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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 78, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2016/78\hfil $p$-Laplacian problems in bounded domains]
{Multiple solutions for $p$-Laplacian problems involving general
 subcritical growth in bounded domains}

\author[N. T. Chung, P. H. Minh, T. H. Nga \hfil EJDE-2016/78\hfilneg]
{Nguyen Thanh Chung, Pham Hong Minh, Tran Hong Nga}

\address{Nguyen Thanh Chung \newline
Department of Mathematics, Quang Binh University,
312 Ly Thuong Kiet, Dong Hoi,
Quang Binh, Viet Nam}
\email{ntchung82@yahoo.com}

\address{Pham Hong Minh \newline
Department of Mathematics, Quang Binh University,
312 Ly Thuong Kiet, Dong Hoi,
Quang Binh, Viet Nam}
\email{phamhongminh24@gmail.com}

\address{Tran Hong Nga \newline
Department of Mathematics, Quang Binh University,
312 Ly Thuong Kiet, Dong Hoi,
Quang Binh, Viet Nam}
\email{tranhongnga0209@gmail.com}

\thanks{Submitted February 26, 2016. Published March 18, 2016.}
\subjclass[2010]{35D05, 35J60}
\keywords{$p$-Laplacian problems; general subcritical growth;
\hfill\break\indent  concave-convex nonlinearities; variational method}

\begin{abstract}
 Using variational methods, we study the existence of multiple solutions for a
 class of $p$-Laplacian problems with concave-convex nonlinearities in
 bounded domains. Our result improves those in \cite{YYLan,LaTa} stated only
 for subcritical growth.
\end{abstract}

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

\section{Introduction}

In this article, we are interested in the existence of solutions
for $p$-Laplacian problems of the form
\begin{equation}\label{e1.1}
\begin{gathered}
- \Delta_p u = g(x,u), \quad x \in \Omega,\\
 u=0, \quad x \in \partial\Omega,
\end{gathered}
\end{equation}
where $\Omega\subset\mathbb{R}^N$ ($N\geq 2$) is a smooth bounded domain,
$g:\Omega\times \mathbb{R}\to \mathbb{R}$ is a continuous function satisfying subcritical
growth condition.

Problem \eqref{e1.1} has been studied extensively for many years.
Since Ambrosetti and Rabinowitz proposed the mountain pass theorem in 1973
(see \cite{AmRa}), critical point theory has become one of the main tools
for finding solutions to elliptic equations and systems
of variational type. To apply this theorem, the authors introduced
one of very important conditions (Ambrosetti and Rabinowitz type condition)
on the nonlinear term $g$ as follows:
\begin{itemize}
\item[(AR)] For some $\theta>p$, and $R>0$, we have
$$
0<\theta G(x,t)\leq g(x,t)t, \quad \forall |t|\geq R, \quad \text{ a.e. }
x\in \Omega,
$$
\end{itemize}
where $G(x,t)=\int_0^tg(x,s)\,ds$. This condition ensures that the energy functional
associated to the problem satisfies the Palais-Smale condition ((PS) condition
for short).
Clearly, if the condition (AR) is satisfied then there exist two positive
constants $d_1, d_2$ such that
$$
G(x,t) \geq d_1|t|^\mu-d_2, \quad \forall (x,t) \in \Omega\times \mathbb{R}.
$$
This means that $g$ is $p$-superlinear at infinity in the sense that
$$
\lim_{|t|\to+\infty}\frac{G(x,t)}{|t|^p} = +\infty.
$$
In recent years, there have been many authors considering problem \eqref{e1.1}
 without the (AR) type condition, we refer to some interesting papers on this
topic \cite{CoMa, ItLoUb, KrLiVa, LiYa, SLiu, MiSo, MZSun,WaTa} and the references
cited there.  Miyagaki et al \cite{MiSo}, studied problem \eqref{e1.1}
in the semilinear case $p=2$ by proposing
the following non-global condition on the superlinear term $g(x,t)$:
There exists $t_0>0$ such that
$$
\frac{g(x,t)}{t} \text{ is increasing for } t\geq t_0 \text{ and decreasing for }
t\leq -t_0,\quad \forall x\in \Omega.
$$
Using the mountain pass theorem with the (PS) condition in \cite{AmRa}, the authors
obtained the existence of a non-trivial weak solution. This result was extended
to the $p$-Laplace operator $-\Delta_pu$ by  Li et al \cite{LiYa}.
It should be noticed that in \cite{LiYa,MiSo}, the authors need the following
subcritical growth condition
\begin{itemize}
\item[(A0')] $|g(x,t)|\leq C(1+|t|^{r-1})$ for all $t\in \mathbb{R}$, a.e.
$x\in \Omega$, $r\in [1,p^\ast)$, where
$p^\ast=\frac{Np}{N-p}$ if $1<p<N$ and $p^\ast=+\infty$ if $p\geq N$.
\end{itemize}

