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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 90, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/90\hfil Infinitely many weak solutions]
{Infinitely many weak solutions for a $p$-Laplacian equation with
nonlinear boundary conditions}

\author[J.-H. Zhao,  P.-H. Zhao\hfil EJDE-2007/90\hfilneg]
{Ji-Hong Zhao,  Pei-Hao Zhao}  % in alphabetical order

\address{Department of Mathematics, Lanzhou University\\
 Lanzhou, 730000,  China}
\email[Ji-Hong Zhao]{zhaojihong2007@yahoo.com.cn}
\email[Pei-Hao Zhao]{zhaoph@lzu.edu.cn}

\thanks{Submitted March 26, 2007. Published June 15, 2007.}
\thanks{This work was partly supported by the Fundamental Research Fund
  for Physics and \hfill\break\indent Mathematics of Lanzhou University}
\subjclass[2000]{35J20, 35J25}
\keywords{$p$-Laplacian; nonlinear boundary conditions;
weak solutions; \hfill\break\indent critical exponent; variational principle}

\begin{abstract}
 We study the following quasilinear problem with
 nonlinear boundary conditions
 \begin{gather*}
 -\Delta _{p}u+a(x)|u|^{p-2} u=f(x,u) \quad \text{in }\Omega, \\
 |\nabla u|^{p-2} \frac{\partial u}{\partial \nu}=g(x,u) \quad
 \text{on } \partial\Omega,
 \end{gather*}
 where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$ with smooth
 boundary and  $\frac{\partial}{\partial \nu}$ is the outer normal
 derivative, $\Delta_{p}u=\mathop{\rm div}(|\nabla u|^{p-2}\nabla u)$ is the
 $p$-Laplacian with $1<p<N$. We consider the above problem under
 several conditions on $f$ and $g$, where $f$ and $g$ are both
 Carath\'{e}odory functions. If $f$ and $g$ are both superlinear
 and subcritical with respect to $u$, then we prove the existence
 of infinitely many solutions of this problem by using ``fountain
 theorem" and ``dual fountain theorem" respectively. In the case,
 where $g$ is superlinear but subcritical and $f$ is critical with
 a subcritical perturbation, namely
 $f(x,u)=|u|^{p^{*}-2}u+\lambda|u|^{r-2}u$, we show that there
 exists at least a nontrivial solution when $p<r<p^{*}$ and there
 exist infinitely many solutions when $1<r<p$, by using
 ``mountain pass theorem'' and ``concentration-compactness principle''
 respectively.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}

\section{Introduction}

Consider a quasilinear elliptic problem
\begin{equation} \label{e1}
\begin{gathered}
-\Delta _{p}u+a(x)|u|^{p-2} u=f(x,u) \quad \text{in }\Omega , \\
|\nabla u|^{p-2} \frac{\partial u}{\partial \nu}=g(x,u) \quad
\text{on }
\partial\Omega,
\end{gathered}
\end{equation}
 where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$ with smooth
 boundary and $\frac{\partial}{\partial \nu}$ is the outer normal derivative,
$\Delta_{p}u=\mathop{\rm div}(|\nabla u|^{p-2}\nabla u)$ is the
$p$-Laplacian with $1<p<N$,
 \begin{equation} \label{e2}
a(x)\in L^{\infty}(\Omega) \text{ satisfying }
\mathop{\rm ess\,inf}_{x\in{\overline{\Omega}}} a(x)>0.
\end{equation}

The study of nonlinear elliptic boundary value problem about
$p$-Laplacian of the form \eqref{e1} is an interesting topic in
recent years. Many results have been obtained on this kind of
problem, for example see \cite{P,CR,BR1,K,MR1,MR2,B} and the
references therein. Such problem appear naturally in the study of
optimal constants for Sobolev trace embedding and it arises in
various applications, e.g. non-Newtonian fluids, reaction-diffusion
problems, glaciology, biology etc(see \cite{D,AD,BR2,BMR,B}). The
first paper that analyzed \eqref{e1} is \cite{BR1}. In that paper,
the authors systematically studied the existence of nontrivial
solutions of \eqref{e1} under $f(u)=|u|^{p-2}u$ and $g$ are
subcritical, critical with a subcritical perturbation and
supercritical with respect to $u$. Using the ideas from \cite{GP},
they established the existence results, nonexistence result,
especially the result of nonlinear eigenvalue problem. In \cite{B},
the author proved the existence of at least three nontrivial
solutions for \eqref{e1} under adequate assumptions on the source
terms $f$ and $g$. On the other hand, when $\Omega$ is unbounded, we
can see \cite{P,CR,K} for some existence and multiplicity results of
solutions to problem \eqref{e1} in some weighted Sobolev spaces. Our
aim in this paper is to prove that the infinitely many solutions
results for the problem \eqref{e1} under various assumptions on
nonlinear terms $f$ and $g$. If $f$ and $g$ are both superlinear and
subcritical with respect to $u$, then we prove the existence of
infinitely many solutions of problem \eqref{e1} by using ``fountain
theorem" and ``dual fountain theorem" respectively. In the case,
where $g$ is superlinear but subcritical and $f$ is critical with a
subcritical perturbation, namely
$f(x,u)=|u|^{p^{*}-2}u+\lambda|u|^{r-2}u$, we show that there exists
at least a nontrivial solution when $p<r<p^{*}$ and there exist
infinitely many solutions when $1<r<p$, by using ``mountain pass
theorem" and ``concentration-compactness principle" respectively.
The main ideas of our paper is from \cite{W,BR1}.

Throughout this paper the following hypotheses are assumed.
\begin{itemize}
\item[(F1)] $f(x,u)$ is a Carath\'{e}odory function and for some
$p<q<p^{*}=\frac{Np}{N-p}$, there exists a constant $C_{1}>0$ such
that
$$
|f(x,u)|\leq C_{1}(1+|u|^{q-1}) \quad \text{for all }x\in \Omega,\;
u\in\mathbb{R}.
$$

\item[(G1)] $g(x,u)$ is a Carath\'{e}odory function and for some
$p<z<p_{*}=\frac{(N-1)p}{N-p}$, there exists a constant $C_{2}>0$
such that
$$
|g(x,u)|\leq C_{2}(1+|u|^{z-1})\quad \text{for all } x\in \partial\Omega,\;
u\in\mathbb{R}.
$$


\item[(F2)] There exists $\alpha_{1}>p$ and $R>0$ such that
$$
|u|\geq R\Longrightarrow 0<\alpha_{1} F(x,u)\leq uf(x,u)\quad
\text{for all } x\in \Omega,
$$
where $F(x,u)=\int_{0}^{u}f(x,t)dt$ is the primitive function of
$f(x,u)$.

\item[(G2)] There exists $\alpha_{2}>p$ and $R>0$ such that
$$
|u|\geq R\Longrightarrow 0<\alpha_{2} G(x,u)\leq ug(x,u)\quad
\text{for all } x\in \partial\Omega,
$$
where
$G(x,u)=\int_{0}^{u}g(x,t)dt$ is the primitive function of
$g(x,u)$.

\item[(F3)] $f(x,u)$ is an odd function with respect to $u$, that
is,
$$
f(x,-u)=-f(x,u) \quad \text{for all } x\in \Omega.
$$

\item[(G3)] $g(x,u)$ is an odd function
with respect to $u$, that is,
$$
g(x,-u)=-g(x,u)\quad \text{for all } x\in \Omega.
$$

\item[(G4)] $\lim_{u\to0} \frac{g(x,u)}{|u|^{p-1}}=0$.

