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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 151, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/151\hfil Existence of solutions to $p$-Laplacian equations]
{Existence of solutions to $p$-Laplacian equations involving general
subcritical growth}

\author[Y.-Y. Lan \hfil EJDE-2014/151\hfilneg]
{Yong-Yi Lan}  % in alphabetical order

\address{Yong-Yi Lan \newline
School of Sciences, Jimei University \\
Xiamen  361021, China}
\email{lanyongyi@jmu.edu.cn}

\thanks{Submitted February 13, 2014. Published June 27, 2014.}
\thanks{Supported by HuiZhen Huang Foundation
for Subject Construction at Jimei University}
\subjclass[2000]{35B33, 35J92, 35J35}
\keywords{$p$-Laplacian equation; subcritical growth;
variational methods; \hfill\break\indent 
(C) condition; mountain-pass lemma}

\begin{abstract}
 In this article, we consider the  quasilinear elliptic equation
 $-\Delta_p u=\mu f(x,u)$  with the Dirichlet boundary coditions, and under
 suitable growth condition on the nonlinear term $f$.
 Existence of solutions is given for all $\mu>0$ via the variational
 method and some analysis techniques.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction and main results}

 In this article, we  consider the  Dirichlet boundary-value problem
 \begin{equation}
 \begin{gathered}
   -\Delta_p u=\mu f(x,u),\quad x\in\Omega,\\
    u=0,\quad x\in\partial\Omega,
 \end{gathered} \label{e1.1}
 \end{equation}
where $\Omega$ is an open bounded domain in $\mathbb{R}^N$ with
smooth boundary $\partial\Omega $, $p>$1,
$-\Delta_p u=\operatorname{div}(|\nabla u|^{p-2}\nabla u)$ is the
$p$-Laplacian of $u$,  $f(x,t)$ is  continuous on 
$\overline{\Omega}\times\mathbb{R}$.


 We look for the weak solutions of \eqref{e1.1} which are the same 
as the critical  points of the functional 
$I_{\mu}:W^{1,p}_0(\Omega)\to\mathbb{R}$ defined by
\begin{equation} \label{e1.2}
I_{\mu}(u)=\frac{1}{p}\int_\Omega|\nabla
u|^p\,\mathrm{d}x-\mu\int_\Omega F(x,u)\,\mathrm{d}x,
\end{equation}
where  $F(x,t)=\int_0^tf(x,s)\,\mathrm{d}s$, and $ W^{1,p}_0(\Omega)$
is the  Sobolev  space with  the usual norm: 
$$
\|u\|^p=\int_\Omega|\nabla u|^p\,\mathrm{d}x. 
$$

In this article,  the hypotheses on the nonlinearity $f(x,t)$ are the following:
\begin{itemize}
\item[(F1)] There exist constants $\theta\geq 1$, $\alpha>0$ such that
$$
\theta G(x,t)+\alpha\geq G(x,st)\quad \text{for all } t\in\mathbb{R},
\; x\in\overline{\Omega},\;  s\in[0,1],
$$
where $G(x,t):=tf(x,t)-pF(x,t)$.

\item[(F2)] 
$$
\lim_{|t|\to\infty}\frac{f(x,t)}{t|t|^{p^{\ast}-2}}=0\quad 
\text{uniformly a.e. } x\in\Omega,
$$
where $p^*=\frac{Np}{N-p}$ if $1<p<N$ and $p^*=+\infty$ if 
$p\geq N$  is the Sobolev critical exponent.


\item[(F3)] 
$$
\lim_{t\to0}\frac{f(x,t)}{t|t|^{p-2}}=0\quad \text{uniformly a.e. }x\in\Omega.
$$

\item[(F4)] 
$$
\lim_{|t|\to+\infty}\frac{F(x,t)}{|t|^{p}}=+\infty\quad \text{uniformly a.e. }
x\in\Omega.
$$
\end{itemize}
Problem \eqref{e1.1} is one of the main quasilinear elliptic problems which 
have been studied extensively for many years, see, for example 
\cite{Ambrosetti}--\cite{Schechter}, \cite{Willem}.
Since Ambrosetti and Rabinowitz  proposed the mountain-pass theorem in 1973 
(see \cite{Ambrosetti}), critical point theory has become one
of the main tools for finding solutions to elliptic equations of variational type.
A standard existence result for \eqref{e1.1} is that for any  $\mu>0$,  
\eqref{e1.1} possesses at least
 a nontrivial solution if $f(x,t)$ satisfies the following conditions:
\begin{itemize}
\item[(1)]  $p$-superlinear at $t=0$:  $\lim_{t\to0}\frac{f(x,t)}{t|t|^{p-2}}=0$
uniformly a.e. $x\in\Omega$.