Recently  Lan et al \cite{YYLan,LaTa}  studied problem \eqref{e1.1}
by introducing a general type of subcritical growth condition, where
$r=p^\ast$. Using mountain pass theorem \cite{AmRa}, they obtained the
existence of at least one nontrivial
weak solution of \eqref{e1.1} without (AR) condition.
In this article, we consider \eqref{e1.1}  when $g(x,u)=\lambda|u|^{q-2}u+f(x,u)$,
i.e.
\begin{equation}\label{e1.2}
\begin{gathered}
- \Delta_p u = \lambda|u|^{q-2}u+f(x,u), \quad x \in \Omega,\\
 u=0, \quad x \in \partial\Omega,
\end{gathered}
\end{equation}
where $\Omega\subset\mathbb{R}^N$ ($N\geq 2$) is a smooth bounded domain, $1<q<p$,
$\lambda$ is a positive parameter, $f:\overline\Omega\times \mathbb{R}\to \mathbb{R}$ is
a continuous function satisfying
the following general subcritical growth condition
\begin{itemize}
\item[(A0)] $\lim_{|t|\to+\infty} f(x,t)/|t|^{p^\ast-1}=0$ uniformly a.e.
 $x\in\Omega$.
\end{itemize}
In particular, as in \cite{YYLan,LaTa}, we do not use the (AR) condition
for the nonlinear term $f$, see condition (A4) as well as some examples and
comments in the papers \cite{YYLan,LaTa}.
Using the mountain pass theorem \cite{AmRa} combined with
Ekeland variational principle \cite{Ekeland}, we will obtain the existence
of at least two nontrivial weak solutions for problem \eqref{e1.1}.
 Our result introduced here is a natural extension from the previous ones
for elliptic problems with concave-convex nonlinearities
\cite{AmBrCe,PaRo}. Regarding this interesting topic, we refer the readers
to the paper \cite{DeGoUb}, in which the authors studied elliptic problems
with local superlinearity and sublinearity.

To state the main result of this paper, let us introduce the following conditions
on the function $f$:
\begin{itemize}
\item[(A1)] There exists a positive constant $\overline t>0$ such that
$F(x,t)\geq 0$ a.e.
$x\in \Omega$ and all $t \in [0,\overline t]$, where $F(x,t):=\int_0^tf(x,s)\,ds$.

\item[(A2)] $\limsup_{|t|\to 0}\frac{F(x,t)}{|t|^{p}}<\lambda_1$ uniformly a.e.
$x\in \Omega$,
where $\lambda_1$ is the first eigenvalue of $-\Delta_p$.

\item[(A3)] $\lim_{|t|\to +\infty}\frac{F(x,t)}{|t|^{p}}=+\infty$
uniformly a.e. $x\in \Omega$.

\item[(A4)] There exist constants $\theta\geq 1$, $C_\ast>0$ such that
$$
\theta H(x,t)+C_\ast\geq H(x,st)
$$
for all $t\in \mathbb{R}$, $x\in \Omega$, $s\in [0,1]$, where $H(x,t)=f(x,t)t-pF(x,t)$.
\end{itemize}

It should be noticed that the function $f(x,t)=|t|^{p-2}t\log(1+|t|)$
satisfies (A1)--(A4). We refer the readers to \cite{YYLan,LaTa} for more details.
In this article, we look for weak solutions to \eqref{e1.2} in the usual
 Sobolev space $W^{1,p}_0(\Omega)$ which is equipped with the norm
$\|u\|=\big(\int_\Omega|\nabla u|^p\,dx\big)^{1/p}$.

\begin{definition}\label{def1.1} \rm
We say that $u\in W^{1,p}_0(\Omega)$ is a weak solution of  \eqref{e1.2} if
$$
\int_\Omega |\nabla u|^{p-2}\nabla u\nabla v\,dx
-\lambda\int_\Omega |u|^{q-2}uv\,dx-\int_\Omega f(x,u)v\,dx=0
$$
for all $v\in W^{1,p}_0(\Omega)$.
\end{definition}

Our main result is given by the following theorem.

\begin{theorem}\label{the1.2}
Suppose that {\rm (A0)--(A4)} are satisfied. Then, there exists
$\lambda^\ast>0$ such that for any $\lambda\in (0,\lambda^\ast)$,
 problem \eqref{e1.2} has two nontrivial weak solutions.
\end{theorem}

\section{Multiple solutions}

In this section, we  prove our main result.
Let us denote by $c_i$ general
positive constants. As we will see, in order to obtain the existence of at least
two weak solutions for problem \eqref{e1.2} we  use variational methods
(mountain pass theorem and Ekeland variational principle).

We look for the weak solutions of \eqref{e1.2} which are the same as
the critical points of the functional $J: W^{1,p}_0(\Omega)\to \mathbb{R}$ defined by
$$
J(u)=\frac{1}{p}\int_\Omega |\nabla u|^p\,dx
-\frac{\lambda}{q}\int_\Omega |u|^q\,dx
-\int_{\Omega}F(x,u)\,dx.
$$
We can see that $J\in C^{1}(W^{1,p}_0(\Omega),\mathbb{R})$ and
$$
J'(u)(v)=\int_\Omega |\nabla u|^{p-2}\nabla u\nabla v\,dx
-\lambda\int_\Omega |u|^{q-2} uv\,dx-\int_{\Omega}f(x,u)v\,dx
$$
for all $u,v\in W^{1,p}_0(\Omega)$.