\end{itemize}
Define $W^{1,p}(\Omega)=\{u\in L^{p}(\Omega): \int_{\Omega}|\nabla
u|^{p}\mathrm{d}x<\infty\}$ with the norm
 \begin{equation} \label{e3}
\| u \|_{1,p}:=(\int_{\Omega}(|\nabla
u|^{p}+a(x)|u|^{p})\mathrm{d}x)^{\frac{1}{p}}.
 \end{equation}
Then $W^{1,p}(\Omega)$ is a Banach space. For a variational approach,
the functional associated to the
problem \eqref{e1} is
\begin{equation} \label{e4}
\varphi(u)=\frac{1}{p}\int_{\Omega}(|\nabla u|^{p}+a(x)|u|^{p})\mathrm{d}x-
 \int_{\Omega}F(x,u)\mathrm{d}x-\int_{\partial\Omega}G(x,u)\mathrm{d}S,
\end{equation}
 where $u \in W^{1,p}(\Omega)$ and $\mathrm{d}S$ is the measure on
the boundary. Since (F1) and (G1) we can easily to obtain
$\varphi\in C^{1}(W^{1,p}(\Omega),\mathbb{R})$ and
\begin{align*}
\langle\varphi'(u);v\rangle
&=\int_{\Omega}(|\nabla u|^{p-2}\nabla
u\nabla v +a(x)|u|^{p-2}uv)\mathrm{d}x\\
&\quad -\int_{\Omega}f(x,u)v\mathrm{d}x -
\int_{\partial\Omega}g(x,u)v\mathrm{d}S
\end{align*}
for all $u,v \in W^{1,p}(\Omega)$. We say that $u$ is a weak solution of the
problem \eqref{e1} if $u$ is the critical point of the functional $\varphi$
on $W^{1,p}(\Omega)$.

\begin{remark} \label{rmk1.1}\rm
 According to the regularity theorem of
\cite{L}, if $\partial\Omega$ is of class
$C^{1,\alpha}(0<\alpha\leq1)$ and $g$ satisfies
\begin{equation*}
|g(x,u)-g(y,v)|\leq C(|x-y|^{\alpha}+|u-v|^{\alpha}),\quad |g(x,u)|\leq C
\end{equation*}
for all $x,y\in\Omega, u,v\in\mathbb{R}$, then the regularity up to
the boundary of \cite[Theorem 2]{L} shows that every weak solution
of \eqref{e1} belongs to $C_{\rm loc}^{1,\beta}(\overline{\Omega})$
for some $0<\beta\leq1$.
\end{remark}

\begin{remark} \label{rmk1.2} \rm
 Under the assumption \eqref{e2} it is easy to
check that the norm \eqref{e3} is equivalent to the usual one, that is the
norm with $a(x)\equiv 1$ in \eqref{e3}.
\end{remark}

 Our main results are as follows.

\begin{theorem}\label{thm1.1}
Under the assumptions
(F1)--(F3) and (G1)--(G3),  problem \eqref{e1} has a sequence of
solutions ${u_{k}}\in W^{1,p}(\Omega)$ such
that $\varphi{(u_{k})}\to \infty $ as $k\to \infty$.
\end{theorem}

For a special $f$, we obtain a sequence of weak solutions with
negative energy.

\begin{theorem}\label{thm1.2}
Let $f(x,u)=\mu|u|^{r-2}u+\lambda|u|^{s-2}u$, where $1<r<p<s<p^{*}$ and
assume $(G_{1})\sim(G_{4})$ are satisfied. Then
\begin{enumerate}

\item for every $\lambda>0$, $\mu\in \mathbb{R}$, problem
\eqref{e1} has a sequence of solutions ${u_{k}}\in W^{1,p}(\Omega)$ such
that $\varphi(u_{k})\to \infty $ as $k\to \infty$,

\item for every $\mu>0$, $\lambda\geq 0$, problem \eqref{e1} has a
sequence of solutions ${v_{k}}\in W^{1,p}(\Omega)$ such that
$\varphi(v_{k})<0$, $\varphi(v_{k})\to 0 $ as
$k\to \infty$.
\end{enumerate}
\end{theorem}
Next we consider the critical growth on $f$. In this case, the
compactness of the embedding $W^{1,p}(\Omega)\hookrightarrow
L^{p^{*}}(\Omega)$ fails, so to recover some sort of compactness, in
spirit of \cite{BN}, we consider a perturbation of the critical
power, that is, $f(x,u)=|u|^{p^{*}-2}u+\lambda|u|^{r-2}u$. We also
need much more assumptions on $g$ around about the origin.

\begin{itemize}
\item[(G2')] there exists $\alpha_{2}>p$ such that
$$
0<\alpha_{2}G(x,u)\leq ug(x,u)\quad \text{for all } x\in \partial\Omega, \;
 u\in \mathbb{R}\setminus\{0\}.
$$
\end{itemize}
Here we use the ``concentration-compactness principle" introduced in
\cite{L1,L2}. We prove the following two theorems.

\begin{theorem}\label{thm1.3}
Let $f(x,u)=|u|^{p^{*}-2}u+\lambda|u|^{r-2}u$ with $p<r<p^{*}$ and
assume (G1), (G2), (G3) and (G4) are satisfied. Then there exists a
constant $\lambda_{0}>0$ depending on $p$, $r$, $N$ and $|\Omega|$
such that if $\lambda>\lambda_{0}$, problem \eqref{e1} has at least
a nontrivial solution in $W^{1,p}(\Omega)$.
\end{theorem}

\begin{theorem}\label{thm1.4}
Let $f(x,u)=|u|^{p^{*}-2}u+\lambda|u|^{r-2}u$ with $1<r<p$ and
assume (G1), (G2), (G3) and (G4) are satisfied. Then their exists a
constant $ \widetilde{\lambda}>0$ depending on $p$, $r$, $N$ and
$|\Omega|$ such that if $0<\lambda< \widetilde{\lambda}$, problem
\eqref{e1} has infinitely many
 nontrivial solutions ${u_{k}}\in W^{1,p}(\Omega)$ such that
$\varphi(u_{k})<0$, $\varphi(u_{k})\to 0 $ as
$k\to \infty$.
\end{theorem}

 This paper is organized as follows. In the second section, we recall some
definitions and preliminary theorems, including the well-known
``fountain theorem" and ``dual fountain theorem". The $(PS)_{c}$
condition and $(PS)_{c}^{*}$ condition are also introduced. In the
third section, we consider the subcritical case and give the proof
of Theorem \ref{thm1.1} and Theorem \ref{thm1.2}. In the last section.
We consider the critical case and give the proof of Theorems
\ref{thm1.3} and \ref{thm1.4}.

\section{Preliminaries}

First we introduced some notations: $X$ denotes Banach space with
the norm $\|\cdot\|_{X}$, $X^{*}$ denotes the conjugate space with
$X$, $L^{p}(\Omega)$ denotes Lebesgue space with the usual norm
$|\cdot|_{p}$, $W^{1,p}(\Omega)$ denotes Sobolev space with the
norm $\|\cdot\|_{1,p}$ defined by \eqref{e3}, $\langle\cdot;\cdot\rangle$
is the dual paring of the space $X^{*}$ and $X$, $|\Omega|$
denotes the Lebesgue measure of the set $\Omega\subset
\mathbb{R}^{N}$, $C_{1},C_{2},\dots$, denote (possibly
different) positive constants.

One important aspect of applying the standard methods of
variational theory is to show that the functional $\varphi$
satisfies the $(PS)_{c}$ or $(PS)_{c}^{*}$ condition which is
introduced the following definition.

\begin{definition}\label{def2.1} \rm
Let $\varphi\in C^{1}(X,\mathbb{R})$ and $c\in \mathbb{R}$. The function
 $\varphi$ satisfies the $(PS)_{c}$ condition if any sequence
$\{u_{n}\}\subset X$ such  that
$$
\varphi(u_{n})\to c,\quad \varphi'(u_{n})\to 0\quad \text{in }
 X^{*}\text{ as } n\to\infty
$$
contains a subsequence converging to a critical point of $\varphi$.
\end{definition}

Let $X$ be a reflexive and separable Banach space, then there are
${e_{j}}\in X$ and ${e_{j}^{*}}\in X^{*}$ such that
\begin{gather*}
X=\overline{\mathop{\rm span}\{e_{j}| j=1,2,\dots}\},\quad
 X^{*}=\overline{\mathop{\rm span}\{{e_{j}^{*}}| j=1,2,\dots}\}, \\
\langle e_{i}^{*};e_{j}\rangle = \begin{cases} 1, \quad i=j,\\
0, \quad i\neq j.
 \end{cases}
\end{gather*}
 For convenience, we write $X_{j}:= \mathop{\rm span} \{e_{j}\}$,
$Y_{k}:=\oplus_{j=1}^{k}X_{j}$,
$Z_{k}:=\overline {\oplus_{j=k}^{\infty}X_{j}}$.
And let $B_{k}:=\{u\in Y_{k}:\|u\|_{X}\leq \rho_{k}\}$,
$N_{k}:=\{u\in Z_{k}:\|u\|_{X}=\gamma_{k}\}$, where
$\rho_{k}>\gamma_{k}>0$.