\item[(2)] subcritical at $t=\infty$: there are positive constants $a$ and
$b$ such that
\[
|f(x,t)|\leq a+b|t|^{q-1}, \quad \forall t\in\mathbb{R},\; x\in\Omega.
\]
where $1\leq q<p^*$.

\item[(3)] the  Ambrosetti--Rabinowitz  condition (AR for short): for some 
$\theta>p$, $C>0$,
\begin{equation} \label{e1.3}
0<\theta F(x,t)\leq f(x,t)t,\quad \forall |t|\geq C,\;x\in\Omega.
\end{equation}
\end{itemize}
The (AR) condition has appeared in most of the studies for quasilinear problems and
plays an important role in studying the existence of nontrivial solutions of many
quasilinear elliptic boundary value problems. It is  quite natural and important
not only to ensure that the Euler-Lagrange functional associated to
problem \eqref{e1.1} has a mountain pass geometry, but
also to guarantee that Palais-Smale sequence of the Euler-Lagrange functional is
bounded. Since then, the (AR) condition has been used extensively in many
literature sources (see \cite{Bartsch,Coti,YDing}).
 But this condition is very restrictive eliminating many nonlinearities.
There are always many functions that do not satisfy the (AR) condition.
 For example,  for the sake of simplicity, we consider the case $p=2$,
$$
f(x,t)=2t\ln(1+|t|).
$$
Many efforts have been made to extend the range of the nonlinearity.
For example, Miyagaki  and Souto \cite{Miyagaki} studied \eqref{e1.1}
for when $p=2$ and replaced the (AR) condition by  some monotonicity arguments.
They assumed that there is $t_0>0$ such that
\begin{equation} \label{e1.4}
\frac{f(x,t)}{t} \text{ is increasing for $t\geq t_0$ and  decreasing for
$t\leq -t_0$, for all }x\in\Omega;
\end{equation}
or a weaker condition is that there exist $C>0$ such that
\begin{equation} \label{e1.5}
tf(x,t)-2F(x,t)\leq sf(x,s)-2F(x,s)+C,\quad
\end{equation}
for all $0<t<s$ or $s<t<0$, for all $x\in\Omega$.

There are some other well known solvability conditions 
(see \cite{Costa,DeFigueiredo,Lam,Schechter,Willem}).
Moreover, in the study of critical points of real-valued functionals,
with or without constraints, the Palais-Smale condition(the (P.S.) 
condition for short) and its variants play a
essential role.

To ensure the global compactness, one needs to impose the subcritical growth
condition on the nonlinearity $f(x,t)$: there exists a constant
$C_0>0$ such that
\[
|f(x,t)|\leq C_0(1+|t|^{p-1}), \quad \forall t\in\mathbb{R},\;x\in\Omega,
\]
where $1<p<p^*$. However, in  the present paper, we consider a class of 
elliptic partial differential equations with more
general growth condition, that is (F2).

Based on variational methods, Miyagaki  and Souto  in \cite{Miyagaki} 
obtained the following theorem:

\begin{theorem}[{\cite[Theorem 1.1]{Miyagaki}}] \label{thmA}
Under hypotheses \eqref{e1.4}, \eqref{e1.2}, {\rm (F3)} and {\rm (F4)}, 
 problem \eqref{e1.1} for when $p=2$ has a nontrivial weak solution,
 for all $\lambda>0$.
\end{theorem}

Many efforts have been made to extend these results 
(see \cite{Li1,Liu,Mao} and the references therein). 
Li and Yang extend the results from $p=2$ to  $p>1$ in
\cite{Li1}, they obtained the following theorem:

 \begin{theorem}[{\cite[Theorem 1.1]{Li1}}] \label{thmB}
Under hypotheses \eqref{e1.4}, \eqref{e1.2}, {\rm (F3)} and {\rm (F4)}, 
 problem \eqref{e1.1} has a nontrivial weak solution,
 for all $\lambda>0$.
\end{theorem}

 The aim of the article is to consider the problem in a different case: 
based on a variant version of   mountain pass theorem, we can prove the same 
result under more generic conditions,  which  generalizes 
Theorems \ref{thmA} and \ref{thmB}.