\begin{lemma}\label{lem2.1}
There exists $\lambda^\ast>0$ such that for any $\lambda\in (0,\lambda^{\ast})$, we can
choose $\alpha, \rho>0$ so that $J(u)\geq \alpha$ for all $u\in W^{1,p}_0(\Omega)$
with $\|u\|=\rho$.
\end{lemma}

\begin{proof}
From (A0) and (A2), for any $\epsilon>0$, there exists $c(\epsilon)>0$ depending on
$\epsilon$, such that
\begin{equation}\label{e2.1}
F(x,t) \leq \frac{1}{p}(\lambda_1-\epsilon)|t|^p+c(\epsilon)|t|^{p^\ast}
\end{equation}
for all $t\in \mathbb{R}$ and a.e. $x\in \Omega$. Hence, using Sobolev's embedding, we have
\begin{align}\label{e2.2}
\begin{split}
J(u)&=\frac{1}{p}\int_\Omega |\nabla u|^p\,dx-\frac{\lambda}{q}\int_\Omega |u|^q\,dx
-\int_{\Omega}F(x,u)\,dx\\
&\geq\frac{1}{p}\|u\|^p-\frac{\lambda}{q}c_1\|u\|^q-\frac{1}{p}(\lambda_1-\epsilon)
\int_\Omega |u|^p\,dx-c(\epsilon)\int_\Omega |u|^{p^\ast}\,dx\\
&\geq \frac{1}{p}\Big(1-\frac{\lambda_1-\epsilon}{\lambda_1}\Big)\|u\|^p
-\frac{\lambda}{q}c_1\|u\|^q-\overline c(\epsilon)\|u\|^{p^\ast}\\
&=\Big(\frac{\epsilon}{p\lambda_1}-\frac{\lambda}{q}c_1\|u\|^{q-p}
-\overline c(\epsilon)\|u\|^{p^\ast-p}\Big)\|u\|^p,
\end{split}
\end{align}
where $\overline c(\epsilon)$ and $c_1$ are positive constants.

For each $\lambda>0$, we consider the function $\gamma_\lambda: (0,+\infty) \to \mathbb{R}$
defined by
\begin{equation}\label{e2.3}
\gamma_\lambda(t) = \frac{\lambda}{q}c_1t^{q-p}-\overline c(\epsilon)t^{p^\ast-p}.
\end{equation}
It is clear that $\gamma_\lambda(t)$ is a continuous function on $(0,+\infty)$. Since
$p^\ast>p>q>1$, it follows that
\begin{equation}\label{e2.4}
\lim_{t\to 0^+}\gamma_\lambda(t) = \lim_{t\to +\infty}\gamma_\lambda(t)=+\infty.
\end{equation}
Hence, we can find $t_\ast>0$ such that $0<\gamma_\lambda(t_\ast)=\min_{t\in (0,
+\infty)}\gamma_\lambda(t)$, in which $t_\ast$ is defined by the equation
$$
0=\gamma_\lambda'(t_\ast)=\frac{\lambda c_1}{q}(q-p)t_\ast^{q-p-1}+\overline
c(\epsilon)(p^\ast-p)t^{p^\ast-p-1}_\ast
$$
or
$$
t_\ast=\Big(\frac{\lambda c_1(p-q)}{q\overline c(\epsilon)(p^\ast-p)}
\Big)^{\frac{1}{p^\ast-q}}.
$$
Some simple computations show that
\begin{equation}\label{e2.5}
\gamma_\lambda(t_\ast)=c_2.\lambda^\frac{p^\ast-p}{p^\ast-q}\to 0 \quad
\text{as } \lambda\to 0^+.
\end{equation}
From relations \eqref{e2.3}, \eqref{e2.4} and \eqref{e2.5}, there exists
$\lambda^{ \ast}>0$ such that for any $\lambda\in (0,\lambda^\ast)$,
we can choose $\alpha>0$
and $\rho > 0$ so that $J(u) \geq \alpha>0$ for all $u \in W^{1,p}_0(\Omega)$ with
$\|u\| = \rho$.
\end{proof}

\begin{lemma}\label{lem2.2}
There exists $\phi \in W^{1,p}_0(\Omega)$, $\phi >0$ such that $J(t\phi)\to -\infty$
as $t\to+\infty$.
\end{lemma}

\begin{proof}
(ii) From (A3), it follows that for any $M>0$ there exists a constant $c_M=
c(M)>0$ depending on $M$, such that
\begin{equation}\label{e2.6}
F(x,t)\geq M|t|^{p^+}-c_M, \quad \text{for a.e. }x\in\Omega,\;\forall t\in\mathbb{R}.
\end{equation}
Take $\phi\in C_0^\infty(\Omega)$ with $\phi>0$, from \eqref{e2.6} and the
definition of $J$, we obtain
\begin{align}\label{e2.7}
\begin{split}
J(t\phi)
&=\frac{1}{p}\int_\Omega|\nabla t\phi|^{p}\,dx-\lambda\int_\Omega
\frac{1}{q}|t\phi|^{q}\,dx-\int_\Omega F(x,t\phi)\,dx \\
& \leq \frac{1}{p}\|t\phi\|^{p}- M \int_\Omega |t\phi|^{p}\,dx-\frac{\lambda}{q}
\int_{\Omega} |t\phi|^{q}\,dx+c_M|\Omega| \\
& \leq t^{p}\Big(\frac{1}{p}\|\phi\|^{p}-M \int_\Omega |\phi|^{p}\,dx\Big)
-\frac{\lambda t^{q}}{q}\int_\Omega |\phi|^{q}\,dx+c_M|\Omega|,
\end{split}
\end{align}
where $t>0$ and $|\Omega|$ denotes the Lebesgue measure of $\Omega$.