\begin{definition}\label{def2.2} \rm
 Let $\varphi\in C^{1}(X,\mathbb{R})$ and $c\in \mathbb{R}$. The function
 $\varphi$ satisfies the $(PS)_{c}^{*}$ condition (with respect to $(Y_{n})$)
if  any sequence $\{u_{n_{j}}\}\subset Y_{n_{j}}$ such that
 $$
\varphi(u_{n_{j}})\to c,\quad
\varphi|_{Y_{n_{j}}}'(u_{n_{j}})\to 0 \quad
\text{in }
 X^{*} \text{ as } n_{j}\to \infty
$$
contains a subsequence converging to a critical point of $\varphi$.
 \end{definition}

\begin{theorem}[{Fountain theorem, \cite[Thm. 3.6]{W}}]\label{thm2.1}
Let $\varphi\in C^{1}(X,\mathbb{R})$ be an even functional. If ,
for every $k\in \mathbb{N}$, there exists $\rho_{k}>\gamma_{k}>0$ such that
\begin{itemize}
\item[(A1)] $a_{k}:=\sup_{u\in Y_{k},\,\|u\|_{X}=\rho_{k}} \varphi(u)\leq 0$,

\item[(A2)] $b_{k}:=\inf_{u\in Z_{k},\,
\|u\|_{X}=\gamma_{k}} \varphi(u)\to \infty$
as $k\to \infty$,

\item[(A3)] $\varphi$ satisfies the $(PS)_{c}$ condition
for every $c>0$.

\end{itemize}
Then $\varphi$ has an unbounded sequence of critical values.
\end{theorem}

\begin{theorem}[{Dual fountain theorem, \cite[Theorem 3.18]{W}}]\label{thm2.2}
 Let  $\varphi\in C^{1}(X,\mathbb{R})$ be an even functional. If , for every
$k\geq  k_{0}$, there exists $\rho_{k}>\gamma_{k}>0$ such that
\begin{itemize}
\item[(B1)]  $a_{k}:=\inf_{u\in Z_{k},\,
\|u\|_{X}=\rho_{k}} \varphi(u)\geq 0$,

\item[(B2)] $b_{k}:=\sup_{u\in Y_{k},\, \|u\|_{X}=\gamma_{k}} \varphi(u)< 0$,

\item[(B3)] $d_{k}:=\inf_{u\in Z_{k},\, \|u\|_{X}\leq\rho_{k}} \varphi(u)\to 0$
 as $k\to \infty$,

\item[(B4)] $\varphi$ satisfies the $(PS)_{c}^{*}$ condition for every
$c\in[d_{k_{0}},0[$.

\end{itemize}
Then $\varphi$ has a sequence of negative critical values
converging to $0$.
\end{theorem}

\section{Proof of Theorem \ref{thm1.1}}
\subsection*{Proof of the $(PS)_{c}$ condition}
Let us introduce the following lemmas which will be helpful in the proof.

\begin{lemma}[{\cite[Lemma 2.1]{MR2}}] \label{lem3.1}
 Let  $A: W^{1,p}(\Omega)\to W^{1,p}(\Omega)^{*}$ be the function given by
 $\langle A(u);v\rangle:=\int_{\Omega}|\nabla u|^{p-2}\nabla u\nabla
v\mathrm{d}x +  \int_{\Omega}a(x)|u|^{p-2}uv\mathrm{d}x$.
 Then $A$ is continuous, odd, $(p-1)$-homogeneous, and continuously invertible.
\end{lemma}

\begin{lemma}[{\cite[Lemma 2.2]{MR2}}] \label{lem3.2}
 Let $B:W^{1,p}(\Omega)\to W^{1,p}(\Omega)^{*}$ be the function given by
 $\langle B(u);v\rangle:=\int_{\partial\Omega}g(x,u)v\mathrm{d}S$, where
 $g(x,u)$ be a Carath\'{e}odory function with subcritical
 growth.
 Then $B$ is continuous and compact.
\end{lemma}

\begin{lemma}[{\cite[Lemma 2.3]{MR2}}] \label{lem3.3}
  Let  $C:W^{1,p}(\Omega)\to W^{1,p}(\Omega)^{*}$ be the function given by
 $\langle C(u);v\rangle=\int_{\Omega}f(x,u)v\mathrm{d}x$, where $f(x,u)$
is a Carath\'{e}odory function
with subcritical growth. Then $C$ is continuous and compact.
\end{lemma}

\begin{lemma}\label{lem3.4}
Under the hypotheses of Theorem \ref{thm1.1}, $\varphi$
satisfies the $(PS)_{c}$ condition with $c>0$.
\end{lemma}

\begin{proof} Suppose that $\{u_{n}\}\subset W^{1,p}(\Omega)$,
for every $c>0$,
$$
\varphi(u_{n})\to c,\text{ } \varphi'(u_{n})\to 0  \text{ in }
 W^{1,p}(\Omega)^{*} \text{ as }n\to\infty.
$$

 First we prove the boundness of
$\{u_{n}\}$. After integrating, we obtain from the assumptions
(F2) and (G2) that there exist $C_{1}$, $C_{2}>0$ such
that
\begin{gather}
C_{1}(|u|^{\alpha_{1}}-1)\leq F(x,u)\quad
\text{for all }x\in\Omega,\; u\in\mathbb{R}, \label{e5}\\
C_{2}(|u|^{\alpha_{2}}-1)\leq G(x,u)\quad
\text{for all }x\in\partial\Omega,\; u\in\mathbb{R}. \label{e6}
\end{gather}
Set $\alpha=\min\{\alpha_{1},\alpha_{2}\}$ and choose
$\frac{1}{\beta}\in(\frac{1}{\alpha},\frac{1}{p})$ , and from \eqref{e5}
and \eqref{e6}, we obtain for $n$ sufficiently large,
\begin{align*}
& c + 1 + \|u_{n}\|_{1,p}\\
&\geq \varphi(u_{n})-\frac{1}{\beta}\langle\varphi'(u_{n}),u_{n}\rangle\\
&\geq (\frac{1}{p}-\frac{1}{\beta})\|u_{n}\|_{1,p}^{p}+(\frac{\alpha_{1}}{\beta}-1)\int_{\Omega}F(x,u_{n})\mathrm{d}x
-(\frac{\alpha_{2}}{\beta}-1)\int_{\partial\Omega}G(x,u_{n})\mathrm{d}S\\
&\geq (\frac{1}{p}-\frac{1}{\beta})\|u_{n}\|_{1,p}^{p}+C_{1}(\frac{\alpha_{1}}{\beta}-1)|u_{n}|_{\alpha_{1}}^{\alpha_{1}}+C_{2}(\frac{\alpha_{2}}{\beta}-1)|u_{n}|_{_{L^{\alpha_{2}}(\partial\Omega)}}^{\alpha_{2}}-C_{3}.
\end{align*}
 Note that
$\frac{\alpha_{i}}{\beta}-1>0 (i=1,2)$, then $\{u_{n}\}$ is
bounded in $W^{1,p}(\Omega)$.

