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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 205, 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 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/205\hfil Existence of solutions]
{Existence of solutions for quasilinear elliptic equations with nonlinear
boundary conditions and indefinite weight}

\author[G. Zhang, X. Liu, S. Liu \hfil EJDE-2012/205\hfilneg]
{Guoqing Zhang, Xiangping Liu, Sanyang Liu}  % in alphabetical order

\address{Guoqing Zhang \newline
College of Science, University of Shanghai for Science and Technology, 
Shanghai 200093, China}
\email{shzhangguoqing@126.com}

\address{Xiangping Liu \newline
College of Science, University of Shanghai for Science and Technology, 
Shanghai 200093, China}
\email{Liuxp83355650@yeah.net}

\address{Sanyang Liu \newline
Department of Applied Mathematics, Xidian University, Xi'an 710071, China}
\email{liusanyang@126.com}

\thanks{Submitted June 18, 2012. Published November 24, 2012.}
\subjclass[2000]{35J60, 35P30}
\keywords{Regularity; existence; nonlinear boundary conditions;
\hfill\break\indent indefinite weight}

\begin{abstract}
 In this article, we establish the existence and non-existence of solutions
 for quasilinear equations with nonlinear boundary conditions and indefinite
 weight. Our proofs are based on variational methods and their geometrical 
 features.  In addition, we prove that all the weak solutions are in
 $C^{1,\beta}(\overline{\Omega})$ for some $\beta\in(0,1)$.
\end{abstract}

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

\section{Introduction}

 In this article, we consider the  problem
\begin{equation}\label{eq1.1}
   \begin{gathered}
\operatorname{div}(a(x)|Du|^{p-2}Du)=|u|^{p-2}u,  \quad \text{in } \Omega,\\
      a(x)|Du|^{p-2}\frac{\partial u}{\partial \nu}+|u|^{q-2}u+h(x)
=\lambda V(x)|u|^{p-2}u, \quad\text{on } \partial\Omega,
\end{gathered}
\end{equation}
where $\Omega$ is a bounded domain in $\mathbb{R}^N$, with a $C^{2,\alpha}$
 boundary for some $\alpha\in(0,1)$, $1<p<N$, $q<p^{\star}=\frac{(N-1)p}{N-p}$,
$\frac {\partial}{\partial \nu}$ is
the outer normal derivative, $0<a_0\leq a(x)\in L^{\infty}(\overline{\Omega})$.
The functions $V(x)$, $h(x)$ are defined on $\partial\Omega$ and satisfy
the assumption
\begin{itemize}
\item[(H1)] $V\in L^{s}(\partial\Omega)$, $V(x)$ is a indefinite weight, i.e.
$$
V^{+}(x)=\max\{V(x),0\}\neq 0, x\in \partial\Omega,
$$
where $s>\frac{N-1}{p-1}$,  and $h(x)\in L^{s}(\partial\Omega)$.
\end{itemize}

Elliptic problems with nonlinear boundary conditions arise in many and diverse
contexts, such as differential geometry (e.g., in the scalar curvature problem
and the Yamabe problem \cite{1}),
Non-Newtonian fluid mechanics \cite{2}, and mathematical biology problem
(e.g., a prototype of pattern formation in biology and the steady-state
problem for a chemotactic aggregation model \cite{3}).
In this paper, we consider the quasilinear problems with mixed nonlinear
boundary condition and the indefinite character; i.e. $V(x)$ may change
 sign on $\partial\Omega$. Some existence and non-existence
results are obtained.

On the other hand, the regularity for elliptic problems with nonlinear boundary
conditions have been studied. For the semilinear elliptic problem, Ebmeyer \cite{4}
obtained that every weak solution belongs to $C^{\beta}(\Omega)(0<\beta<1)$.
Using the result of Dibenedetto \cite{5}, Anane, Chakrone, Moradi \cite{6}
obtained that the eigenfunction of  the first eigenvalue is in
$C^{1,\beta}(\overline{\Omega})(0<\beta<1)$ for the linear eigenvalue problem
of the  $p$-Laplacian. In this paper, for problem \eqref{eq1.1}
with nonlinear boundary conditions
and indefinite weight, we obtain that all weak solutions are in
 $L^{\infty}(\partial\Omega)\cap L^{\infty}(\Omega)$ and
$C^{1,\beta}(\overline{\Omega})$ for some $\beta\in(0,1)$.