From \eqref{e2.7} and the fact that $1<q<p$, if $M$ is large enough such that
$$
\frac{1}{p}\|\phi\|^{p}-M \int_\Omega |\phi|^{p}\,dx < 0,
$$
then we have $\lim_{t\to +\infty} J(t\phi) = -\infty$.
\end{proof}

\begin{lemma}\label{lem2.3}
There exists $\psi \in W^{1,p}_0(\Omega)$, $\psi >0$ such that $J(t\psi) <0$ for
all $t>0$ small enough.
\end{lemma}

\begin{proof}
Take $\psi\in C_0^\infty(\Omega)$ with $\psi>0$, from the definition of $J$ and
condition (A1), for all $t\in \left(0,\frac{\overline t}{\|\psi\|_{L^\infty
(\Omega)}}\right)$ small enough, we obtain
\begin{align}\label{e2.8}
\begin{split}
J(t\psi)
& = \frac{1}{p}\int_\Omega|\nabla t\psi|^{p}\,dx-\frac{\lambda}{q}
\int_\Omega |t\psi|^{q}\,dx-\int_\Omega F(x,t\psi)\,dx \\
&\leq\frac{t^{p}}{p}\|\psi\|^{p}-\frac{\lambda t^{q}}{q}\int_\Omega|\psi|^{q}\,dx.
\end{split}
\end{align}
From this inequality, taking
$$
0< \delta < \frac{\lambda p\int_\Omega |\psi|^{q}\,dx}{q\|\psi\|^{p}}
$$
we conclude that $J(t\psi)<0$ for all $0<t<\min\{\delta^\frac{1}{p-q},
\frac{\overline t}{\|\psi\|_{L^\infty(\Omega)}}\}$.
The proof of Lemma \ref{lem2.3} is complete.
\end{proof}

\begin{lemma}\label{lem2.4}
The functional $J$ satisfies the (Ce) condition.
\end{lemma}

\begin{proof}
Let $\{u_m\}\subset W^{1,p}_0(\Omega)$ be a $(C_c)$ sequence of the
functional $J$, that is,
$$
J(u_m) \to \overline c, \quad \|J'(u_m)\|_\ast(1+\|u_m\|)\to 0
\quad \text{as } m\to \infty,
$$
which shows that
\begin{equation}\label{e2.9}
\overline c = J(u_m) +o(1), \quad J'(u_m)(u_m) = o(1),
\end{equation}
where $o(1) \to 0$ as $m\to \infty$.

We prove that $\{u_m\}$ is bounded in $W^{1,p}_0(\Omega)$. Indeed, by
contradiction, we assume that $\|u_m\|\to +\infty$ as $m\to \infty$.
Let $w_m =\frac{u_m}{\|u_m\|}$ we obtain $w_m\in W^{1,p}_0(\Omega)$ with
$\|w_m\|=1$.
Then there exists $w\in W^{1,p}_0(\Omega)$ such that $\{w_m\}$ converges
weakly to $w$ in $W^{1,p}_0(\Omega)$ and
\begin{gather}\label{e2.10}
w_m(x) \to w(x), \quad \text{a.e. in } \Omega, \; m\to \infty, \\
w_m \to w\quad \text{strongly in } L^{r}(\Omega), \; m \to \infty, \;
1 \leq r<p^\ast, \label{e2.11}\\
|w_m|_{p^\ast}^{p^\ast}\leq c_3. \label{e2.12}
\end{gather}
Let $\Omega_{\ne}: = \{x\in \Omega:~w(x) \ne 0\}$.
If $x\in \Omega_{\ne}$ then
it follows from \eqref{e2.10} that $\lim_{m\to\infty}w_m(x)=\lim_{m\to\infty}
\frac{u_m(x)}{\|u_m\|}=w(x)$ and thus $|u_m(x)|=|w_m(x)|\|u_m\|\to +\infty$ as
$m\to \infty$ for a.e. $x\in \Omega_{\ne}$.