 Next we show that the strongly
convergence of $\{u_{n}\}$ in $W^{1,p}(\Omega)$. Since $\{u_{n}\}$
is bounded, up to a subsequence (which we still denote by
$\{u_{n}\}$), we may assume  that there exists $u\in W^{1,p}(\Omega)$
such that  $u_{n}\rightharpoonup u$ weakly in $W^{1,p}(\Omega)$ as
 $n\to \infty$. Note that
$\varphi'(u_{n})=A(u_{n})-B(u_{n})-C(u_{n})\to  0$.
 By the compactness of $B$, $C$ and the continuity of $A^{-1}$, we have
$$
u_{n}\to A^{-1}(a(x)B(u)-C(u)) \text{ in } W^{1,p}(\Omega)\text{ as
} n\to \infty.
$$
Thus $u_{n}\to u$ in $W^{1,p}(\Omega)$.
\end{proof}

To prove Theorem \ref{thm1.1} we also need the following two lemmas.

\begin{lemma}[\cite{FH}] \label{lem3.5}
 If $1\leq q<p^{*}$=$\frac{Np}{N-p}$, then
$$
\beta_{k}:=\sup_{u\in Z_{k},\, \|u\|_{1,p}=1} | u|_{q}\to 0\text{ as
} k\to\infty.
$$
\end{lemma}

\begin{lemma}[\cite{FH}] \label{lem3.6}
 If $1\leq z<p_{*}$=$\frac{(N-1)p}{N-p}$, then
$$
\sigma_{k}:=\sup_{u\in Z_{k},\,\|u\|_{1,p}=1}
|u|_{L^{z}(\partial\Omega)}\to 0 \text{ as }  k\to \infty.
$$
\end{lemma}

\begin{remark} \label{rmk3.1} \rm
 In \cite{FH}, the authors gave a
more general form of two above lemmas. Here the key step of the
proof of these two lemmas is that the Sobolev embedding
$W^{1,p}(\Omega)\hookrightarrow L^{q}(\Omega)$ is compact for
$1\leq q<p^{*}$ and the Sobolev trace embedding
$W^{1,p}(\Omega)\hookrightarrow L^{z}(\partial\Omega)$ is compact
for $1\leq z<p_{*}$.
\end{remark}

\subsection*{The proof of Theorem \ref{thm1.1}} \quad

 Assumptions (F1) and
(G1) and the Lemma \ref{lem3.4} imply that $\varphi$ is continuously
differentiable on $W^{1,p}(\Omega)$ and satisfies the $(PS)_{c}$
condition for every $c>0$. So from Theorem \ref{thm2.1}, we need only to
verify $\phi$ satisfying the condition (A1) and (A2).

 As for \eqref{e5} and \eqref{e6} in
Lemma \ref{lem3.4}, we have
$$
\varphi(u)\leq\frac{\|u_{n}\|_{1,p}^{p}}{p}-C_{1}|
u|_{\alpha_{1}}^{\alpha_{1}}-C_{2}|u|_{L^{\alpha_{2}}
(\partial\Omega)}^{\alpha_{2}}-C_{1}|\Omega|-C_{2}|\partial\Omega|.
$$
Since on the finite-dimensional space $Y_{k}$ all norms are
equivalent, so $\alpha_{i}>p$ ($i=1,2$) implies that (A1) is
satisfied for $\rho_{k}>0$ large enough.

 After integrating, we obtain from the
assumptions (F1) and (G1) that there exist constants
$C_{1}, C_{2}>0$ such that
$$
F(x,u)\leq C_{1}(1+|u|^{q}),\quad G(x,u)\leq C_{2}(1+|u|^{z}).
$$
Let us define
$$
\beta_{k}:=\sup_{u\in Z_{k},\, |u\|_{1,p}=1} |u|_{q},\quad\,d\sigma_{k}
:=\sup_{u\in Z_{k},\, \|u\|_{1,p}=1} |u|_{L^{z}(\partial\Omega)}.
$$
On $Z_{k}$, we have
\begin{align*}
\varphi(u)&=\frac{1}{p}\|u\|_{1,p}^{p}-\int_{\Omega}F(x,u)\mathrm{d}x
-\int_{\partial\Omega}G(x,u)\mathrm{d}S\\
&\geq \frac{1}{p}\|u\|_{1,p}^{p}-C_{1}|
u|_{q}^{q}-C_{1}|\Omega|-C_{2}|u|_{L^{z}(\partial\Omega)}^{z}-C_{2}|\partial\Omega|\\
&\geq \frac{1}{p}\|u\|_{1,p}^{p}-C_{1}\beta_{k}^{q}\|u\|_{1,p}^{q}-C_{2}\sigma_{k}^{z}\|u\|_{1,p}^{z}
-C_{3}.
\end{align*}
Let
\begin{gather*}
\frac{1}{4p}\rho^{p}-C_{1}\beta_{k}^{q}\rho^{q}=0, \\
\frac{1}{4p}\rho^{p}-C_{2}\sigma_{k}^{z}\rho^{z}=0. % 7,8
\end{gather*}
 From these two  equations, we have
$\rho_{k}:=(4pC_{1}\beta_{k}^{q})^{\frac{1}{p-q}}$,
$\rho_{k}':=(4pC_{2}\sigma_{k}^{z})^{\frac{1}{p-z}}$.
 From Lemmas \ref{lem3.5} and \ref{lem3.6} we know that
$\beta_{k}\to 0$, $\sigma_{k}\to 0$ as $k\to \infty$. So, we know
\begin{equation} \label{e9}
\rho_{k}\to \infty  \text{ as } k\to \infty,\quad
 \rho_{k}'\to \infty  \text{ as }
k\to \infty.
\end{equation}
Let
\begin{equation} \label{e10}
\gamma_{k}=\min\{\rho_{k},\rho_{k}'\},
\end{equation}
we obtain, if $u\in Z_{k}$ and $\|u\|_{1,p}=\gamma_{k}$, then
$\varphi(u)\geq \frac{1}{2p}\gamma_{k}^{p}-C_{3}$.
From \eqref{e9} and \eqref{e10}, so (A2)
is proved. It suffices then to use the fountain theorem to complete the
proof.

 Here, we show two examples for readers for special cases of $f$ to
understand our theorem.

\begin{example} \label{exa3.1} \rm
 Let $p<q<p^{*}$ and assumed (G1)--(G3) are satisfied. We consider the
quasilinear elliptic equation % \label{e11}
\begin{gather*}
-\Delta_{p}u+a(x)|u|^{p-2} u=|u|^{q-2}u \quad \text{in } \Omega,\\
|\nabla u|^{p-2} \frac{\partial u}{\partial \nu}=g(x,u)
\quad \text{on } \partial\Omega.
\end{gather*}
This problem has a sequence of solutions $\{u_{k}\}$ such
that $\varphi(u_{k})\to \infty$ as $k\to \infty$.
\end{example}

\begin{example} \label{exa3.2} \rm
Let $1<r<p<s<p^{*}$ and assumed
(G1)--(G3) are satisfied. We consider the
quasilinear elliptic equation %\label{12}
\begin{gather*}
-\Delta_{p}u+a(x)|u|^{p-2} u=\mu |u|^{r-2}u+\lambda| u|^{s-2}u \quad
\text{in } \Omega,\\
|\nabla u|^{p-2} \frac{\partial u}{\partial \nu}=g(x,u)
\quad \text{on } \partial\Omega.
\end{gather*}
Then  for every $\lambda >0$, $\mu \in \mathbb{R}$,
this problem  has a sequence of solutions $\{u_{k}\}$ such that
$\varphi(u_{k})\to \infty$ as $k\to \infty$.
\end{example}

\subsection*{Proof of Theorem \ref{thm1.2}}

  The first conclusion of Theorem \ref{thm1.2} is
just example \ref{exa3.2}. We shall
 prove Theorem \ref{thm1.2} by using Theorem \ref{thm2.2},
so we need to verify the condition
 (B1)--(B4). Now we assume that $\mu>0$ .

