

We will use the fountain theorem to prove Theorem \ref{Fountain}. Since
$X$ is a reflexive and separable Banach space, there exist
$\{e_j\}\subset X$ and $\{e^*_j\}\subset X^*$ such that
\[
X=\overline{\operatorname{span}\{e_j:j=1,2,\dots \}},\quad
X^*=\overline{\operatorname{span}\{e^*_j:j=1,2,\dots \}}
\]
in which
\[
\langle e_i,e_j^*\rangle=\begin{cases}
1,&i=j,\\
0,&i\neq j,
\end{cases}
\]
We will write $X_j=\operatorname{span}\{e_j\}$, $Y_k=\oplus_{j=1}^{k}
X_j$, $X_k=\overline{\oplus_{j=k}^{\infty} X_j}$.

\begin{lemma}[\cite{Willem}] 
Let $q<p^*$, and denote
\[
\beta_k=\sup\{|v|_q:\|v\|=1,v\in Z_k\}.
\]
Then $\lim_{k\to +\infty}\beta_k=0$.
\end{lemma}
Next, we have the Fountain Theorem, see \cite{Willem}.

\begin{lemma} \label{FT}
 Assume
\begin{itemize}
\item[(A1)] $X$ is a Banach space, $\Phi\in C^1(X,\mathbb{R})$ is an
even functional.
\end{itemize}
For each $k\in\mathbb{N}$, there exist $\rho_k>r_k>0$ such that
\begin{itemize}
\item[(A2)] $\inf_{v\in Z_k,\|v\|=r_k}\Phi(v)\to +\infty$ as $k\to +\infty$.
\item[(A3)]$\max_{v\in Y_k,\|v\|=\rho_k}\Phi(v)\leq0$
\item[(A4)] $\Phi$ satisfies $(PS)_c$ condition for every $c>0$.
\end{itemize}
Then $\Phi$ admits a sequence of critical values tending to
$+\infty$. 
\end{lemma}

\begin{proof}[Proof of Theorem \ref{Fountain}]
By  assumption (F4) and since the translation of $g$ 
defined in section \ref{1} is odd and increasing, $\mathcal{F}$ is
even, which implies $\Phi=\varphi-\mathcal{F}$ is also even. Further
more, by Lemma \ref{PS}, $\Phi$ satisfies the $(PS)_c$ condition.
We need only to prove that there exist $\rho_k>r_k>0$ such that
condition (A2) and (A3) in Lemma \ref{FT} hold.

(A2) Let $v\in Z_k$, $\|v\|=r_k:=(C_1qK_0^{2q}\beta_k^q)^{1/(p-q)}$, 
in which $K_0$
is the same one in Lemma \ref{g}. By (F1) and Lemma \ref{g}, we have
\begin{align*}
\Phi(v)&=\frac{1}{p}\int_\Omega|\nabla v|^p\,\mathrm{d} x-\int_\Omega
F(x,g(v))\,\mathrm{d} x\\
&\geq\frac{1}{p}\|v\|^p-C_1\int_\Omega |g(v)|^{2q} \,\mathrm{d} x-C_2\\
&\geq\frac{1}{p}\|v\|^p-C_1K_0^{2q}\int_\Omega |v|^{q} \,\mathrm{d} x-C_2\\
&\geq\frac{1}{p}\|v\|^p-C_1K_0^{2q}\beta_k^q\|v\|^{q} \,\mathrm{d} x-C_2\\
&=\frac{1}{p}(C_1qK_0^{2q}\beta_k^q)^{\frac{p}{p-q}}-C_1K_0^{2q}
\beta_k^q(C_1qK_0^{2q}\beta_k^q)^{\frac{q}{p-q}}-C_2\\
&=\Big(\frac{1}{p}-\frac{1}{q}\Big)(C_1qK_0^{2q}\beta_k^q)^{\frac{p}{p-q}}-C_2
\to +\infty, \quad \text{as }k\to +\infty
\end{align*}
since $p^*>q>p$ and $\beta_k\to 0$.