Using (A3) we have
\begin{equation}\label{e2.13}
\lim_{m\to\infty}\frac{F(x,u_m(x))}{|u_m(x)|^p}=+\infty, \quad \text{ a.e. } x
\in \Omega_{\ne}.
\end{equation}
This implies
\begin{equation}\label{e2.14}
\lim_{m\to\infty}\frac{F(x,u_m(x))}{|u_m(x)|^p}|w_m(x)|^p=+\infty, \quad
\text{a.e. } x\in \Omega_{\ne}.
\end{equation}
Using condition (A3) again, there exists $t_0>0$ such that
\begin{equation}\label{e2.15}
\frac{F(x,t)}{|t|^p} > 1
\end{equation}
for all $x\in \Omega$ and $|t| > t_0>0$. Since $F(x,t)$ is continuous
on $\overline\Omega \times [-t_0,t_0]$, there exists a positive constant
$c_4$ such that
\begin{equation}\label{e2.16}
|F(x,t)| \leq c_4
\end{equation}
for all $(x,t) \in \overline\Omega \times [-t_0,t_0]$. From \eqref{e2.15} and
\eqref{e2.16} there exists $c_5 \in \mathbb{R}$ such that
\begin{equation}\label{e2.17}
F(x,t) \geq c_5
\end{equation}
for all $(x,t) \in \overline\Omega \times \mathbb{R}$. From \eqref{e2.17}, for all
$x \in \Omega$ and $m$, we have
$$
\frac{F(x,u_m(x))-c_5}{\|u_m\|^p} \geq 0
$$
or
\begin{equation}\label{e2.18}
\frac{F(x,u_m(x))}{|u_m(x)|^p}|w_m(x)|^p-\frac{c_5}{\|u_m\|^p}\geq 0,
\quad \forall x\in \Omega, \; \forall m.
\end{equation}
Using \eqref{e2.9} and the Sobolev embedding, there exists $c_6>0$ such
that
\begin{align*}
\begin{split}
\overline c & = J(u_m)+o(1) \\
&=\frac{1}{p}\int_\Omega |\nabla u_m|^p\,dx-\frac{\lambda}{q}\int_\Omega
|u_m|^q\,dx-\int_{\Omega} F(x,u_m)\, dx+o(1) \\
&\geq\frac{1}{p}\|u_m\|^p-\frac{\lambda c_6}{q}\|u_m\|^q-\int_{\Omega}
F(x,u_m)\, dx+o(1);
\end{split}
\end{align*}
since $1<q<p$, this implies
\begin{equation}\label{e2.19}
\int_\Omega F(x,u_m)\,dx \geq \frac{1}{p}\|u_m\|^p-\frac{\lambda c_6}{q}
\|u_m\|^q- \overline c + o(1) \to + \infty \quad\text{as } m \to \infty.
\end{equation}
Also we have
\begin{equation}\label{e2.20}
\begin{split}
\|u_m\|^p
&= p\int_\Omega F(x,u_m)\,dx+\frac{\lambda p}{q}\int_\Omega
|u_m|^q\,dx+p\overline c- o(1) \\
& \geq p\int_\Omega F(x,u_m)\,dx +p\overline c- o(1) > 0 \quad
\text{for $m$ large enough}.
\end{split}
\end{equation}
Next, we  claim that $|\Omega_{\ne}|=0$. In fact, if $|\Omega_{\ne}| \ne 0$,
then by relations \eqref{e2.18}, \eqref{e2.19}, \eqref{e2.20} and the Fatou lemma,
we have
\begin{equation}\label{e2.21}
\begin{split}
+\infty
& = (+\infty)|\Omega_{\ne}| \\
& = \int_{\Omega_{\ne}}\liminf_{m\to \infty}\frac{F(x,u_m(x))}{|u_m(x)|^p}
|w_m(x)|^p\,dx-\int_{\Omega_{\ne}}\limsup_{m\to\infty}\frac{c_5}{\|u_m\|^p}
\,dx \\
&=\int_{\Omega_{\ne}}\liminf_{m\to \infty}\Big(\frac{F(x,u_m(x))}{|u_m(x)|^p}
|w_m(x)|^p-\frac{c_5}{\|u_m\|^p}\Big)\,dx \\
& \leq \liminf_{m\to \infty}\int_{\Omega_{\ne}}\Big(\frac{F(x,u_m(x))}{|u_m(x)|^p}
|w_m(x)|^p-\frac{c_5}{\|u_m\|^p}\Big)\,dx \\
& \leq \liminf_{m\to \infty}\int_\Omega\Big(\frac{F(x,u_m(x))}{|u_m(x)|^p}
|w_m(x)|^p-\frac{c_5}{\|u_m\|^p}\Big)\,dx \\
& = \liminf_{m\to \infty}\int_{\Omega}\frac{F(x,u_m(x))}{\|u_m\|^p}\,dx
-\limsup_{m\to\infty}\int_{\Omega}\frac{c_5}{\|u_m\|^p}\,dx \\
& = \liminf_{m\to \infty}\int_{\Omega}\frac{F(x,u_m(x))}{\|u_m\|^p}\,dx\\
& \leq \liminf_{m\to \infty}\frac{\int_{\Omega}F(x,u_m(x))\,dx}{p\int_\Omega
F(x,u_m)\,dx +p\overline c- o(1)}.
\end{split}
\end{equation}
From \eqref{e2.19} and \eqref{e2.21}, we obtain
$$
+\infty \leq \frac{1}{p},
$$
which is a contradiction. This shows that $|\Omega_{\ne}|=0$ and thus $w(x)=0$
a.e. in $\Omega$.