This article is organized as follows:
In Section 2, we state our main results. In section 3, we obtain
some existence and non-existence results. Section 4 is devoted to
proving the regularity of the solutions for the problem \eqref{eq1.1}.

\section{Main results}

 Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^N$, and $V(x)$
 satisfies (H1). We denote the Sobolev space
\begin{equation}\label{eq2.1}
   L^p(\partial\Omega;V)=\{u:\partial\Omega\to \mathbb{R};
 \int_{\partial\Omega}V(x)|u|^pd\sigma<+\infty\},
\end{equation}
and the norm
$\|u\|_{L^p(\partial\Omega;V)}=(\int_{\partial\Omega}V(x)|u|^pd\sigma)
^{1/p}$. Consider the Sobolev trace embedding
$W^{1,p}(\Omega)\hookrightarrow L^p(\partial\Omega;V)$, we obtain that
the embedding is compact when $V(x)$ satisfies (H1) (see \cite{7}),
where the norm in $W^{1,p}(\Omega)$ is defined as
$$
\|u\|_{W^{1,p}(\Omega)}=\Big(\int_{\Omega}[|\nabla u|^p+|u|^p]dx\Big)^{1/p}.
$$
As the function $a(x)$ satisfies
$0<a_0\leq a(x)\in L^{\infty}(\overline{\Omega})$, we define the space $E$
is the reflexive Banach space under the  norm
$$
\|u\|_{a,\Omega}=\Big(\int_{\Omega}[a(x)|Du|^p+|u|^p]dx\Big)^{1/p}.
$$
Of course, $E\sim W^{1,p}(\Omega)$, we obtain that  the embedding
$E\hookrightarrow L^p(\partial\Omega;V)$ is compact and there exists
a $\widetilde{C}=\widetilde{C}(\overline{\Omega}, V(x), p)>0$ such that
\begin{equation}\label{eq2.2}
   \widetilde{C}\|v\|^p_{L^p(\partial\Omega;V)}\leq\|v\|^p_{a,\Omega}\quad
\text{for any } v\in E.
\end{equation}

Now, we state the main results in this article.

\begin{theorem} \label{thm1.1}
 If $p<q<p^{\star}$ and $\int_{\partial\Omega}h\varphi\,d\sigma\geq0$
for all $\varphi\in E$ with $\varphi|_{\partial\Omega}>0$,
then there exists  $\lambda_0>0$ such that
\begin{itemize}
\item[(1)] if $\lambda<\lambda_0$, then  \eqref{eq1.1}
 does not have any weak solutions,
\item[(2)] if $\lambda>\lambda_0$, then  \eqref{eq1.1} has at least
 one weak solution.
\end{itemize}
\end{theorem}

We remark that there are functions $h$ such that $\int_{\partial\Omega}h\varphi\,d\sigma\geq0$ for all $\varphi\in E$ with $\varphi|_{\partial\Omega}>0$:
For $p=2$ and $\Omega$ is a unit circle, let $x=e^{i\alpha}$,
 $x\in \partial\Omega$, and
$$
h=\begin{cases}
    1+\alpha^{2},  &0<\alpha\leq 2\pi,\\
     -1, & \alpha=0.
\end{cases}
$$

\begin{theorem} \label{thm1.2}
If $u$ is a weak solution of  \eqref{eq1.1} and $q<\frac{p^{2}-2p+N}{N-p}$,
then $u$ has the following properties:
\begin{itemize}
\item[(1)] $u\in L^{\infty}(\Omega)\cap L^{\infty}(\partial\Omega)$,

\item[(2)] $u\in C^{1,\beta}(\overline{\Omega})$ for  some $\beta\in (0,1)$, and $\|u\|_{C^{1,\beta}(\overline{\Omega})}\leq K$,
where
\begin{gather*}
K=K\big(p,N,G,\|u\|_{L^{s'q_0}(\partial\Omega)}, \|V\|_{L^{s}(\partial\Omega)}\big),\\
G=\Big(\int_{\partial\Omega}|(|u|^{q-2}u+h)|^{s}d\sigma\Big)^{1/s},
\end{gather*}
$s>\frac{N-1}{p-1}$, $s'q_0\in [s'p,p^{\star}]$, and $s'$ is the conjugate of $s$.
\end{itemize}
\end{theorem}

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

For this proof we use direct methods in variational methods.