(A3) From  assumption (F2) and Lemma \ref{g}, there exists a
constant $C_1>0$ such that
\[
F(x,g(t))\geq C_1|g(t)|^{2\theta}\geq C_2|t|^\theta,\quad
\text{for }|t|\geq M.
\]
For any $w\in Y_k$ with $\|w\|=1$ and $\rho_k=t>1$, we have
\begin{align*}
\Phi(tw)&=\frac{1}{p}\int_\Omega|t\nabla w|^p\,\mathrm{d} x-\int_\Omega
F(x,g(tw))\,\mathrm{d} x\\
&\leq\frac{1}{p}\|tw\|^p-C\int_\Omega |tw|^{\theta} \,\mathrm{d} x+C\\
&\leq\frac{t^p}{p}-Ct^\theta|w|_\theta^\theta+C.
\end{align*}
Since all norms in a finite dimensional space $Y_k$ are equivalent, we
have $\Phi(tw)\to -\infty$ by $\theta>p$.
The conclusion of Theorem \ref{Fountain} is obtained by Lemma
\ref{FT}.
\end{proof}

Also by the fine properties of the $g$, we give the solution
existence result for that the nonlinear term is ``concave and convex
nonlinearities'' by Dual Fountain Theorem. More precisely we have
the following theorem.

\begin{theorem} \label{Dual} 
Assume $\gamma, \beta>0$ such that $p<\gamma<p^*$,
$\beta<p$ and $f(x,t)=\lambda|t|^{2\gamma-2}t+\delta|t|^{2\beta-2}t$. Then
\begin{itemize}
\item[(i)] for every $\lambda>0$, $\delta\in\mathbb{R}$, \eqref{(P)}
has a sequence of weak solutions $\{v_k\}_{k=1}^{+\infty}$, such
that $\Phi(\pm v_k)\to +\infty$ as $k\to +\infty$;

\item[(ii)]for every $\delta>0$, $\lambda\in\mathbb{R}$, \eqref{(P)}
has a sequence of weak solutions $\{w_k\}_{k=1}^{+\infty}$, such
that $\Phi(\pm w_k)\to 0$ as $k\to +\infty$.
\end{itemize}
\end{theorem}

To prove Theorem \ref{Dual}, we will need the following 
 ``Dual Fountain Theorem'', see \cite{Willem}.

\begin{lemma} 
Assume {\rm (A1)} is satisfied, and there is a $k_0>0$ such that for 
each $k\geq k_0$, there exist $\rho_k>r_k>0$ such that
\begin{itemize}
\item[(B1)] $\inf_{v\in Z_k,\|v\|=\rho_k}\Phi(v)\geq0$.
\item[(B2)] $b_k:=\max_{v\in Y_k,\|v\|=r_k}\Phi(v)<0$.
\item[(B3)] $d_k:=\inf_{v\in Z_k,\|v\|\leq \rho_k}\Phi(v)\to 0$ as 
  $k\to +\infty$.
\item[(B4)] $\Phi$ satisfies the $(PS)_c^*$ condition for every 
$c\in[d_{k_0},0)$.
\end{itemize}
Then $\Phi$ has a sequence of negative critical values converging to
$0$. 
\end{lemma}

\begin{definition} \rm
We say that $\Phi$ satisfies the $(PS)_c^*$ condition with
respect to $\{Y_n\}_{n=1}^{\infty}$, if any sequence
$\{v_{n_j}\}\subset X$ such that $n_j\to +\infty,v_{n_j}\in
Y_{n_j}$, $\Phi(v_{n_j})\to  c$ and
$\Phi|'_{Y_{n_j}}(v_{n_j})\to 0$, contains a subsequence
converging to a critical point of $\Phi$. 
\end{definition}