Since the function $t\mapsto J(tu_m)$ is continuous in $t \in [0,1]$,
for each $m$ there exists $t_m \in [0,1]$ such that
\begin{equation}\label{e2.22}
J(t_mu_m) : = \max_{t\in [0,1]}J(tu_m), \quad m=1, 2, \dots.
\end{equation}
It is clear that $t_m>0$ and $J(t_mu_m)\geq \overline c >0=J(0)=J(0.u_m)$. If $t_m<1$
then $\frac{d}{dt}J(tu_m)|_{t=t_m}=0$ which gives $J'(t_mu_m)(t_mu_m) = 0$.
If $t_m = 1$, then $J'(u_m)(u_m)=o(1)$. So we always have
\begin{equation}\label{e2.23}
J'(t_mu_m)(t_mu_m) = o(1).
\end{equation}
Now, we fix a big integer $k \geq 1$ and define the sequence $\{v_m\}$ by
\begin{equation}\label{e2.24}
v_m=\left(2p\|u_k\|^p\right)^{1/p}w_m, \quad m = 1, 2, \dots.
\end{equation}
From the dominated convergence theorem and since $w=0$ we have
\begin{equation}\label{e2.25}
\lim_{m\to\infty}\int_\Omega |v_m|^q\,dx=0.
\end{equation}
Furthermore, by (A0), for every $\epsilon>0$, there exists $c(\epsilon)>0$ such
that
$$
|F(x,t)|\leq \frac{1}{c_3}\epsilon|t|^{p^\ast}+c(\epsilon), \quad \forall t\in \mathbb{R},
\text{ a.e. } x\in \Omega.
$$
Let $\delta=\frac{\epsilon}{2c(\epsilon)}>0$, $E\subseteq \Omega$, $|E|<\delta$
we have
\begin{align*}
\big|\int_E F(x,v_m)\,dx\big|
&\leq \int_E |F(x,v_m)|\,dx\\
&\leq \int_E c(\epsilon)\,dx + \frac{1}{2c_3}\epsilon\int_E |v_m|^{p^\ast}\,dx\\
&\leq \frac{\epsilon}{2}+\frac{\epsilon}{2},
\end{align*}
hence $\{\int_\Omega F(x,v_m)\,dx:m\in \mathbb N\}$ is
equi-absolutely-continuous. It follows easily from Vitali convergence
theorem that
$$
\int_\Omega F(x,v_m)\,dx\to \int_\Omega F(x,0)\,dx=0 \quad \text{ as } m\to \infty.
$$
Since $\|u_m\|\to +\infty$ as $m\to\infty$, we can find $m_k \geq k$ such that
\begin{equation}\label{e2.26}
0<\frac{(2p\|u_k\|^p)^{1/p}}{\|u_m\|}< 1, \quad \forall m > m_k.
\end{equation}
Hence, using relations \eqref{e2.22}, \eqref{e2.24}-\eqref{e2.26}, it follows that
\begin{equation}\label{e2.27}
\begin{split}
& J(t_mu_m) \\
& \geq J\Big(\frac{\left(2p\|u_k\|^{p}\right)^{1/p}}{\|u_m\|}u_m\Big)\\
&=J(v_m) \\
&=\frac{1}{p}\int_\Omega |\nabla v_m|^{p}\,dx - \frac{\lambda}{q}
\int_\Omega|v_m|^{q}\,dx-\int_{\Omega} F(x,v_m)\,dx \\
& \geq \frac{1}{p}\int_\Omega \left(\|u_k\|^{p}.(2p)^\frac{p}{p}.|\nabla w_m
|^{p}\right)\,dx-\frac{\lambda}{q}\int_\Omega|v_m|^{q}\,dx-\int_\Omega
F(x,v_m)\,dx \\
& \geq 2\|u_k\|^{p}-\frac{\lambda}{q}\int_\Omega|v_m|^{q}\,dx-\int_\Omega
F(x,v_m)\,dx \\
& \geq \|u_k\|^{p}
\end{split}
\end{equation}
for any $m>m_k \geq k$ large enough.

On the other hand, using  condition (A4) and relation \eqref{e2.23}, for
all $m >m_k>k$ large enough, we have
\begin{equation}\label{e2.28}
\begin{split}
&J(t_mu_m)\\
&=J(t_mu_m) - \frac{1}{p}J'(t_mu_m)(t_mu_m) +o(1) \\
&=\frac{1}{p}\int_\Omega |\nabla t_mu_m|^{p}\,dx-\frac{\lambda}{q}\int_{
\Omega}|t_mu_m|^{q}\,dx-\int_\Omega F(x,t_mu_m)\,dx \\
& \quad -\frac{1}{p}\int_{\Omega} |\nabla t_mu_m|^{p}\,dx
+\frac{\lambda}{p}\int_{\Omega} |t_mu_m|^{q}\,dx\\
&\quad +\frac{1}{p}\int_\Omega f(x,t_mu_m)t_mu_m\,dx +o(1)\\
& = \lambda\big(\frac{1}{p}-\frac{1}{q}\big)\int_{\Omega}|t_mu_m|^{q}\,dx
+ \frac{1}{p} \int_{\Omega} H(x,t_mu_m)\,dx\\
& \leq \frac{1}{p} \int_{\Omega} \big(\theta H(x,u_m)+C_\ast\big)\,dx + o(1) \\
& = \theta\Big(\frac{1}{p}\int_\Omega|\nabla u_m|^{p}\,dx-\frac{\lambda}{q}
\int_\Omega|u_m|^{q}\,dx-\int_\Omega F(x,u_m)\,dx\Big)\\
&\quad -\frac{\theta}{p}\Big(\int_\Omega |\nabla u_m|^{p}\,dx-\lambda
\int_\Omega |u_m|^{q}\,dx-\int_\Omega f(x,u_m)u_m\,dx\Big) \\
& \quad +\lambda\theta\big(\frac{1}{q}-\frac{1}{p}\big)\int_\Omega
|u_m|^{q}\,dx +\frac{\theta C_\ast|\Omega|}{p}+o(1) \\
&=\theta J(u_m)-\frac{\theta}{p}J'(u_m)(u_m)+\lambda\theta
\big(\frac{1}{q}-\frac{1}{p}\big)\int_\Omega|u_m|^{q}\,dx
 +\frac{\theta C_\ast|\Omega|}{p}+o(1) \\
& \leq \theta J(u_m)-\frac{\theta}{p}J'(u_m)(u_m)+\lambda \theta c_7
\big(\frac{1}{q}-\frac{1}{p}\big) \|u_m\|^{q} +\frac{\theta C_\ast|\Omega|}{p}+o(1).
\end{split}
\end{equation}
From \eqref{e2.27} and \eqref{e2.28}, we deduce that for all $m>m_k>k$ large
enough,
$$
\|u_k\|^{p} \leq \theta J(u_m)-\frac{\theta}{p}J'(u_m)(u_m)+\lambda \theta c_7
\big(\frac{1}{q}-\frac{1}{p}\big) \|u_m\|^{q} +\frac{\theta C_\ast|\Omega|}{p}+o(1)
$$
or
\begin{equation}\label{e2.29}
\|u_k\|^{p}-\lambda \theta c_7\big(\frac{1}{q}-\frac{1}{p}\big) \|u_m\|^{q} \leq
\theta J(u_m)-\frac{\theta}{p}J'(u_m)(u_m)+\frac{\theta C_\ast|\Omega|}{p}+o(1)
\end{equation}
Recall that $k\geq 1$ is an arbitrarily big integer and $m>m_k>k$. In \eqref{e2.29},
let $k\to \infty$ we have $m\to \infty$ and the left hand side of \eqref{e2.29} tends
to $+\infty$ since $q<p$. In the right hand side of \eqref{e2.29}, $J(u_m)\to c$
and $\frac{\theta}{p}J'(u_m)(u_m)\to 0$ as $m \to \infty$. Thus, we have a contradiction.
This proves that the sequence $\{u_m\}$ is bounded in $W^{1,p}_0(\Omega)$.