 To verify (B1), we define
$\beta_{k}:=\underset{\substack {u\in Z_{k}\\
\|u\|_{1,p}=1}}{\sup}| u|_{r}$. From the assumptions (G1) and
(G4), we have $G(x,u)\leq \varepsilon|u|^{p}+C|u|^{z}$, where
$\varepsilon\to 0$ as $|u|\to 0$. So from the
Sobolev trace embedding, we have
\begin{align*}
\varphi(u)&=\frac{1}{p}\|u\|_{1,p}^{p}-\frac{\mu}{r}|u|_{r}^{r}-\frac{\lambda}{s}|u|_{s}^{s}
-\int_{\partial\Omega}G(x,u)\mathrm{d}S\\
&\geq \frac{1}{p}\|u\|_{1,p}^{p}-\frac{\mu}{r}\beta_{k}^{r}\|u\|_{1,p}^{r}-\frac{\lambda}{s}C_{1}\|u\|_{1,p}^{s}
-\varepsilon|u|_{L^{p}(\partial\Omega)}^{p}-C_{2}|u|_{L^{z}(\partial\Omega)}^{z}\\
&\geq \frac{1}{p}\|u\|_{1,p}^{p}-\frac{\mu}{r}\beta_{k}^{r}\|u\|_{1,p}^{r}-\frac{\lambda}{s}C_{1}\|u\|_{1,p}^{s}
-\varepsilon C_{3}\|u\|_{1,p}^{p}-C_{4}\|u\|_{1,p}^{z}.\\
&\geq (\frac{1}{p}-\varepsilon
C_{3})\|u\|_{1,p}^{p}-\frac{\mu}{r}\beta_{k}^{r}\|u\|_{1,p}^{r}-\frac{\lambda}{s}C_{1}\|u\|_{1,p}^{s}
-C_{4}\|u\|_{1,p}^{z}.
\end{align*}
 Since $p<s$ and $p<z$, there exists $R>0$ such that
$\|u\|_{1,p}\leq R$. We have
\begin{gather*}
\frac{1}{4p}\|u\|_{1,p}^{p}-\frac{\lambda}{s}C_{1}\|u\|_{1,p}^{s}\geq
0,\\
\frac{1}{4p}\|u\|_{1,p}^{p}-C_{4}\|u\|_{1,p}^{z}\geq 0.
\end{gather*}
From these two inequalities, it follows that
\begin{equation} \label{e15}
\varphi(u)\geq(\frac{1}{2p}-\varepsilon
C_{3})\|u\|_{1,p}^{p}-\frac{\mu}{r}\beta_{k}^{r}\|u\|_{1,p}^{r} .
\end{equation}
Choose $\varepsilon$ so small such that
$\frac{1}{2p}-\varepsilon C_{3}>0$  and let
$\rho_{k}:=[(\frac{1}{2p}-\varepsilon
C_{3})^{-1}\frac{\mu}{r}\beta_{k}^{r}]^{\frac{1}{p-r}}$,
by Lemma \ref{lem3.5}, $\beta_{k}\to 0$ as $k\to \infty$,
it follows that $\rho_{k}\to 0$ as $k\to \infty$. So there
exists $k_{0}$ such that $\rho_{k}\leq R$ when $k\geq k_{0}$. Thus,
for $k\geq k_{0}$, $u\in Z_{k}$ and $\|u\|_{1,p}=\rho_{k}$, we have
$\varphi(u)\geq 0$ and (B1) is proved.

 From (G2) we know there exists $C>0$
such that $ C(|u|^{\alpha_{2}}-1)\leq G(x,u)$. Then, we have
\begin{align*}
\varphi(u)&=\frac{1}{p}\|u\|_{1,p}^{p}-\frac{\mu}{r}|u|_{r}^{r}
-\frac{\lambda}{s}|u|_{s}^{s}
-\int_{\partial\Omega}G(x,u)\mathrm{d}S\\
&\leq \frac{1}{p}\|u\|_{1,p}^{p}-\frac{\mu}{r}|u|_{r}^{r}
 -\frac{\lambda}{s}|u|_{s}^{s}
-C|u|_{L^{\alpha_{2}}(\partial\Omega)}^{\alpha_{2}}-C|\partial\Omega|.
\end{align*}
Since on the finite dimensional space $Y_{k}$ all norms are
equivalent, as $r<p$, so if $\mu>0$ then (B2) is satisfied
for every $r_{k}>0$ small enough.

We obtain from \eqref{e15}, for $k\geq k_{0}$, $u\in
Z_{k},\|u\|_{1,p}\leq\rho_{k}$, $\varphi(u)\geq
-\frac{\mu}{r}\beta_{k}^{r}\rho_{k}^{r}$, since $\beta_{k}\to 0$ and
$\rho_{k}\to 0$ as $k\to \infty$, (B3) is also satisfied.

 Finally we proved the $(PS)_{c}^{*}$
condition. Consider a sequence $u_{n_{j}}\in Y_{n_{j}}$ such that
$$
\varphi(u_{n_{j}})\to c,\text{ }
 \varphi|_{Y_{n_{j}}}'(u_{n_{j}})\to 0 \text{ in }
W^{1,p}(\Omega)^{*} \text{ as }  n_{j}\to \infty.
$$
For $n_{j}$ big enough, let $\zeta=\min\{s,\alpha_{2}\}$
and choose $\frac{1}{\beta}\in (\frac{1}{\zeta},\frac{1}{p})$. Now
as $\lambda\geq0$ we have
\begin{align*}
 c + 1 + \|u_{n_{j}}\|_{1,p}
&\geq \varphi(u_{n_{j}})-\frac{1}{\beta}\langle\varphi'(u_{n_{j}});
 u_{n_{j}}\rangle\\
&\geq (\frac{1}{p}-\frac{1}{\beta})\|u_{n_{j}}\|_{1,p}^{p}
  -\mu(\frac{1}{r}-\frac{1}{\beta})|u_{n_{j}}|_{r}^{r}.
\end{align*}
We can obtain the boundness of $(u_{n_{j}})$ in
$W^{1,p}(\Omega)$ since $1<r<p$. Going if necessary to a
subsequence, we can assume that $u_{n_{j}}\rightharpoonup u$ in
$W^{1,p}(\Omega)$. As in Lemma \ref{lem3.4}, it is easy to conclude,
$u_{n_{j}}\to u$ in $W^{1,p}(\Omega)$ and $\varphi'(u)=0$.
%\end{proof}

To obtain much more general conclusion of Theorem \ref{thm1.2}, we make the
following assumption.
\begin{itemize}
\item[(G2'')]  There exists $\alpha_{2}\geq s$ such that
$$
|u|\geq R\Longrightarrow 0<\alpha_{2} G(x,u)\leq ug(x,u)\quad
\text{for all } x\in \partial\Omega.
$$
\end{itemize}

\begin{corollary}\label{coro3.1}
Let $1<r<p<s<p^{*}$ and under the assumptions (G1), (G2''),
(G3) and (G4). We consider the  quasilinear
elliptic equation
\begin{equation} \label{e16}
\begin{gathered}
-\Delta_{p}u+a(x)|u|^{p-2} u=\mu |u|^{r-2}u+\lambda| u|^{s-2}u
\quad \text{in }\Omega,\\
|\nabla u|^{p-2} \frac{\partial u}{\partial \nu}=g(x,u)
\quad \text{on } \partial\Omega.
\end{gathered}
\end{equation}
Then
\begin{enumerate}
\item  for every $\lambda>0$, $\mu\in \mathbb{R}$, problem
\eqref{e16} has a sequence of solutions ${u_{k}}\in W^{1,p}(\Omega)$ such
that $\varphi(u_{k})\to \infty $ as $k\to \infty$,

\item for every $\mu>0$, $\lambda\in\mathbb{R}$, problem
\eqref{e16} has a sequence of solutions ${v_{k}}\in W^{1,p}(\Omega)$ such
that $\varphi(v_{k})<0$, $\varphi(v_{k})\to 0 $ as
$k\to \infty$.
\end{enumerate}
\end{corollary}

\begin{proof} We need only to prove the boundness of
$\{u_{n_{j}}\}$ in $(PS)_{c}^{*}$ sequence. Consider a sequence
$u_{n_{j}}\in Y_{n_{j}}$ such that
$$
\varphi(u_{n_{j}})\to c,\text{ }
 \varphi|_{Y_{n_{j}}}'(u_{n_{j}})\to 0 \text{ in } W^{1,p}(\Omega)^{*}
\text{ as } n_{j}\to\infty.
$$
For $n_{j}$ big enough, from (G2'') we have
\begin{align*}
&  c + 1 + \|u_{n_{j}}\|_{1,p}\\
&\geq \varphi(u_{n_{j}})-\frac{1}{s}\langle\varphi'(u_{n_{j}});u_{n_{j}}
 \rangle\\
&= (\frac{1}{p}-\frac{1}{s})\|u_{n_{j}}\|_{1,p}^{p}
  -\mu(\frac{1}{r}-\frac{1}{s})|u_{n_{j}}|_{r}^{r}
+\int_{\partial\Omega}(\frac{1}{s}g(x,u_{n_{j}})u_{n_{j}}-G(x,u_{n_{j}}))\mathrm{d}S\\
&\geq (\frac{1}{p}-\frac{1}{s})\|u_{n_{j}}\|_{1,p}^{p}
  -\mu(\frac{1}{r}-\frac{1}{s})|u_{n_{j}}|_{r}^{r}.
\end{align*}
We obtain the boundness of $\{u_{n_{j}}\}$ in $W^{1,p}(\Omega)$
since $1<r<p$.
\end{proof}