Our main results reads as follows:

\begin{theorem} \label{thm1}
 Suppose that {\rm (F1)--(F4)} hold.  Then  \eqref{e1.1} has a weak 
nontrivial solution, for all $\lambda>0$.
\end{theorem}


Note that (F3) implies that  problem \eqref{e1.1} has a trivial solution 
$u =0$ and we are interested in the existence of
nontrivial solutions.

\begin{remark} \rm
Theorem \ref{thm1} improves Theorem \ref{thmB} in two aspects. To show this, 
it suffices to compare condition  (F1) with \eqref{e1.4} and \eqref{e1.5},
and  to compare condition  (F2) with \eqref{e1.2}.
\end{remark}

 At first, we can easily prove that  (F1) is equivalent to \eqref{e1.5}
 when $\theta=1$, and (F1) gives some general sense  of monotony when $\theta>1$.
There are functions satisfying our condition (F1)
and not satisfying the condition \eqref{e1.5}.  For example,  
for the sake of simplicity, we consider the case $p=2$, let
$$
F(x,t)=t^{2}\ln(1+t^{2})+t\sin t,
$$
then
$$
f(x,t)=2t \ln(1+t^{2})+t^{2} \cdot \frac{2t}{1+t^{2}}+ \sin t+t \cos t,
$$
it follows that
$$
G(x,t)=tf(x,t)-2F(x,t)=2(t^{2}-1)+\frac{2}{1+t^{2}}+(t^{2} \cos t-t\sin t).
$$
Let $\theta$ = 1000, we can prove by some simple computation that $G$ 
satisfies (F1) but does not satisfy the condition \eqref{e1.5} any more.

Secondly, it is obvious that \eqref{e1.2} implies (F2).
There are functions satisfying our growth condition (F2)
and not satisfying the subcritical growth
condition \eqref{e1.2}. For example,  for the sake of simplicity, we consider 
the case $p=2$,  let
$$
F(t)=\frac{t^{2^{\ast}}}{\ln(e+t^{2})}
$$
then
$$
f(t)=\frac{2^{\ast}t^{2^{\ast}-1}(e+t^{2})\ln(e+t^{2})
-2t^{2^{\ast}+1}}{(e+t^{2})(\ln(e+t^{2}))^{2}}.
$$

The proof of  Theorem \ref{thm1} is much easier than that of the main 
results in \cite{Li1,Miyagaki}.

\begin{remark} \rm 
In assumption (F2), we are dealing with functionals satisfying the so-called 
non-standard growth conditions.  Due to the lack of compactness of the embeddings in
$W^{1,p}_0(\Omega)\hookrightarrow L^{p^{\ast}}(\Omega)$,
we cannot use the standard variational argument directly. 
We overcome the   difficulty
by Vitali convergence theorem and some analysis techniques.
\end{remark}

This paper is organized as follows. 
In section 2, we give the proof of the Theorem \ref{thm1}.  
 In the following discussion, we denote various
positive constants as $c$ or $c_i$ $(i=0, 1,2,  \dots)$ for
convenience.

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

The proof consists of three steps. We prove Theorem \ref{thm1} only when
$\mu=1$. The case of a general $\mu>0$ will follow immediately. 
In fact, if $\mu>0$  and $\mu\neq1$, we only let
$g(x,t)=\mu f(x,t)$. Then   \eqref{e1.1}  becomes
\begin{gather*}
 -\Delta_p u=g(x,u),\quad x\in\Omega,\\
    u=0\quad x\in\partial\Omega.
\end{gather*}
The nonlinear term $g$ also satisfies
condition (F1)--(F4), Then the same conclusion as in the case $\mu=1$ holds.
\smallskip 

\noindent\textbf{First step:}  The (C) condition.
 Let $\{u_{n}\}$ be any sequence in  $W^{1,p}_0(\Omega)$ such that $I(u_{n})$ 
is bounded and $\|I'(u_{n})\|(1+\|u_{n}\|)$
converges to zero; that is,
 $$
I(u_{n})\to c,\quad \|I'(u_{n})\|(1+\|u_{n}\|)\to0
$$
which shows that
\begin{equation} \label{e2.1}
c=I(u_{n})+o(1),\quad \langle I'(u_{n}),u_{n}\rangle=o(1)
\end{equation}
where $o(1)\to 0$ as $n\to\infty$.