Now, since the Banach space $W^{1,p}_0(\Omega)$ is reflexive, there exists $u \in
W^{1,p}_0(\Omega)$ such that passing to a subsequence, still denoted by $\{u_m\}$,
it converges weakly to $u$ in $W^{1,p}_0(\Omega)$ and converges strongly to $u$ in
$L^r(\Omega)$, $1\leq r<p^\ast$. Moreover, $\{u_m\}$ converges weakly to $u$ in
$L^{p^\ast}(\Omega)$ and we have $|u_m|_{p^\ast}^{p^\ast}\leq c_8$. From (A0),
for every $\epsilon>0$, there exists $c(\epsilon)>0$ such that
$$
|f(x,t)t|\leq \frac{1}{2c_8}\epsilon|t|^{p^\ast}+c(\epsilon), \quad \forall t\in \mathbb{R}, \quad
\text{ a.e. } x\in \Omega.
$$
Let $\delta=\frac{\epsilon}{2c(\epsilon)}>0$, $E\subseteq \Omega$, $|E|<\delta$
we have
\begin{align*}
\big|\int_E f(x,u_m)u_m\,dx\big|
&\leq \int_E |f(x,u_m)u_m|\,dx\\
&\leq \int_E c(\epsilon)\,dx + \frac{1}{2c_8}\epsilon\int_E |u_m|^{p^\ast}\,dx\\
&\leq \frac{\epsilon}{2}+\frac{\epsilon}{2},
\end{align*}
hence $\{\int_\Omega f(x,u_m)u_m\,dx:m\in \mathbb N\}$ is equi-absolutely-continuous.
 It follows easily from Vitali convergence theorem that
\begin{equation}\label{e2.30}
\int_\Omega f(x,u_m)u_m\,dx\to \int_\Omega f(x,u)u\,dx \quad\text{as } m\to \infty.
\end{equation}
Using (A0) again, for any $\epsilon>0$ there exists $c(\epsilon)>0$ such that
$$
|f(x,t)|\leq \frac{1}{2c_9c_{10}}\epsilon|t|^{p^\ast-1}+c(\epsilon), \quad
\forall t\in \mathbb{R}, \; \text{ a.e. } x\in \Omega,
$$
where
$$
c_9\geq \Big(\int_\Omega|u_m|^{p^\ast}\,dx\Big)^{\frac{p^\ast-1}{p^\ast}}, \quad
\forall m; \quad
 c_{10}:=\Big(\int_\Omega|u|^{p^\ast}\,dx\Big)^{1/p^\ast}.
$$
From the H\"{o}lder inequality, for every $E\subseteq \Omega$, we have
\begin{gather*}
\int_Ec(\epsilon)|u|\,dx
\leq c(\epsilon)|E|^\frac{p^\ast-1}{p^\ast}\Big(\int_E|u|^{
p^\ast}\,dx\Big)^{1/p^\ast}
= c(\epsilon)|E|^\frac{p^\ast-1}{p^\ast}c_{10},
\\
\int_E|u_m|^{p^\ast-1}|u|\,dx
\leq \Big(\int_E|u_m|^{p^\ast}\,dx\Big)^{\frac{p^\ast-1}{p^\ast}}
\Big(\int_E|u|^{p^\ast}\,dx\Big)^{1/p^\ast}\leq c_9c_{10}.
\end{gather*}
Let $\delta=(\frac{\epsilon}{2c_{10}c(\epsilon)})^\frac{p^\ast}{p^\ast-1}>0$,
$E\subseteq \Omega$, $|E|<\delta$ we have
\begin{align*}
\big|\int_E f(x,u_m)u\,dx\big|
&\leq \int_E |f(x,u_m)u|\,dx\\
&\leq \int_E c(\epsilon)|u|\,dx
+ \frac{1}{2c_9c_{10}}\epsilon\int_E |u_m|^{p^\ast-1}|u|\,dx\\
&\leq \frac{\epsilon}{2}+\frac{\epsilon}{2},
\end{align*}
hence $\{\int_\Omega f(x,u_m)u_m\,dx:m\in \mathbb N\}$ is
equi-absolutely-continuous. It follows easily from Vitali convergence
theorem that
\begin{equation}\label{e2.31}
\int_\Omega f(x,u_m)u\,dx\to \int_\Omega f(x,u)u\,dx \quad \text{as } m\to \infty.
\end{equation}
From \eqref{e2.30} and \eqref{e2.31} we have
\begin{equation}\label{e2.32}
\int_\Omega f(x,u_m)(u_m-u)\,dx\to 0 \quad \text{as } m\to \infty.
\end{equation}
We also have
\begin{equation} \label{e2.33}
\begin{split}
\int_\Omega |u_m|^{q-2}u_m(u_m-u)\,dx
&\leq \int_\Omega |u_m|^{q-1}|u_m-u|\,dx\\
&\leq\Big(\int_\Omega |u_m|^q\,dx\Big)^\frac{q-1}{q}
\Big(\int_\Omega |u_m-u|^q\,dx\Big)^{1/q}\to 0
\end{split}
\end{equation}
 as $ m\to \infty$.
Since $J'(u_m)(u_m-u)\to 0$ as $m\to \infty$, we deduce from \eqref{e2.32} and
\eqref{e2.33} that
$$
\int_\Omega|\nabla u_m|^{p-2}\nabla u_m(\nabla u_m-\nabla u)\,dx\to 0 \quad
\text{as } m\to \infty,
$$
which gives us that $\{u_m\}$ converges strongly to $u$ in $W^{1,p}_0(\Omega)$ and
the functional $J$ satisfies the (Ce) condition.
\end{proof}