\begin{proof}[Proof of Theorem \ref{Dual}]
The proof of this part (i) is similar to
that of Theorem \ref{Fountain}, if we specify
$f(x,t):=\lambda|t|^{2\gamma-2}t+\delta|t|^{2\beta-2}t$ and
$F(x,t):=\frac{\lambda}{2\gamma}|t|^{2\gamma}+\frac{\delta}{2\beta}|t|^{2\beta}$.
We only verify the (PS) condition here. Suppose
\[
\{v_n\}\subset X, \quad |\Phi(v_n)|\leq C,\quad
\Phi'(v_n)\to 0\quad\text{as }n\to +\infty.
\]
for $\|v\|>1$ and large $n$, by Lemma \ref{g}, we have
\begin{align*}
\quad c+1+\|v_{n}\|
&\geq \Phi(v_{n})-\frac{1}{\gamma}\langle\Phi'(v_{n}),v_{n}\rangle\\
&=\frac{1}{p}\|v_{n}\|^{p}-\frac{1}{\gamma}\|v_{n}\|^{p}+\int_{\Omega}\big(\frac{1}{\gamma}
f(x,g(v_{n}))g'(v_{n})v_n-F(x,g(v_{n}))\big)\, \mathrm{d}x\\
&\geq
\frac{1}{p}\|v_{n}\|^{p}-\frac{1}{\gamma}\|v_{n}\|^{p}+\int_{\Omega}\Big(
\frac{1}{2}\frac{1}{\gamma} f(x,g(v_{n}))g(v_{n})-F(x,g(v_{n}))\Big)\, \mathrm{d}x\\
&\geq\Big(\frac{1}{p}-\frac{1}{\gamma}
\Big)\|v_{n}\|^{p}+\frac{\delta}{2}(\frac{1}{\gamma}-\frac{1}{\beta})\int_{\Omega}|g(v_{n})|^{2\beta}\, \mathrm{d}x\\
&\geq
\Big(\frac{1}{p}-\frac{1}{\gamma}\Big)\|v_{n}\|^{p}-C_1\int_\Omega|v_{n}|^{\beta}\,\mathrm{d} x,\\
&\geq
\Big(\frac{1}{p}-\frac{1}{\gamma}\Big)\|v_{n}\|^{p}-C_1|v_{n}|_{\beta}^{\beta},\\
&\geq\Big(\frac{1}{p}-\frac{1}{\gamma}\Big)\|v_{n}\|^{p}-C_2\|v_n\|^\beta,
\end{align*}
since $\gamma>p>\beta$, we know that $\{v_n\}$ is bounded in $X$.

(ii) From the odd and increasing properties of the function $g$ in
Lemma \ref{g}, we know the functional $\Phi$ is even, i.e. (A1) is
satisfied.

 To verify (B1), we define
\[
\beta_k:=\sup\{|v|_\beta:\|v\|=1,v\in Z_k\}
\]
For any $v\in Z_k$, $\|v\|=1$ and $0<t<1$, we have
\begin{equation}\label{kkkkkkk}
\begin{aligned}
\Phi(tv)&=\frac{t^p}{p}\|v\|^p-\int_\Omega F(x,g(tv))\,\mathrm{d} x\\
&\geq\frac{t^p}{p}\|v\|^p-\frac{|\lambda|}{2\gamma}\int_\Omega|g(tv)|^{2\gamma}\,\mathrm{d}
x-\frac{\delta}{2\beta}\int_\Omega|g(tv)|^{2\beta}\,\mathrm{d} x \\
&\geq\frac{t^p}{p}-K_0^{2\gamma}\frac{|\lambda|}{2\gamma}\int_\Omega|tv|^{\gamma}\,\mathrm{d}
x-K_0^{2\beta}\frac{\delta}{2\beta}\int_\Omega|tv|^{\beta}\,\mathrm{d} x \\
&\geq\frac{t^p}{p}-CK_0^{2\gamma}\frac{|\lambda|}{2\gamma}t^\gamma\|v\|^{\gamma}
-K_0^{2\beta}\beta_k^\beta\frac{\delta}{2\beta}t^\beta\|v\|^{\beta}\\
&\geq\frac{t^p}{p}-CK_0^{2\gamma}\frac{|\lambda|}{2\gamma}t^\gamma
-K_0^{2\beta}\beta_k^\beta\frac{\delta}{2\beta}t^\beta,
\end{aligned}
\end{equation}
for big $k$ such that $\beta_k\leq1$,
where $C$ is the Sobolev constant.
 Since $\gamma>p$, there exists
a $0<\rho_1<1$ such that $\frac{\rho_1^p}{2p}\geq
CK_0^{2\gamma}\frac{|\lambda|}{2\gamma}\rho_1^\gamma$. Let
$0<t\leq\rho_1$. From the inequality \eqref{kkkkkkk}, we have
\begin{equation}\label{kkk}
\Phi(tv)\geq\frac{t^p}{2p}-K_0^{2\beta}\beta_k^\beta\frac{\delta}{2\beta}t^\beta.
\end{equation}

Let $t=\rho_k=\big(\frac{p\delta
K_0^{2\beta}\beta_k^\beta}{\beta}\big)^{\frac{1}{p-\beta}}$ for big
$k$ such that $\rho_k\leq\rho_1$ and $\beta_k\leq1$, we have
$\Phi(tv)\geq0$, i.e. for big $k\in\mathbb{N}$,
\[
\inf_{v\in Z_k,\|v\|=\rho_k}\Phi(tv)\geq0,
\]
which implies (B1) holds.