 We now prove that $\{u_{n}\}$ is bounded in  $W^{1,p}_0(\Omega)$. 
By contradiction, we assume $\|u_{n}\|\to\infty$ as $n\to\infty$.
Let $w_{n}=\frac{u_{n}}{\|u_{n}\|}$, then $w_{n}\in W^{1,p}_0(\Omega)$ 
with $\|w_{n}\|=1$. Then there exists a $w\in W^{1,p}_0(\Omega)$  such that
\begin{equation} \label{e2.2}
\begin{gathered}
w_{n}\rightharpoonup w\quad \text{in }W^{1,p}_0(\Omega),\\
w_{n}\to w\quad \text{a.e. in } \Omega,\\
w_{n}\to w\quad \text{in }L^r(\Omega),\text{ with }1\leq r<p^{\ast},\\
\|w_{n}\|_{2^{\ast}}^{2^{\ast}}\leq C_1<\infty.
\end{gathered}
\end{equation}
Let $\Omega_{\neq}=\{x\in\Omega,\,w(x)\neq0\}$; then one has
$$
\lim_{n\to\infty}w_{n}(x)=\lim_{n\to\infty}\frac{u_{n}(x)}{\|u_{n}\|}
=w(x)\neq0\quad \text{in }\Omega_{\neq}.
$$
So we have\begin{equation} \label{e2.3}
|u_{n}(x)|\to+\infty\,\,\,\text{a.e. in }\Omega_{\neq}.
\end{equation}
Using (F4), we have
\begin{equation} \label{e2.4}
\lim_{n\to\infty}\frac{F(x,u_{n}(x))}{|u_{n}(x)|^{p}}=+\infty,\quad
\text{a.e. in }\Omega_{\neq}.
\end{equation}
This means that
\begin{equation} \label{e2.5}
\lim_{n\to\infty}\frac{F(x,u_{n}(x))}{|u_{n}(x)|^{p}}|w_{n}(x)|^{p}=+\infty,
\quad \text{a.e. in }\Omega_{\neq}.
\end{equation}
By (F4) again, there is an $C_0>0$ such that
\begin{equation} \label{e2.6}
\frac{F(x,t)}{|t|^{p}}>1,
\end{equation}
for any $x\in\Omega$ and  $|t|\geq C_0$.
 Since $F(x,t)$ is continuous on $\overline{\Omega}\times[-C_0,C_0]$,
there is an $M>0$ such that
\begin{equation} \label{e2.7}
|F(x,t)|\leq M, \quad \text{for all }(x,t)\in\overline{\Omega}\times[-C_0,C_0].
\end{equation}
From \eqref{e2.6}, \eqref{e2.7}, we see that there is a constant $C$
such that for any $(x,t)\in\overline{\Omega}\times\mathbb{R}$, we have
\begin{equation} \label{e2.8}
F(x,t)\geq C
\end{equation}
which shows that
\[
\frac{F(x,u_{n}(x))-C}{\|u_{n}\|^{p}}\geq0.
\]
This implies that
\begin{equation} \label{e2.9}
\frac{F(x,u_{n}(x))}{|u_{n}(x)|^{p}}|w_{n}(x)|^{p}-\frac{C}{\|u_{n}\|^{p}}\geq0.
\end{equation}
Using \eqref{e2.1} we have
$$
c=I(u_{n})+o(1)=\frac{1}{p}\|u_{n}\|^{p}-\int_{\Omega}F(x,u_{n})\,\mathrm{d}x+o(1).
$$
So we see that
\begin{equation} \label{e2.10}
\|u_{n}\|^{p}=pc+p\int_{\Omega}F(x,u_{n})\,\mathrm{d}x+o(1).
\end{equation}
By \eqref{e2.1} and \eqref{e2.10}, we obtain
\begin{equation} \label{e2.11}
\int_{\Omega}F(x,u_{n})\,\mathrm{d}x\to+\infty.
\end{equation}
We claim that $|\Omega_{\neq}|=0$. In fact, if  $|\Omega_{\neq}|\neq0$,
then combining  \eqref{e2.5} and \eqref{e2.9} with  Fatou's lemma, one has
\begin{equation} \label{e2.12}
\begin{aligned}
+\infty
&= \int_{\Omega_{\neq}}\liminf_{n\to+\infty}\frac{F(x,u_{n}(x))}{|u_{n}(x)|^{p}}|w_{n}(x)|^{p}\,\mathrm{d}x-
\int_{\Omega_{\neq}}\limsup_{n\to+\infty}\frac{C}{\|u_{n}\|^{p}}\,\mathrm{d}x\\
&\leq \int_{\Omega_{\neq}}\liminf_{n\to+\infty}\Big(\frac{F(x,u_{n}(x))}{|u_{n}(x)|^{p}}|w_{n}(x)|^{p}-
\frac{C}{\|u_{n}\|^{p}}\Big)\,\mathrm{d}x\\
&\leq \liminf_{n\to+\infty}\int_{\Omega_{\neq}}\Big(\frac{F(x,u_{n}(x))}{|u_{n}(x)|^{p}}|w_{n}(x)|^{p}-
\frac{C}{\|u_{n}\|^{p}}\Big)\,\mathrm{d}x\\
&\leq \liminf_{n\to+\infty}\int_{\Omega}\Big(\frac{F(x,u_{n}(x))}{|u_{n}(x)|^{p}}|w_{n}(x)|^{p}-
\frac{C}{\|u_{n}\|^{p}}\Big)\,\mathrm{d}x\\
&= \liminf_{n\to+\infty}\int_{\Omega}\frac{F(x,u_{n}(x))}{\|u_{n}\|^{p}}\,\mathrm{d}x\\
&\leq \liminf_{n\to+\infty}\frac{\int_{\Omega}F(x,u_{n}(x))
\,\mathrm{d}x}{pc+p\int_{\Omega}F(x,u_{n})\,\mathrm{d}x+o(1)}.
\end{aligned}
\end{equation}
So by \eqref{e2.11} and \eqref{e2.12} we deduce a contradiction.
This shows that $|\Omega_{\neq}|=0$.
Hence $w(x)=0$  a.e. in $\Omega$.