\section{Critical cases}
\subsection*{Critical case 1}
In this subsection and next subsection, we study that $f$ has the
critical growth with superlinear or sublinear perturbation in
problem \eqref{e1}. In these cases, we know all of conditions of Theorem
\ref{thm1.2} are satisfied and we need only to prove the $(PS)_{c}$
condition. However, noticing that the inclusion
  $W^{1,p}(\Omega)\hookrightarrow L^{p^{*}}(\Omega)$ is only continuous but
 not compact, we can no longer expect the $(PS)_{c}$ condition to
 be
 hold. Thanks to the concentration-compactness principle in \cite{L1,L2}. We can prove a local $(PS)_{c}$ condition
  that will hold true for $\varphi(u)$ below a certain value of energy. The
proof of Theorem \ref{thm1.3} and Theorem \ref{thm1.4}
 is similar to \cite{BR1}. We
write here for the readers convenience. Now, let
$f(x,u)=|u|^{p^{*}-2}u+\lambda|u|^{r-2}u$ with $p<r<p^{*}$ and
$\lambda$ is a positive parameter.

\begin{lemma}[\cite{L1,L2}]\label{lem4.1}
 Let $\{u_{j}\}$ be a weakly convergent sequence in
$W^{1,p}(\Omega)$ with weak limit $u$ such that
$|\nabla u_{j}|^{p}\rightharpoonup d\mu$,
$|u_{j}|^{p^{*}}\rightharpoonup \,d\sigma$ weakly convergent
in the sense of measures. Then there exist
$x_{1},x_{2},\dots, x_{l}\in \Omega$
such that
\begin{gather*}
\,d\sigma=|u|^{p^{*}} + \sum_{j=1}^{l}\sigma_{j}\delta_{x_{j}},\quad
\sigma_{j}>0,\\
d\mu\geq|\nabla u|^{p}+\sum_{j=1}^{l}\mu_{j}\delta_{x_{j}},\quad \mu_{j}>0,\\
(\sigma_{j})^{\frac{p}{p^{*}} }\leq \frac{\mu_{j}}{S}.
\end{gather*}
\end{lemma}

Now, we can prove a local $(PS)_{c}$ condition by using Lemma \ref{lem4.1}.

\begin{lemma}\label{lem4.2}
Let $\{u_{j}\}\subset W^{1,p}(\Omega)$ be a $(PS)_{c}$
sequence for $\varphi$ with energy level $c$. If
$c<(\frac{1}{p}-\frac{1}{p^{*}})S^{\frac{p^{*}}{p^{*}-p}}$, where
$S$ is the best constant in the Sobolev embedding
$W^{1,p}(\Omega)\hookrightarrow L^{p^{*}}(\Omega)$, then there
exists a subsequence that converges strongly in $W^{1,p}(\Omega)$.
\end{lemma}

\begin{proof}
Let $\{u_{j}\}$ be a $(PS)_{c}$ sequence,
it follows that $\{u_{j}\}$ is bounded in $W^{1,p}(\Omega)$(see
Lemma \ref{lem3.4}). By Lemma \ref{lem4.1}, there exists a subsequence, that we
still denote $\{u_{j}\}$ such that
\begin{gather}
u_{j}\rightharpoonup u  \text{ weakly in }
W^{1,p}(\Omega), \nonumber \\
u_{j}\to u \text{ strongly in }
L^{r}(\Omega),\quad 1<r<p^{*},\nonumber \\
u_{j}\to u  \text{ a.e. in } \Omega, \nonumber\\
|\nabla u_{j}|^{p}\rightharpoonup d\mu\geq|\nabla
u|^{p}+\sum_{k=1}^{l}\mu_{k}\delta_{x_{k}}, \quad\mu_{k}>0, \label{e17}\\
|u_{j}|^{p^{*}}\rightharpoonup \,d\sigma=|u|^{p^{*}}+
\sum_{k=1}^{l}\sigma_{k}\delta_{x_{k}}, \quad \sigma_{k}>0. \label{e18}
\end{gather}
Choose $\phi\in C_{0}^{\infty}(\mathbb{R^{N}})$ such that
$$
\phi \equiv 1 \text{ in } B(x_{k},\varepsilon), \quad \phi \equiv 0
\text{ in } B(x_{k},2\varepsilon)^{c},\quad |\nabla\phi|\leq
\frac{2}{\varepsilon},
$$
 where $x_{k}$ belongs to the support of $\,d\sigma$.
Considering $\{{u_{j}\phi}\}$, it is easy to see this sequence is
bounded in $W^{1,p}(\Omega)$. Since $\varphi'(u_{j})\to 0$
in $W^{1,p}(\Omega)^{*}$ as $j\to \infty$, we obtain that
\begin{equation} \label{e19}
\lim_{j\to\infty} \langle\varphi'(u_{j});\phi
u_{j}\rangle=0.
\end{equation}
Then from \eqref{e17} and \eqref{e18}, we obtain
\begin{align*}
& \lim_{j\to \infty} \int_{\Omega}|\nabla u_{j}|^{p-2}\nabla
u_{j}\nabla\phi u_{j}\mathrm{d}x\\
&=\int_{\Omega}\phi \,d\sigma+\lambda\int_{\Omega}|u|^{r}\phi
\mathrm{d}x+\int_{\partial\Omega}ug(x,u)\phi
\mathrm{d}S-a(x)\int_{\Omega}|u|^{p}\phi
\mathrm{d}x-\int_{\Omega}\phi d\mu.
\end{align*}
 Now, by H\"{o}lder inequality and weak convergence, we obtain
\begin{align*}
0&\leq\lim_{j\to \infty} |\int_{\Omega}|\nabla
u_{j}|^{p-2}\nabla u_{j}\nabla\phi
u_{j}\mathrm{d}x|\\
&\leq \lim_{j\to \infty} (\int_{\Omega}|\nabla
u_{j}|^{p}\mathrm{d}x)^{\frac{p-1}{p}}(\int_{\Omega}|\nabla\phi|^{p}|
u_{j}|^{p}\mathrm{d}x)^{\frac{1}{p}}\\
&\leq C(\int_{B(x_{k},2\varepsilon)\cap \Omega}|\nabla\phi|^{p}|
u|^{p}\mathrm{d}x)^{\frac{1}{p}}\\
&\leq C(\int_{B(x_{k},2\varepsilon)\cap
\Omega}|\nabla\phi|^{N}\mathrm{d}x)^{\frac{1}{N}}(\int_{B(x_{k},2\varepsilon)\cap
\Omega}| u|^{\frac{Np}{N-p}}\mathrm{d}x)^{\frac{N-p}{Np}}\\
&\leq  C(\int_{B(x_{k},2\varepsilon)\cap \Omega}|
u|^{\frac{Np}{N-p}}\mathrm{d}x)^{\frac{N-p}{Np}}\to 0 \quad
\text{as}\quad\varepsilon \to 0.
\end{align*}
Then from \eqref{e19} we have
\begin{align*} %20
&\lim_{\varepsilon\to 0}\big[\int_{\Omega}\phi \,d\sigma+\lambda\int_{\Omega}|
u|^{r}\phi \mathrm{d}x+\int_{\partial\Omega}ug(x,u)\phi
\mathrm{d}S-a(x)\int_{\Omega}| u|^{p}\phi
\mathrm{d}x-\int_{\Omega}\phi d\mu\big]\\
&= \sigma_{k}-\mu_{k}=0.
\end{align*}
Then from Lemma \ref{lem4.1}, we have that
$(\sigma_{k})^{p/p^*}S\leq \mu_{k}$.
 Therefore by the above equality,
\begin{equation*}
(\sigma_{k})^{p/p^*}S\leq \sigma_{k}.
\end{equation*}
Then, either $\sigma_{k}=0$ or
\begin{equation*} %21
\sigma_{k}\geq S^{\frac{p*}{p*-p}}.
\end{equation*}
If this inequality occurs for some $k_{0}$, then, from the fact that
$\{u_{j}\}$ is a $(PS)_{c}$ sequence and from (G2) we obtain
\begin{align*}
c&=\lim_{j\to \infty} \varphi(u_{j}) =
\lim_{j\to \infty} \varphi(u_{j})-\frac{1}{p}\langle\varphi'(u_{j});u_{j}\rangle\\
&\geq (\frac{1}{p}-\frac{1}{p^{*}})\int_{\Omega}|
u|^{p^{*}}\mathrm{d}x
+(\frac{1}{p}-\frac{1}{p^{*}})S^{\frac{p*}{p*-p}}+\lambda
(\frac{1}{p}-\frac{1}{r})\int_{\Omega}|
u|^{r}\mathrm{d}x\\
&\geq (\frac{1}{p}-\frac{1}{p^{*}})S^{\frac{p*}{p*-p}},
\end{align*}
 which contradicts our hypothesis. Since
$c<(\frac{1}{p}-\frac{1}{p^{*}})S^{\frac{p*}{p*-p}}$, it follows
that
$$
\int_{\Omega}|u_{j}|^{p^{*}} \mathrm{d}x\to
\int_{\Omega}|u|^{p^{*}}\mathrm{d}x,
$$
so we have $u_{j}\to u$ in $L^{p^{*}}(\Omega)$. Now the proof is
complete with the continuity of the operator
$A^{-1}$.
\end{proof}