(B2) For $v\in Y_k$ such that $\|v\|\leq 1$, by Lemma \ref{g} we
have
\[
\begin{aligned}
\Phi(v)&=\frac{1}{p}\|v\|^p-\int_\Omega F(x,g(v))\,\mathrm{d} x\\
&\leq\frac{1}{p}\|v\|^p+\frac{|\lambda|}{2\gamma}\int_\Omega|g(v)|^{2\gamma}\,\mathrm{d}
x-\frac{\delta}{2\beta}\int_\Omega|g(v)|^{2\beta}\,\mathrm{d} x\\
&\leq\frac{1}{p}\|v\|^p+\frac{|\lambda|}{2\gamma}K_0\int_\Omega|v|^{\gamma}\,\mathrm{d}
x-\frac{\delta}{2\beta}C\int_\Omega|v|^{\beta}\,\mathrm{d} x.
\end{aligned}
\]
Then $\operatorname{dim}Y_k<\infty$, $\beta<p$ and $\gamma>p$ imply that there
exists a $0<r_k<\rho_k$ small enough such that $\Phi(v)<0$ for
$\|u\|=r_k$, i.e.,
\[
b_k:=\max_{v\in Y_k,\|v\|=r_k}\Phi(v)<0,
\]
which implies (B2).

(B3) Since $Y_k\cup Z_k\neq \emptyset$, and $r_k<\rho_k$, we have
\[
d_k:=\inf_{v\in Z_k,\|u\|\leq \rho_k}\Phi(v)\leq
b_k=\max_{v\in Y_k,\|v\|=r_k}\Phi(v)<0.
\]
By \eqref{kkk}, for $v\in Z_k$, $\|v\|=1$, $0\leq t\leq\rho_k$, we
have
\[
\Phi(tv)\geq\frac{t^p}{2p}-K_0^{2\beta}\beta_k^\beta
\frac{\delta}{2\beta}t^\beta
\geq -K_0^{2\beta}\beta_k^\beta\frac{\delta}{2\beta}t^\beta\\
\to 0,\quad\text{as }  k\to +\infty,
\]
which implies that $d_k\to 0$, i.e., (B3) holds.