Since $I(tu_{n})$ is continuous in $t\in[0,1]$, there exists $t_{n}\in[0,1]$ 
such that
$$
I(t_{n}u_{n})=\max_{t\in[0,1]}I(tu_{n}).
$$
Clearly, $t_{n}>0$ and $I(t_{n}u_{n})\geq 0=I(0)$. 
If $t_{n}<1$ we have that $\frac{\mathrm{d}}{\mathrm{d}t}I(tu_{n})|_{t=t_{n}}=0$,
which gives
$\langle I'(t_{n}u_{n}), t_{n}u_{n}\rangle=0$.
If $t_{n}=1$, then \eqref{e2.1} gives that
$\langle I'(u_{n}), u_{n}\rangle=o(1)$.
So we always have 
$$
\langle I'(t_{n}u_{n}), t_{n}u_{n}\rangle=o(1).
$$
From (F1), for $t\in[0,1]$ we have
\begin{equation} \label{e2.13}
\begin{aligned}
p I(tu_{n})
&\leq p I(t_{n}u_{n})\\
&= p I(t_{n}u_{n})-\langle I'(t_{n}u_{n}), t_{n}u_{n}\rangle+o(1)\\
&= \int_{\Omega}[t_{n}u_{n}f(x,t_{n}u_{n})-p F(x,t_{n}u_{n})]\,\mathrm{d}x+o(1)\\
&\leq \int_{\Omega}[\theta(u_{n}f(x,u_{n})-p F(x,u_{n}))+\alpha]\,\mathrm{d}x+o(1)\\
&\leq \theta(\|u_{n}\|^{p }+p c-\|u_{n}\|^{p }+o(1))+\alpha|\Omega|+o(1)\\
&\leq p \theta c+\alpha|\Omega|+o(1).
\end{aligned}
\end{equation}
where we used \eqref{e2.1} and \eqref{e2.10}, $\theta$ and $\alpha$ as in (F1).