\subsection*{Proof of Theorem \ref{thm1.3}}

 We want to obtain our result by using
mountain pass theorem. First from the assumption (G1) and
(G4), we have
\begin{equation} \label{e22}
G(x,u)\leq \varepsilon|u|^{p}+C|u|^{z},
\end{equation}
where $\varepsilon\to 0$ as $|u|\to 0$. From the Sobolev
embedding theorem and Sobolev trace inequality, we have
\begin{align*}
\varphi(u)
&\geq \frac{1}{p}\|u\|_{1,p}^{p}-\frac{1}{p^{*}}\int_{\Omega}|
u|^{p^{*}}\mathrm{d}x-\frac{\lambda}{r}\int_{\Omega}|
u|^{r}\mathrm{d}x-\varepsilon\int_{\partial\Omega}|u|^{p}\mathrm{d}S-C_{1}\int_{\partial\Omega}|u|^{z}\mathrm{d}S\\
&\geq \frac{1}{p}\|u\|_{1,p}^{p}-\frac{1}{p^{*}}\int_{\Omega}|
u|^{p^{*}}\mathrm{d}x-\frac{\lambda}{r}\int_{\Omega}|
u|^{r}\mathrm{d}x-\varepsilon C_{2}\|u\|_{1,p}^{p}-
C_{3}\|u\|_{1,p}^{z}\\
&\geq (\frac{1}{p}-\varepsilon
C_{2})\|u\|_{1,p}^{p}-\frac{1}{p^{*}}\int_{\Omega}|
u|^{p^{*}}\mathrm{d}x-\frac{\lambda}{r}\int_{\Omega}|
u|^{r}\mathrm{d}x-
C_{3}\|u\|_{1,p}^{z}\\
&\geq (\frac{1}{p}-\varepsilon
C_{2})\|u\|_{1,p}^{p}-\frac{1}{p^{*}}S^{p^{*}}\|u\|_{1,p}^{p^{*}}-\frac{\lambda}{r}C_{4}\|u\|_{1,p}^{r}-
C_{3}\|u\|_{1,p}^{z}.
\end{align*}
Choose $\varepsilon>0$
sufficiently small such that $\frac{1}{p}-\varepsilon C_{2}>0$
and let
\begin{equation}g(t)=(\frac{1}{p}-\varepsilon
C_{2})t^{p}-\frac{1}{p^{*}}S^{p^{*}}t^{p^{*}}
-\frac{\lambda}{r}C_{4}t^{r}-C_{3}t^{z},
\end{equation}
it is easy to check that $g(R)>r>0$ for some $R$ sufficiently small
since $p<\min\{r, p^{*}, z\}$. On the other hand, since $p<\min\{r,
p^{*}, z\}$, so for fixed $\omega\in {W^{1,p}(\Omega)}$ with
$\omega|_{\Omega}\neq 0$, we have
$\lim_{t\to \infty} \varphi(t\omega)=-\infty$. Then there exists $v_{0}\in
W^{1,p}(\Omega)$ such that $\|v_{0}\|_{1,p}>R$ and
$\varphi(v_{0})<r$. So according to the mountain pass Theorem, we
know the critical value is
$c:=\inf_{\phi\in\Gamma} \sup_{t\in [0,1]}
\varphi(\phi(t))$, where $ \Gamma=\{\phi:[0,1]\to
W^{1,p}(\Omega)$ is continuous and $\phi(0)=0$, $\phi(1)=v_{0} \}$.
Now the problem is to show that
$c<(\frac{1}{p}-\frac{1}{p^{*}})S^{\frac{p^{*}}{p^{*}-p}}$ and we
want to apply the local $(PS)_{c}$ condition. For this purpose we
fix $\omega\in {W^{1,p}(\Omega)}$ with $\|\omega\|_{p*}=1$ and
define $h(t)=\varphi(t\omega)$. Let us calculate the maximum of $h$.
Since $\underset{t\to \infty}{\lim}h(t)=-\infty$, it follows
that there exists a $t_{\lambda}>0$  such that
$\sup_{t>0}\varphi(t\omega)=h(t_{\lambda})$.
Differentiating $h$, we obtain
\begin{equation} \label{e24}
0=h'(t_{\lambda})=t_{\lambda}^{p-1}\|\omega\|_{1,p}^{p}-
t_{\lambda}^{p^{*}-1}-t_{\lambda}^{r-1}\lambda|\omega|_{r}^{r}-
\int_{\partial\Omega}g(x,t_{\lambda}\omega)\omega \mathrm{d}S.
\end{equation}
 From assumptions (G1) and (G4), we obtain
\begin{align*}
|\int_{\partial\Omega}g(x,t_{\lambda}\omega)\omega \mathrm{d}S|
&\leq \int_{\partial\Omega}|g(x,t_{\lambda}\omega)||\omega|\mathrm{d}S\\
&\leq \varepsilon t_{\lambda}^{p-1}\int_{\partial\Omega}|\omega|^{p}
  \mathrm{d}S+C_{1}t_{\lambda}^{z-1}\int_{\partial\Omega}|\omega|^{z}
  \mathrm{d}S\\
&=\varepsilon t_{\lambda}^{p-1}|\omega|_{L^{p}(\partial\Omega)}^{p}
  +C_{1}t_{\lambda}^{z-1}|\omega|_{L^{z}(\partial\Omega)}^{z}.
\end{align*}
From \eqref{e24},
$$
t_{\lambda}^{p-1}\|\omega\|_{1,p}^{p}-
t_{\lambda}^{p^{*}-1}-t_{\lambda}^{r-1}\lambda|\omega|_{r}^{r}-\varepsilon
t_{\lambda}^{p-1}|\omega|_{L^{p}(\partial\Omega)}^{p}
-C_{1}t_{\lambda}^{z-1}|\omega|_{L^{z}(\partial\Omega)}^{z}\leq 0.
$$
Then
\begin{equation} \label{e25}
t_{\lambda}^{p^{*}-p}+t_{\lambda}^{r-p}\lambda|\omega|_{r}^{r}
+C_{3}t_{\lambda}^{z-p}\|\omega\|_{1,p}^{z}\leq
(1-\varepsilon C_{2})\|\omega\|_{1,p}^{p}.
\end{equation}
Hence,
$t_{\lambda}\leq C\|\omega\|_{1,p}^{\frac{p}{p^{*}-p}}$. So from
\eqref{e25},
$t_{\lambda}^{p^{*}-r}+\lambda|\omega|_{r}^{r}+C_{3}t_{\lambda}^{z-r}\|\omega\|_{1,p}^{z}\to
\infty$ as $\lambda\to \infty$. we obtain
\begin{equation} \label{e26}
\underset{\lambda\to \infty}{\lim}t_{\lambda}=0.
\end{equation}
On the other hand, it is easy to check that if
$\lambda>\overline{\lambda}$ we could have
$\varphi(t_{\overline{\lambda}}\omega)\geq
\varphi(t_{\lambda}\omega)$. So by \eqref{e26}, we get
$\lim_{\lambda\to \infty} \varphi(t_{\lambda}\omega)=0$.
But this equality means that there exists a constant $\lambda_{0}>0$
such that if $\lambda>\lambda_{0}$, then
$\underset{t\geq 0}{\sup}\varphi(t\omega)
 <(\frac{1}{p}-\frac{1}{p^{*}})S^{\frac{p^{*}}{p^{*}-p}}$.
We choose $v_{0}=t_{0}\omega$ with $t_{0}$ sufficiently large to
have $\varphi(t_{0}\omega)<0$. This completes  the
proof.