 (1) We  prove  only that \eqref{eq1.1} does not have any weak solutions
for $\lambda$ small enough. Indeed, assume that $u\in E$ is a weak solution of
\eqref{eq1.1}; then we have
\begin{equation}\label{eq3.1}
\begin{split}
&\int_{\Omega}a(x)|Du|^{p-2}DuD\varphi\,dx +\int_{\Omega}|u|^{p-2}u\varphi\,dx
+\int_{\partial\Omega}|u|^{q-2}u\varphi\,d\sigma
+\int_{\partial\Omega}h\varphi\,d\sigma\\
&= \lambda\int_{\partial\Omega}V(x)|u|^{p-2}u\varphi\,d\sigma,
\end{split}
\end{equation}
for any $\varphi\in E$. Taking $\varphi=u$ in \eqref{eq3.1}, we obtain
\begin{equation}\label{eq3.2}
   \|u\|^p_{a,\Omega}+\|u\|^q_{L^q(\partial\Omega)}
+\int_{\partial\Omega}hu\,d\sigma
=\lambda\|u\|^p_{L^p(\partial\Omega;V)}.
\end{equation}
Clearly, for $p<q<p^{\star}$,  problem \eqref{eq1.1} does not have
non-trivial solution whenever $\lambda\leq0$.

Furthermore, by \eqref{eq2.2} and \eqref{eq3.2}, we have
$$
\lambda\|u\|^p_{L^p(\partial\Omega;V)}
\geq\|u\|^p_{a,\Omega}\geq \widetilde{C}\|u\|^p_{L^p(\partial\Omega;V)}.
$$
i.e., $\lambda\geq \widetilde{C}$, which implies that when
$\lambda_0\leq \widetilde{C}$, problem \eqref{eq1.1} still does not have
weak solution. This completes the proof of (1) of Theorem \ref{thm1.1}.

(2) Let the functional $J_{\lambda}:E\to \mathbb{R}$ be
\begin{equation}\label{eq3.3}
   J_{\lambda}(u)=\frac{1}{p}\|u\|^p_{a,\Omega}
+\frac{1}{q}\|u\|^q_{L^q(\partial\Omega)}
+\int_{\partial\Omega}hud\sigma
-\frac{\lambda}{p}\|u\|^p_{L^p(\partial\Omega;V)}.
\end{equation}
By (H1), we obtain the weak solution of the problem \eqref{eq1.1}
 is the critical point of the functional $J_{\lambda}$.

Firstly, we prove that the functional $J_{\lambda}$ is coercive.
Indeed, fix a $w\in E\backslash \{0\}$, by ({\ref{eq2.2}}) and $p<q$, we have
\begin{equation}\label{eq3.4}
   \begin{aligned}
     J_{\lambda}(tw)
& =\frac{t^p}{p}\|w\|_{a,\Omega}^p+\frac{t^q}{q}\|w\|_{L^q
(\partial\Omega)}^q
 +t\int_{\partial\Omega}hwd\sigma-\frac{\lambda t^p}{p}\|w\|_{L^p
 (\partial\Omega;V)}^p\\
& \geq\frac{t^p}{p}(1-\frac{\lambda}{p\widetilde{C}})\|w\|_{a,\Omega}^p
+\frac{t^q}{q}\|w\|_{L^q(\partial\Omega)}^q+t\int_{\partial\Omega}hwd\sigma.
   \end{aligned}
\end{equation}
Obviously we have $J_{\lambda}(tw)\to +\infty$ when $t\to +\infty$.
So the coercivity of the functional $J_{\lambda}$ is obtained.

Let $\{u_n\}^{\infty}_{n=1}$ be a minimizing sequence of $J_{\lambda}$
in $E$, which is bounded in $E$ by the coercivity of $J_{\lambda}$.
 By the non-negativity of the norm and
$\int_{\partial\Omega}h\varphi\,d\sigma\geq0$ for all $\varphi\in E$,
 we assume that $\{u_n\}^{\infty}_{n=1}$ is non-negative, converges
 weakly to some $u \in E$ and pointwise converges to $u$.