Furthermore, by (F2), for every $\varepsilon>0$, there exists 
$a(\varepsilon)>0$, such that
\[
|F(x,t)|\leq \frac{1}{2C_1}\varepsilon|t|^{p^{\ast}}+a(\varepsilon),
\quad \text{for $t\in\mathbb{R}$ a.e. $x\in\Omega$}.
\]
Let $\delta=\varepsilon/(2a(\varepsilon))>0$,  
$E\subseteq\Omega$, $\operatorname{meas}E<\delta$, we have
\begin{align*}
\big|\int_{E}F(x,w_{n})\,\mathrm{d}x\big|
&\leq \int_{E}|F(x,w_{n})|\,\mathrm{d}x\\
&\leq \int_{E}a(\varepsilon)\,\mathrm{d}x
 +\frac{1}{2C_1}\varepsilon\int_{E}|w_{n}|^{p^{\ast}}\,\mathrm{d}x\\
&\leq \frac{\varepsilon}{2}+\frac{\varepsilon}{2}=\varepsilon,
\end{align*}
hence $\{\int_{\Omega}F(x,w_{n})\,\mathrm{d}x,\,n\in N\}$ 
is equi-absolutely-continuous. It follows easily from Vitali Convergence 
Theorem that
\[
\int_{\Omega}F(x,w_{n})\,\mathrm{d}x\to\int_{\Omega}F(x,0)\,\mathrm{d}x=0;
\]
So, for any $R_0>0$,
\begin{equation} \label{e2.14}
p I(R_0w_{n})=\|R_0w_{n}\|^{p}-p\int_{\Omega}F(x,R_0w_{n})\,\mathrm{d}x
=R_0^{p}+o(1).
\end{equation}
From \eqref{e2.13}, we obtain
\begin{equation} \label{e2.15}
p I(tu_{n}) \leq p \theta c+\alpha|\Omega|+o(1),
\end{equation}
for $t\in[0,1]$. So combining  \eqref{e2.14} with \eqref{e2.15},
\[
R_0^{p}+o(1)=p I(R_0w_{n})
\leq p \theta c+\alpha|\Omega|+o(1).
\]
Letting $n\to\infty$  we obtain
\[
R_0^{p} \leq p\theta c+\alpha|\Omega|+o(1).
\]
Letting $R_0\to\infty$, we obtain a contradiction. Hence
$ \| u_{n}\|$ is bounded.