\subsection{Critical case 2}

In this subsection we study $f$ has critical and sublinear terms
in problem \eqref{e1}, that is, $f(x,u)=|u|^{p^{*}-2}u+\lambda|u|^{r-2}u$
with $1<r<p$ and $\lambda$ a positive parameter. By applying the
variational approach, we will show the existence of
 infinitely many nontrivial critical points of the associated functional
$\varphi$  when $\lambda$ is small enough. First we use Lemma \ref{lem4.1}
to prove local $(PS)_{c}$ condition.

\begin{lemma}\label{lem4.3}
Let $\{u_{j}\}\subset W^{1,p}(\Omega)$ be a $(PS)_{c}$ sequence
for $\varphi$ with energy level $c$. If
$c<(\frac{1}{p}-\frac{1}{p^{*}})S^{\frac{p^{*}}{p^{*}-p}}
-K\lambda^{\frac{p^{*}}{p^{*}-r}}$,
where $K$ depend only on $p,r,N$ and $|\Omega|$, then there exists
a subsequence that converges strongly in $W^{1,p}(\Omega)$.
\end{lemma}

\begin{proof}
Let $\{u_{j}\}$ be a $(PS)_{c}$ sequence; that is,
$$
\varphi(u_{j})\to c,\text{ } \varphi'(u_{j})\to 0  \text{ in }
W^{1,p}(\Omega)^{*}\text{ as } j\to \infty.
$$
From Lemma \ref{lem3.4} it follows immediately that
$\{u_{j}\}$ is bounded in $W^{1,p}(\Omega)$. Then there exists a
subsequence (we still denote $\{u_{j}\}$) which is weakly convergent
to $u$ in $W^{1,p}(\Omega)$. We need to prove $\{u_{j}\}$ is
strongly convergent to $u$ in $L^{p^{*}}(\Omega)$. For this purpose,
suppose $\{u_{j}\}$ is not strongly convergent to $u$ in
$L^{p^{*}}(\Omega)$, then from (G2) we have
\begin{align*}
c&=\lim_{j\to \infty} \varphi(u_{j})
=\lim_{j\to \infty} \varphi(u_{j})-\frac{1}{p}\langle\varphi'(u_{j});u_{j}\rangle\\
&=\lim_{j\to \infty} \Big(\big(\frac{1}{p}\int_{\Omega}| \nabla
u_{j}|^{p}+a(x)| u_{j}|^{p}\mathrm{d}x
-\frac{1}{p^{*}}\int_{\Omega}| u_{j}|^{p^{*}}\mathrm{d}x
-\frac{\lambda}{r}\int_{\Omega}| u_{j}|^{r}\mathrm{d}x\\
&\quad -\int_{\partial\Omega}G(x,u_{j})\mathrm{d}S\big)
-\big(\frac{1}{p}\int_{\Omega}| \nabla u_{j}|^{p}+a(x)|
u_{j}|^{p}\mathrm{d}x\\
&\quad -\frac{1}{p}\int_{\Omega}| u_{j}|^{p^{*}}\mathrm{d}x
-\frac{\lambda}{p}\int_{\Omega}| u_{j}|^{r}\mathrm{d}x
-\int_{\partial\Omega}\frac{1}{p}g(x,u_{j}u_{j})\mathrm{d}S\big)\Big)\\
&\geq (\frac{1}{p}-\frac{1}{p^{*}})\int_{\Omega}|
u|^{p^{*}}\mathrm{d}x
+(\frac{1}{p}-\frac{1}{p^{*}})S^{\frac{p*}{p*-p}}+\lambda
(\frac{1}{p}-\frac{1}{r})\int_{\Omega}| u|^{r}\mathrm{d}x.
\end{align*}
 Now, applying H\"{o}lder inequality, we find
\begin{align*}
c&\geq (\frac{1}{p}-\frac{1}{p^{*}})\int_{\Omega}|
u|^{p^{*}}\mathrm{d}x
+(\frac{1}{p}-\frac{1}{p^{*}})S^{\frac{p*}{p*-p}}+\lambda
(\frac{1}{p}-\frac{1}{r})|\Omega|^{1-\frac{r}{p^{*}}}(\int_{\Omega}|
u|^{p^{*}}\mathrm{d}x)^{\frac{r}{p^{*}}}\\
&=(\frac{1}{p}-\frac{1}{p^{*}})S^{\frac{p*}{p*-p}}+(\frac{1}{p}-\frac{1}{p^{*}})\|u\|_{L^{p^{*}}(\Omega)}^{p^{*}}
+\lambda
(\frac{1}{p}-\frac{1}{r})|\Omega|^{1-\frac{r}{p^{*}}}(\int_{\Omega}|
u|^{p^{*}}\mathrm{d}x)^{\frac{r}{p^{*}}}.
\end{align*}
Let
\[ %27
g(x)=C_{1}x^{p^{*}}-\lambda C_{2}x^{r}.
\]
This function reaches its absolute
minimum at $x_{0}=(\frac{\lambda rC_{2}}{p^{*}C_{1}})^{\frac{1}{p^{*}-r}}$,
that is
$$
g(x)\geq g(x_{0})=-K\lambda^{\frac{p^{*}}{p^{*}-r}},\text{ where }
K=K(p,r,N,|\Omega|).
$$
Hence,
$c\geq(\frac{1}{p}-\frac{1}{p^{*}})S^{\frac{p^{*}}{p^{*}-p}}
-K\lambda^{\frac{p^{*}}{p^{*}-r}}$
which contradicts our hypothesis. So we know
$\int_{\Omega}|u_{j}|^{p^{*}} \mathrm{d}x\to \int_{\Omega}|
u|^{p^{*}}\mathrm{d}x$, and therefore $u_{j}\to u$ in
$L^{p^{*}}(\Omega)$. Now the rest of the proof  is as
that of Lemma \ref{lem4.2}.
\end{proof}

\begin{lemma}\label{lem4.4}
There exists $\widetilde{\lambda}>0$ such that if
$0<\lambda<\widetilde{\lambda}$, then $\varphi$ satisfies a local
$(PS)_{c}$ condition for $c\leq0$.
\end{lemma}

\begin{proof}
We need only to check the local $(PS)_{c}$ condition.
Obviously observe that every $(PS)_{c}$ sequence for
$\varphi$ with energy level $c\leq 0$ must be bounded.
Therefore by Lemma \ref{lem4.3} if $\lambda$ verifies
$$
0<\lambda<(\frac{1}{p}-\frac{1}{p^{*}})S^{\frac{p^{*}}{p^{*}-p}}
-K\lambda^{\frac{p^{*}}{p^{*}-r}},
$$
then their exists a convergent subsequence.
\end{proof}

\subsection*{Proof of Theorem \ref{thm1.4}}

 The proof is analogous to that of Theorem \ref{thm1.2}.
Here we use Lemma \ref{lem4.3} and Lemma \ref{lem4.4}
 respectively to work with the
functional $\varphi$ and complete the proof.


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