Secondly, we prove that the non-negative limit $u\in E$ is a weak solution
of  \eqref{eq1.1}. Indeed, We already know that
 $\lim_{n\to\infty}J_{\lambda}(u_n)
=\inf_{u'\in X}J_{\lambda}(u')$; i.e.,
\begin{equation}\label{eq3.5}
 \lim_{n\to\infty}J_{\lambda}(u_n)\leq J_{\lambda}(u'),\quad
\text{for all } u'\in E.
\end{equation}
So we only need to prove
\begin{equation}\label{eq3.6}
J_{\lambda}(u)\leq\lim_{n\to\infty}\inf J_{\lambda}(u_n).
\end{equation}
By (H1), we have
$$
\int_{\partial\Omega}hu\,d\sigma
=\lim_{n\to\infty}\int_{\partial\Omega}hu_n\,d\sigma,
$$
and by the weak lower semicontinuity of the norm, we have
$$
\frac{1}{p}\|u\|^p_{a,\Omega}+\frac{1}{q}\|u\|^q_{L^q(\partial\Omega)}
\leq\lim_{n\to\infty}\inf(\frac{1}{p}\|u_n\|^p_{a,\Omega}
+\frac{1}{q}\|u_n\|^q_{L^q(\partial\Omega)}).
$$
On the other hand, the boundedness of $\{u_n\}^{\infty}_{n=1}$ and the
compact imbedding $E\hookrightarrow L^p(\partial\Omega;V)$ implies that
$$
\|u\|_{L^p(\partial\Omega;V)}=\lim_{n\to\infty}\|u_n\|_{L^p(\partial\Omega;V)}.
$$
So \eqref{eq3.6} is established. Then by \eqref{eq3.5} and \eqref{eq3.6} we have
$$
J_{\lambda}(u)=\inf_{u'\in E} J_{\lambda}(u').
$$
Thus, $u$ is a global minimizer of $J_{\lambda}$ in $E$.

Thirdly, we show that the weak limit $u$ is a non-trivial weak solution
of  \eqref{eq1.1} if $\lambda>0$ is large enough. Indeed, $J_{\lambda}(0)=0$.
Hence, we only need to prove that there exists $\lambda^{0}>0$, such that
$$
\inf_{u'\in E}J_{\lambda}(u')<0 \quad\text{for  all } \lambda>\lambda^{0}.
$$
Consider the  minimization problem
\begin{equation}\label{eq3.7}
   \lambda^{0}:=\inf\{\frac{1}{p}\|\phi\|^p_{a,\Omega}
+\frac{1}{q}\|\phi\|^q_{L^q(\partial\Omega)}
+\int_{\partial\Omega}h\phi\,d\sigma:\phi\in E \text{ and }
   \|\phi\|^p_{L^p(\partial\Omega;V)}=p\}.
\end{equation}
Let $\{\kappa_n\}^{\infty}_{n=1}\in E$ be a minimizing  sequence
of \eqref{eq3.7}, which is obviously bounded in $E$.
Hence, without loss of generality, we assume that it converges weakly to some
$\kappa\in E$, with $\|\kappa\|^p_{L^p(\partial\Omega;V)}=p$.
By the weak lower semicontinuity of $\|\cdot\|$, We can deduce that
$$
\lambda^{0}=\frac{1}{p}\|\kappa\|^p_{a,\Omega}
 +\frac{1}{q}\|\kappa\|^q_{L^q(\partial\Omega)}
+\int_{\partial\Omega}h\kappa\,d\sigma.
$$
So $J_{\lambda}(\kappa)=\lambda^{0}-\lambda<0$ for any $\lambda>\lambda^{0}$.
Now we denote
\begin{gather*}
\lambda_0 :=\sup\{\lambda>0:\text{ problem \eqref{eq1.1} does  not have
 weak solutions}\},\\
 \lambda_1 :=\inf\{\lambda>0:\text{ problem \eqref{eq1.1} admits  a weak
solution}\}.
\end{gather*}
Of course $\lambda_1\geq\lambda_0>0$.

Lastly, we prove two facts:
(i) problem \eqref{eq1.1} has a weak solution for any $\lambda>\lambda_1$;
(ii) $\lambda_0=\lambda_1$.