By the continuity of the embedding, we have 
$\|u_{n}\|_{2^{\ast}}^{2^{\ast}}\leq C_{2}<\infty$ for all $n$. 
If necessary going to a subsequence, one obtains
\[
u_{n}\rightharpoonup u\text{ in }W^{1,p}_0(\Omega),\quad
\text{and}\quad u_{n}\to u\text{ in }L^r(\Omega),\quad
\text{where }1\leq r<p^{\ast}.
\]
Using  (F2), for every $ \varepsilon>0$, there exists $a(\varepsilon)>0$, such that
$$
|f(x,t)t|\leq \frac{1}{2C_{2}}\varepsilon|t|^{p^{\ast}}+a(\varepsilon),
\quad \text{for } t\in\mathbb{R}, \text{ a.e. }x\in\Omega.
$$
Let $\delta=\varepsilon/(2a(\varepsilon))>0$,  $E\subseteq\Omega$,
$\operatorname{meas}E<\delta$, we have
\begin{align*}
\big|\int_{E}f(x,u_{n})u_{n}\,\mathrm{d}x\big|
&\leq \int_{E}|f(x,u_{n})u_{n}|\,\mathrm{d}x\\
&\leq \int_{E}a(\varepsilon)\,\mathrm{d}x
 +\frac{1}{2C_{2}}\varepsilon\int_{E}|u_{n}|^{p^{\ast}}\,\mathrm{d}x\\
&\leq \frac{\varepsilon}{2}+\frac{\varepsilon}{2}=\varepsilon,
\end{align*}
hence $\{\int_{\Omega}f(x,u_{n})u_{n}\,\mathrm{d}x,\,n\in N\}$ 
is equi-absolutely-continuous. It follows easily from Vitali Convergence 
Theorem that
\begin{equation} \label{e2.16}
\int_{\Omega}f(x,u_{n})u_{n}\,\mathrm{d}x\to\int_{\Omega}f(x,u)u\,\mathrm{d}x.
\end{equation}
From (F2), for any $\varepsilon>0$ there exists $a(\varepsilon)>0$ such that
 $$
|f(x,t)|\leq \frac{1}{2c_1c_{2}}\varepsilon|t|^{p^{\ast}-1}+a(\varepsilon)
\quad \text{for } t\in\mathbb{R},\,x\in\Omega.
$$
where
\[
c_1\geq\Big(\int_{\Omega}|u_{n}|^{p^{\ast}}\,\mathrm{d}x\Big)
^{\frac{p^{\ast}-1}{p^{\ast}}}\quad\forall n;\quad
c_{2}:=\Big(\int_{\Omega}|u|^{p^{\ast}}\,
\mathrm{d}x\Big)^{\frac{1}{p^{\ast}}}\,.
\]
From  H\"{o}lder's inequality, for every $E\subseteq\Omega$, we have
\begin{gather*}
\int_{E}a(\varepsilon)|u|\,\mathrm{d}x
\leq a(\varepsilon)(\operatorname{meas}E)^{\frac{p^{\ast}-1}{p^{\ast}}}
\Big(\int_{E}|u|^{p^{\ast}}\,\mathrm{d}x\Big)^{\frac{1}{p^{\ast}}}
\leq a(\varepsilon)(\operatorname{meas}E)^{\frac{p^{\ast}-1}{p^{\ast}}}c_1;
\\
\int_{E}|u_{n}|^{p^{\ast}-1}|u|\,\mathrm{d}x
\leq\Big(\int_{E}|u_{n}|^{p^{\ast}}\,\mathrm{d}x\Big)^{\frac{p^{\ast}-1}{p^{\ast}}}
\Big(\int_{E}|u|^{p^{\ast}}\,
\mathrm{d}x\Big)^{\frac{1}{p^{\ast}}}\leq c_1c_{2}.
\end{gather*}
Let
$\delta=\big(\frac{\varepsilon}{2c_1a(\varepsilon)}\big)
^{\frac{p^{\ast}}{p^{\ast}-1}}>0$,  $E\subseteq\Omega$,
$\operatorname{meas}E<\delta$, we have
\begin{align*}
\big|\int_{E}f(x,u_{n})u\,\mathrm{d}x\big|
&\leq \int_{E}|f(x,u_{n})u|\,\mathrm{d}x\\
&\leq \int_{E}a(\varepsilon)|u|\,\mathrm{d}x
 +\frac{1}{2c_1c_{2}}\varepsilon\int_{E}|u_{n}|^{p^{\ast}-1}|u|\,\mathrm{d}x\\
&\leq \frac{\varepsilon}{2}+\frac{\varepsilon}{2}=\varepsilon,
\end{align*}
hence $\{\int_{\Omega}f(x,u_{n})u\,\mathrm{d}x,\,n\in N\}$ is also
equi-absolutely-continuous. It follows  from Vitali Convergence Theorem that
\begin{equation} \label{e2.17}
\int_{\Omega}f(x,u_{n})u\,\mathrm{d}x\to\int_{\Omega}f(x,u)u\,\mathrm{d}x.
\end{equation}
Since
\begin{equation} \label{e2.18}
\langle I'(u_{n}),u\rangle=\int_{\Omega}\left(|\nabla u_{n}|^{p-2}
\nabla u_{n}\cdot\nabla u-f(x,u_{n})u\right)\,\mathrm{d}x\to0;
\end{equation}
\begin{equation} \label{e2.19}
\langle I'(u_{n}),u_{n}\rangle=\int_{\Omega}
\left(|\nabla u_{n}|^{p-2}\nabla u_{n}\cdot\nabla u_{n}-f(x,u_{n})u_{n}\right)\,
\mathrm{d}x\to0.
\end{equation}
It follows from \eqref{e2.16}-\eqref{e2.19} that
 $\|u_{n}\|\to\|u\|.$ By  Kadec-Klee property,
we have
\[
u_{n}\to u \quad  \text{in } W^{1,p}_0(\Omega).
\]
\smallskip