Finally we verify the $(PS)_c^*$ condition. Consider a sequence
$v_{n_j}\in Y_{n_j}$ such that
\[
\Phi(v_{n_j})\to  c,\quad
\Phi|'_{Y_{n_j}}(v_{n_j})\to 0\quad \text{in } X^* \text{ as }n_j\to \infty.
\]
If $\lambda>0$, as in the proof Lemma \ref{PS}, it is easy to
get the boundedness of $\|v_{n_j}\|$. If $\lambda<0$, for $\|v\|>1$
and large $n$, by Lemma \ref{g}, we have
\begin{align*}
&c+1+\|v_{n_j}\|\\
&\geq \Phi(v_{n_j})-\frac{1}{\gamma}\langle\Phi'(v_{n_j}),v_{n_j}\rangle\\
&=\frac{1}{p}\|v_{n_j}\|^{p}-\frac{1}{\gamma}\|v_{n_j}\|^{p}+\int_{\Omega}\big(\frac{1}{\gamma}
f(x,g(v_{n_j}))g'(v_{n_j})v_{n_j}-F(x,g(v_{n_j}))\big)\, \mathrm{d}x\\
&\geq
\frac{1}{p}\|v_{n_j}\|^{p}-\frac{1}{\gamma}\|v_{n_j}\|^{p}+\int_{\Omega}\Big(
\frac{1}{2}\frac{1}{\gamma} f(x,g(v_{n_j}))g(v_{n_j})-F(x,g(v_{n_j}))\Big)\, \mathrm{d}x\\
&\geq\Big(\frac{1}{p}-\frac{1}{\gamma}
\Big)\|v_{n_j}\|^{p}+\frac{\delta}{2}(\frac{1}{\gamma}-\frac{1}{\beta})\int_{\Omega}|g(v_{n_j})|^{2\beta}\, \mathrm{d}x\\
&\geq
\Big(\frac{1}{p}-\frac{1}{\gamma}\Big)\|v_{n_j}\|^{p}-C_1\int_\Omega|v_{n_j}|^{\beta}\,\mathrm{d} x,\\
&\geq
\Big(\frac{1}{p}-\frac{1}{\gamma}\Big)\|v_{n_j}\|^{p}-C_1|v_{n_j}|_{\beta}^{\beta},\\
&\geq\Big(\frac{1}{p}-\frac{1}{\gamma}\Big)\|v_{n_j}\|^{p}-C_2\|v_{n_j}\|^\beta.
\end{align*}
Since $\gamma>p>\beta$, we can see $\{v_{n_j}\}$ is bounded in $X$.
We can select, if necessary, a subsequence, we assume
$v_{n_j}\rightharpoonup v$ in $X$. As
$X=\overline{\cup_{n_j}Y_{n_j}}$, we can choose $w_{n_j}\in Y_{n_j}$
such that $w_{n_j}\to  u$. Hence
\begin{align*}
\lim_{n_j\to \infty}\langle\Phi'(v_{n_j}),v_{n_j}-v\rangle
&= \lim_{n_j\to \infty}\langle\Phi'(v_{n_j}),v_{n_j}-w_{n_j}\rangle
+\lim_{n_j\to \infty}\langle\Phi'(v_{n_j}),w_{n_j}-u\rangle\\
&=\lim_{n_j\to \infty}\langle\Phi|_{Y_{n_j}}'(v_{n_j}),v_{n_j}
-w_{n_j}\rangle
=0.
\end{align*}
Sice $\Phi'$ is of type $(S_+)$, we get $v_{n_j}\to  v$,
which implies $\Phi'{v_{n_j}}\to  \Phi'(v)$.

The last step is to verify $\Phi'(v)=0$. For any $u_k\in Y_k$, when
$n_j\geq k$ we have
\begin{align*}
\langle\Phi'(v),u_{k}\rangle&=
\langle\Phi'(v)-\Phi'(v_{n_j}),u_{k}\rangle
+\langle\Phi'(v_{n_j}),u_{k}\rangle\\
&=\langle\Phi'(v)-\Phi'(v_{n_j}),u_{k}\rangle
+\langle\Phi|'_{Y_{n_j}}(v_{n_j}),u_{k}\rangle.
\end{align*}
Going to the limit in the right side of above equation we get
\[
\langle\Phi'(v),u_{k}\rangle=0,\quad \forall u_k\in Y_k,
\]
which means $\Phi'(v)=0$. Thus $\Phi$ satisfies the $(PS)_c^*$
condition.
\end{proof}

When $p=2$, we can have the corresponding theorems in this paper for
the existence of solutions to the following equation for more
physical meanings as we mentioned in section \ref{00}:
\begin{equation}\label{p_2}
\begin{gathered}
        -\Delta u-u\Delta(u^2)=f(x,u),  \quad  \text{in }  \Omega, \\
         u=0, \quad \text{on } \partial\Omega.
\end{gathered}
\end{equation}
In fact, in the most literature such as in \cite{Liu_Wang_Wang,
Liu1, Liu2, Coli, Fang}, the authors consider problem \eqref{p_2} in
$\mathbb{R}^{N}$, the technic they used there include Nehari method,
Mountain Pass theorem and some other topological Mini-Max methods.
In the case for $p=2$, the existence results in this paper are
problems considered in a bounded domain in $\mathbb{R}^{N}$. We
close this section by pointing out that, recently, in an interesting
paper \cite{LiuXiangQing}, the authors developed the existence of a
positive solution by mountain pass theorem, and the existence of a
sequence solutions by symmetric mountain pass theorem under similar
odd condition (F4). The method they used there is an approximation
of the original functional, but without changing of variables.

\subsection*{Acknowledgments}
The authors would like to thank the referee for a careful reading of
an earlier version of the paper and valuable suggestions.

This research was supported by the National Natural Science Foundation of China
(NSFC 10971088 and NSFC 10971087), and by the Fundamental Research
Funds for the Central Universities.


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