Now, we fix $\lambda>\lambda_1$, by the definition of $\lambda_1$,
there exists $\mu\in(\lambda_1,\lambda)$, such that $J_{\mu}$ has a
non-trivial critical point $u_{\mu}\in E$; i.e.,
$$
\|u_{\mu}\|_{a,\Omega}^p+\|u_{\mu}\|_{L^q(\partial\Omega)}^q
+\int_{\partial\Omega}hu_{\mu}d\sigma=\mu\|u_{\mu}\|_{L^p(\partial\Omega;V)}^p,
$$
Clearly, $u_{\mu}$ is a sub-solution of problem \eqref{eq1.1}.
So next we need to find a super-solution of problem \eqref{eq1.1}
 which is greater than $u_{\mu}$.

Consider the  minimization problem
$$
\inf\{\frac{1}{p}\|\phi\|^p_{a,\Omega}
+\frac{1}{q}\|\phi\|^q_{L^q(\partial\Omega)}
+\int_{\partial\Omega}h\phi\,d\sigma
-\frac{\lambda}{p}\|\phi\|^p_{L^p(\partial\Omega;V)}: \phi\in E
\text{ and } \phi\geq u_{\mu}\}.
$$
From above argument, we can know that the minimization problem has
 a solution $u_{\lambda}\geq u_{\mu}$, which is also a weak
solution of  \eqref{eq1.1} provided $\lambda>\lambda_1$.
So for the fixed $\lambda$, we have a sub-solution $u_{\mu}$
and a super-solution $u_{\lambda}$ with $u_{\lambda}\geq u_{\mu}$,
 using \cite[Theorem 2.4]{8}, we obtain a weak solution.
Let us recall the definition of $\lambda_1$, we obtain that \eqref{eq1.1}
does not have solutions for any $\lambda<\lambda_1$.
Then by the define of $\lambda_0$, immediately we have $\lambda_1\leq\lambda_0$,
so $\lambda_1=\lambda_0$.

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

  This is an adaptation of the proof in \cite{6}, and is presented here,
for the reader's convenience.
Let $g=-|u|^{q-2}u-h$, then by $q<\frac{p^{2}-2p+N}{N-p}$, we have
$g\in L^{s}(\partial\Omega)$.

\begin{lemma} \label{lem4.1}
If $u\in E$ is a weak solution of  \eqref{eq1.1}, then there exists a constant
 $C>0$, such that
$$
(\|u\|^{q_n}_{L^{q_n}(\Omega)}+\|u\|^{s'q_n}_{L^{s'q_n}
(\partial\Omega)})^{1/q_n}\leq C,\quad \text{for all } n>n_0,
$$
where the sequence $\{q_n\}^{\infty}_{n=0}$ is defined as
$$
s'q_0\in [s'p,p^{\star}],\quad
p^{\star}=\frac{(N-1)p}{N-p},\quad
q_{n+1}=\frac{q_0}{p}q_n.
$$
Furthermore, $u\in L^{q_n}(\Omega)$ and $u\in L^{s'q_n}(\partial\Omega)$
for all $n\geq 0$, where $s'=s/(s-1)$.
\end{lemma}

\begin{proof}
 Assume that $u\in E$ is a weak solution of  \eqref{eq1.1}.
 By $E\sim W^{1,p}(\Omega)$, $u$ is also in $W^{1,p}(\Omega)$.
Since $s>\frac{N-1}{p-1}$,
we have $1<s'=\frac{s}{s-1}<\frac{N-1}{N-p}$, and
$[p,p^{\star}]\cap[s'p,s'p^{\star}]=[s'p,p^{\star}]\neq\emptyset$.

Let $q_0\in [p,{p^{\star}}/{s'}]$. Then
$$
W^{1,p}(\Omega)\hookrightarrow L^{q_0}(\Omega)\quad\text{and}\quad
W^{1,p}(\Omega)\hookrightarrow L^{q_0s'}(\partial\Omega).
$$
Obviously, $u\in L^{q_0}(\Omega)$ and $u\in L^{s'q_0}(\partial\Omega)$.
 Of course, $u$ is also in $L^{q_0}(\partial\Omega)$.
Suppose that $\|u\|_{L^{s'q_0}(\partial\Omega)}\geq 1$,
if not we consider $u_0=u/\|u\|_{L^{s'q_0}(\partial\Omega)}$,
which is a solution of
\begin{gather*}
 \operatorname{div}(a(x)|Du|^{p-2}Du)=|u|^{p-2}u  \quad\text{in }\Omega,\\
  a(x)|Du|^{p-2}\frac{\partial u}{\partial \nu}=\lambda V(x)|u|^{p-2}u+g'
\quad\text{on } \partial\Omega,
\end{gather*}
with $g'=(\|u\|_{L^{s'q_0}(\partial\Omega)})^{p-1}g\in L^{s}(\partial\Omega)$.