\noindent\textbf{Second step:} 
Mountain-pass geometric structure.
$I$ has a mountain pass geometry; i.e., there exist 
$u_1\in W^{1,p}_0(\Omega)$  and
constants $r, \rho > 0$ such that $I(u_1) < 0$, $\|u_1\| > r$ and
\begin{equation} \label{e2.20}
I(u) \geq \rho,\quad \text{when }\| u\| = r.
\end{equation}
Indeed, By (F3), we have
$t_0 > 0$ and $\lambda \in (0, \lambda_1)$ such that
$$
\frac{f(x,t)}{t|t|^{p-2}}<\lambda,\quad \text{for }|t|<t_0,
$$
where
$$
\lambda_1=\inf_{u\in W^{1,p}_0(\Omega),\,u\neq0}\frac{\|u\|^{p}}{\|u\|_p^{p}}>0
$$
is the first eigenvalue of the operator $-\Delta_p $ with the Dirichlet
boundary value in $\Omega$. This implies that
$$
F(x,t)\leq\frac{\lambda}{p}|t|^{p},\quad \text{for }|t|\leq t_0.
$$
This inequality with (F2) shows that
$$
F(x,t)\leq\frac{\lambda}{p}|t|^{p}+C|t|^{p^{\ast}},\quad
\text{for }t\in\mathbb{R}
$$
with some $C > 0$. Since $\lambda_1>0$ denotes the first eigenvalue of the
operator $-\Delta_p $ with the Dirichlet boundary value in $\Omega$,
it follows that $\| u\|^{p}\geq\lambda_1\| u\|_p^{p}$
for $u\in W^{1,p}_0(\Omega)$. Then $I$ is estimated as
$$
I(u)\geq \frac{1}{p}\| u\|^{p}-\frac{\lambda}{p}\| u\|_p^{p}
-C\| u\|_{p^{\ast}}^{p^{\ast}}\geq
 \frac{\lambda_1-\lambda}{p\lambda_1}\| u\|^{p}-C'\| u\|^{p^{\ast}}.
$$
This shows the existence of $r$ and $\rho$  satisfying:
\[
I(u) \geq \rho,\quad \text{when }\| u\| = r.
\]
From (F4) follows that, for all $M>0$ there exists $C_{M}>0$, such that
\begin{equation} \label{e2.21}
F(x,t)\geq M|t|^{p}-C_{M},\quad \forall\,x\in\Omega,\; t>0.
\end{equation}
Let $\phi$ be a function such that
$\phi\in W^{1,p}_0(\Omega)$, $\phi\geq0$, $\phi\not\equiv 0$. From \eqref{e2.21}
 we obtain
\begin{align*}
I(t\phi)
&=\frac{|t|^{p}}{p}\| \phi\|^{p}-\int_{\Omega}F(x,t\phi)\,\mathrm{d}x\\
&\leq \frac{|t|^{p}}{p}\| \phi\|^{p}-t^{p}\int_{\Omega}M\phi^{p}
\,\mathrm{d}x+c|\Omega|\to-\infty\quad \text{as }t\to\infty.
\end{align*}
We fix $t > 0$ large so that $I(t\phi) < 0$ and $t\| \phi\| > r$.
Let $u_1:= t\phi\in W^{1,p}_0(\Omega)$  and then
constants $r,\, \rho > 0$ such that $I(u_1) < 0,\,\|u_1\| > r$ and
satisfies \eqref{e2.20}, i.e. $I$ has a mountain pass geometry.
\smallskip

\noindent\textbf{Third step:}  Critical value of $I$.
For $u_1$ in  second step, we define
\begin{gather*}
\Gamma:=\{\gamma: C[0,1]\to W^{1,p}_0(\Omega):\gamma(0)=0,\; \gamma(1)=u_1\},\\
c_0:=\inf_{\gamma\in  \Gamma}\max_{0\leq t\leq1}I(\gamma(t))
\end{gather*}
As shown in \cite{Schechter1}, a deformation lemma can be proved
with the (C) condition, replacing the usual Palais-Smale condition, 
and it turns out that the Mountain Pass Theorem still holds.
Then $c_0$ is a critical value of $I$.  For the proof, we refer the
reader to \cite{Rabinowitz,Struwe,Willem1}.


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\end{document}