\begin{proof}[Proof Theorem \ref{the1.2}]
By Lemmas \ref{lem2.1}, \ref{lem2.2} and \ref{lem2.4}, there exists
$\lambda^\ast>0$ such that for any $\lambda \in (0,\lambda^\ast)$,
the functional $J$ satisfies all the
assumptions of the mountain pass theorem. Then we deduce $u_1$ as a
non-trivial critical point of the functional $J$ with
$J(u_1) = \overline c>0$ and thus a non-trivial weak solution of
problem \eqref{e1.2}.

We now prove that there exists a second weak solution $u_2 \in W^{1,p}_0(\Omega)$
such that $u_2 \ne u_1$. Indeed, by \eqref{e2.2}, the functional $J$ is bounded from
below on the ball $\overline B_\rho(0)$.

Applying the Ekeland variational principle in \cite{Ekeland} to the functional
 $J: \overline B_\rho(0)  \to \mathbb{R}$, it follows that there exists
 $u_\epsilon \in \overline B_\rho(0)$ such that
\begin{gather*}
J(u_\epsilon)  < \inf_{u \in \overline B_\rho(0)}J(u)+\epsilon, \\
J(u_\epsilon)  < J(u) + \epsilon\|u-u_\epsilon\|, \quad u \ne u_\epsilon.
\end{gather*}
By Lemmas  \ref{lem2.1} and \ref{lem2.2}, we have
$$
\inf_{u \in \partial B_\rho(0)} J(u) \geq R > 0 \quad\text{and} \quad
\inf_{u \in \overline B_\rho(0)}J(u) < 0.
$$
Let us choose $\epsilon > 0$ such that
$$
0 < \epsilon<\inf_{u \in \partial B_\rho(0)} J(u)
- \inf_{u \in \overline B_\rho(0)}J(u).
$$
Then, $J(u_\epsilon) <  \inf_{u \in \partial B_\rho(0)}J(u)$ and thus, $u_\epsilon
\in B_\rho(0)$.

Now, we define the functional $I : \overline B_\rho(0) \to \mathbb{R}$ by
$I(u)=J(u)+\epsilon \|u-u_\epsilon\|$. It is clear that $u_\epsilon$
is a minimum point of $I$ and thus
$$
\frac{I(u_\epsilon + tv)-I(u_\epsilon)}{t} \geq 0
$$
for all $t > 0$ small enough and all $v \in B_\rho(0)$.
The above information shows that
$$
\frac{J(u_\epsilon+tv)-J(u_\epsilon)}{t}+\epsilon \|v\| \geq 0.
$$
Letting $t \to 0^+$, we deduce that
$\langle {J'(u_\epsilon),v} \rangle \geq -\epsilon \|v\|$.
It should be noticed that $-v$ also belongs to $B_\rho(0)$, so replacing $v$ by
$-v$, we obtain
\begin{gather*}
\left\langle {J'(u_\epsilon),-v} \right\rangle \geq-\epsilon\|-v\|,
\left\langle {J'(u_\epsilon),v} \right\rangle \leq \epsilon \|v\|,
\end{gather*}
which helps us to deduce that $\|J'(u_\epsilon)\|_{\ast} \leq \epsilon$.

Then, there exists a sequence $\{u_m\} \subset B_\rho(0)$ such that
\begin{equation}\label{e2.34}
J(u_m) \to \underline c  = \inf_{u \in \overline B_\rho(0)}J(u) < 0,\quad
J'(u_m) \to 0 \quad \text{in } W^{-1,p}(\Omega) \text{ as } m \to \infty.
\end{equation}
From Lemma \ref{lem2.4}, the sequence $\{u_m\}$ converges strongly to some
$u_2\in W^{1,p}_0(\Omega)$ as $m \to \infty$. Moreover, since
$J\in C^1 (W^{1,p}_0(\Omega),\mathbb{R})$, by \eqref{e2.9} it follows that
 $J(u_2) = \underline c$
and $J'(u_2) = 0$. Thus, $u_2$ is a non-trivial weak solution of \eqref{e1.2}.

Finally, we point out  that $u_1 \ne u_2$ since $J(u_1) = \overline c > 0 >
\underline c = J(u_2)$. The proof of Theorem \ref{the1.2} is complete.
\end{proof}

\subsection*{Acknowledgments}
The authors would like to thank the referees for their suggestions and helpful
comments which improved the presentation of the original manuscript.
This work is supported by Quang Binh University (Grant N. CS.05.2016).


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\end{document}