Using mathematical induction, suppose that $u\in L^{q_n}(\Omega)$,
$u\in L^{s'q_n}(\partial\Omega)$
and $\|u\|_{L^{s'q_n}(\partial\Omega)}\geq 1$, we show that
$$
u\in L^{q_{n+1}}(\Omega),\quad
u\in L^{q_{n+1}}(\partial\Omega),\quad
u\in L^{s'q_{n+1}}(\partial\Omega),\quad
\|u\|_{L^{s'q_{n+1}}(\partial\Omega)}\geq 1.
$$
Define a sequence $\{\omega_{k}\}^{\infty}_{k=0}$ in $E$ by
$$
\omega_{k}(x)=\begin{cases}
  k,     & \text{if }  u(x)\geq k;\\
  u(x),  & \text{if } -k\leq  u(x)\leq k,\; \forall x \in \overline{\Omega};\\
  -k,    & \text{if }  u(x)\leq -k;
\end{cases}
$$
Obviously, $\{\omega_{k}\}^{\infty}_{k=0}$ is in $W^{1,p}(\Omega)$.
Set $\delta=q_n-p>0$, then take the test function
$|\omega_{k}|^{\delta}\omega_{k}$ in \eqref{eq3.1}, we obtain
\begin{equation}\label{eq4.1}
\begin{split}
    \langle \operatorname{div}(a(x)|Du|^{p-2}Du),
 |\omega_{k}|^{\delta}\omega_{k}\rangle
&=\int_{\Omega}|u|^{p-2}u|\omega_{k}|^{\delta}\omega_{k}dx\\
&\geq\int_{\Omega}|\omega_{k}|^{\delta +p}dx=\int_{\Omega}|\omega_{k}|^{q_n}dx,
\end{split}
\end{equation}
and
\begin{equation}\label{eq4.2}
   \begin{aligned}
&\langle \operatorname{div}(a(x)|Du|^{p-2}Du),
 |\omega_{k}|^{\delta}\omega_{k}\rangle\\
& = -\int_{\Omega}a(x)|Du|^{p-2}DuD(|\omega_{k}|^{\delta}\omega_{k})dx+
    \lambda\int_{\partial\Omega}(V(x)|u|^{p-2}u+g)|\omega_{k}|^{\delta}
    \omega_{k}d\sigma\\
& \leq\lambda\int_{\partial\Omega}|u|^{q_n}|V(x)|d\sigma
  +G\|\omega^{\delta+1}_{k}\|_{L^{s'}(\partial\Omega)}
  -B_n\|D(|\omega_{k}|^{\frac{\delta}{p}}\omega_{k})\|^p_{L^p(\Omega)}\\
& \leq\lambda\|u\|^{q_n}_{L^{s'q_n}(\partial\Omega)}
   \|V\|_{L^{s}(\partial\Omega)}+G\|\omega_{k}\|^{\delta+1}_{L^{(\delta+1
   )s'}(\partial\Omega)}
   -B_n\|D(|\omega_{k}|^{\frac{\delta}{p}}\omega_{k})\|^p_{L^p(\Omega)},
  \end{aligned}
\end{equation}
where
$$
G=\Big(\int_{\partial\Omega}||u|^{q-2}u+h|^{s}d\sigma\Big)^{1/s},\quad
B_n=a_0(\delta+1)(\frac{p}{q_n})^p.
$$
Then by \eqref{eq4.1} and \eqref{eq4.2}, we have
\begin{equation}\label{eq4.3}
\begin{split}
&\int_{\Omega}|\omega_{k}|^{q_n}dx\\
&\leq\lambda\|u\|^{q_n}_{L^{s'q_n}(\partial\Omega)}\|V\|_{L^{s}(\partial\Omega)}
+G\|\omega_{k}\|^{\delta+1}_{L^{(\delta+1  )s'}(\partial\Omega)}
-B_n\|D(|\omega_{k}|^{\frac{\delta}{p}}\omega_{k})\|^p_{L^p(\Omega)}.
\end{split}
\end{equation}
Since $W^{1,p}(\Omega)\hookrightarrow L^{q_0}(\Omega)$, there exists
 $C_1=C_1(\Omega,p,q_0)>0$, such that
\begin{equation}\label{eq4.4}
   \begin{aligned}
       \|D(|\omega_{k}|^{\frac{\delta}{p}}\omega_{k})\|^p_{L^p(\Omega)}
& \geq C_1\||\omega_{k}|^{\frac{\delta+p}{p}}\|^p_{L^{q_0}(\Omega)}-
       \||\omega_{k}|^{\frac{\delta+p}{p}}\|^p_{L^p(\Omega)}\\
& \geq C_1\|\omega_{k}\|^{q_n}_{L^{q_{n+1}}(\Omega)}
 -\|\omega_{k}\|^{\delta+p}_{L^{\delta+p}(\Omega)}.
   \end{aligned}
\end{equation}
By \eqref{eq4.3} and \eqref{eq4.4}, we have
\begin{equation}\label{eq4.5}
\begin{split}
&\|\omega_{k}\|^{q_n}_{L^{q_{n+1}}(\Omega)}\\
&\leq A_n(\lambda\|u\|^{q_n}_{L^{s'q_n}(\partial\Omega)}
\|V\|_{L^{s}(\partial\Omega)}+   G\|\omega_{k}\|^{\delta+1}_{L^{(\delta+1)s'}
(\partial\Omega)}+D_n\|\omega_{k}\|^{q_n}_{L^{q_n}(\Omega)}),
\end{split}
\end{equation}
where
$A_n=\frac{1}{B_nC_1}$ and $D_n=B_n-1$.
By $\delta+1<q_n$, we have
$$
\|\omega_{k}\|^{\delta+1}_{L^{(\delta+1)s'}(\partial\Omega)}
\leq\|u\|^{\delta+1}_{L^{(\delta+1)s'}(\partial\Omega)}
\leq
\|u\|^{\delta+1}_{L^{s'q_n}(\partial\Omega)}
(\operatorname{meas}_{\sigma}(\partial\Omega)^{\frac{p-1}{s'q_n}}).
$$
Suppose that $\operatorname{meas}_{\sigma}(\partial\Omega)\leq 1$
and with the assumption $\|u\|_{L^{s'q_n}(\partial\Omega)}\geq1$, we obtain
\begin{equation}\label{eq4.6}
    \|\omega_{k}\|^{\delta+1}_{L^{(\delta+1)s'}(\partial\Omega)}
\leq\|u\|^{\delta+1}_{L^{s'q_n}(\partial\Omega)}
\leq\|u\|^{q_n}_{L^{s'q_n}(\partial\Omega)}.
\end{equation}
So by \eqref{eq4.5} and \eqref{eq4.6}, we obtain
\begin{align*}
   \|\omega_{k}\|^{q_n}_{L^{q_{n+1}}(\Omega)}
&\leq A_n[(\lambda\|V\|_{L^{s}(\partial\Omega)}+G)\|u\|^{q_n}_{L^{s'q_n}
 (\partial\Omega)}+|D_n|\|u\|^{q_n}_{L^{q_n}(\Omega)}]\\
& \leq A_n\max(R,\ |D_n|)(\|u\|^{q_n}_{L^{s'q_n}(\partial\Omega)}
+\|u\|^{q_n}_{L^{q_n}(\Omega)}),
\end{align*}
where
$R=\lambda\|V\|_{L^{s}(\Omega)}+G$.
Then we deduce that
\begin{equation}\label{eq4.7}
    \begin{aligned}
       \|u\|^{q_n}_{L^{q_{n+1}}(\Omega)}
& \leq \lim_{|k|\to +\infty}\inf(\|\omega_{k}\|^{q_n}_{L^{q_{n+1}}(\Omega)})\\
& \leq A_n\max(R,\ |D_n|)(\|u\|^{q_n}_{L^{s'q_n}(\partial\Omega)}
 +\|u\|^{q_n}_{L^{q_n}(\Omega)})
    \end{aligned}
\end{equation}
Thus $u\in L^{q_{n+1}}(\Omega)$.

Next we prove $u\in L^{s'q_{n+1}}(\partial\Omega)$
(so $u\in L^{q_{n+1}}(\partial\Omega)$), and
$\|u\|_{L^{s'q_{n+1}}(\partial\Omega)}\geq1$.
By \eqref{eq4.3} and \eqref{eq4.6}, we have
\begin{equation}\label{eq4.8}
   \int_{\Omega}|\omega_{k}|^{q_n}dx
+ B_n\|D(|\omega_{k}|^{\frac{\delta}{p}}\omega_{k})\|^p_{L^p(\Omega)}
\leq R \|u\|^{q_n}_{L^{s'q_n}(\partial\Omega)}.
\end{equation}
The embedding $W^{1,p}(\Omega)\hookrightarrow L^{s'q_0}(\partial\Omega)$
implies the existence of $C_2=C_2(\overline{\Omega},p,s'q_0)>0$ such that
\begin{equation}\label{eq4.9}
   \begin{aligned}
      \|D(|\omega_{k}|^{\frac{\delta}{p}}\omega_{k})\|^p_{L^p(\Omega)}
 &\geq C_2\||\omega_{k}|^{\frac{\delta+p}{\delta}}\|^p_{L^{s'}q_0(\partial\Omega)}-
      \||\omega_{k}|^{\frac{\delta+p}{p}}\|^p_{L^p(\Omega)}\\
 & \geq C_2\|\omega_{k}\|^{q_n}_{L^{s'q_{n+1}}(\partial\Omega)}
 -\|\omega_{k}\|^{\delta+p}_{L^{\delta+p}(\Omega)}
   \end{aligned}
\end{equation}
Then by \eqref{eq4.8} and \eqref{eq4.9}, we obtain
$$
B_n(C_2\|\omega_{k}\|^{q_n}_{L^{s'q_{n+1}}(\partial\Omega)}
 -\|\omega_{k}\|^{\delta+p}_{L^{\delta+p}(\Omega)})
\leq R\|u\|^{q_n}_{L^{s'q_n}(\partial\Omega)}-
\int_{\Omega}|\omega_{k}|^{q_n}dx.
$$
Then
\begin{align*}
   \|\omega_{k}\|^{q_n}_{L^{s'q_{n+1}}(\partial\Omega)}
&\leq B'_n(R\|u\|^{q_n}_{L^{s'q_n}(\partial\Omega)}
  +|D_n|\|\omega_{k}\|^{q_n}_{L^{q_n}(\Omega)})\\
& \leq B'_n\max(R,\ |D_n|)(\|u\|^{q_n}_{L^{s'q_n}(\partial\Omega)}
  +\|u\|^{q_n}_{L^{q_n}(\Omega)}),
\end{align*}
where $B'_n=1/(C_2B_n)$. Then
\begin{equation}\label{eq4.10}
    \begin{aligned}
        \|u\|^{q_n}_{L^{s'q_{n+1}}(\partial\Omega)}
&\leq \lim_{|k|\to +\infty}\inf(\|\omega_{k}\|^{q_n}_{L^{s'q_{n+1}}
  (\partial\Omega)})\\
& \leq B'_n\max(R,\ |D_n|)(\|u\|^{q_n}_{L^{s'q_n}(\partial\Omega)}
 +\|u\|^{q_n}_{L^{q_n}(\Omega)}).
    \end{aligned}
\end{equation}
Consequently, $u\in L^{s'q_{n+1}}(\partial\Omega)$ and
$\|u\|_{L^{s'q_{n+1}}(\partial\Omega)}>\|u\|_{L^{s'q_n}(\partial\Omega)}\geq1$.
Thus
$$
u\in L^{q_n}(\Omega),\quad
u\in L^{s'q_n}(\partial\Omega),\quad
\|u\|_{L^{s'q_n}(\partial\Omega)}\geq 1,\quad \text{for all } n\geq0
$$

Lastly, we have to show that there exists  $C>0$ such that
$$
(\|u\|^{q_n}_{L^{q_n}(\Omega)}+\|u\|^{s'q_n}_{L^{s'q_n}(\partial\Omega)})
^{1/q_n}\leq C,\quad\text{for all } n>n_0,
$$
By \eqref{eq4.7} and \eqref{eq4.10}, we have
$$
\|u\|^{q_{n+1}}_{L^{s'q_{n+1}}(\partial\Omega)}
+\|u\|^{q_{n+1}}_{L^{q_{n+1}}(\Omega)}
\leq T_n(\max(R,\ |D_n|)(\|u\|^{q_n}_{L^{s'q_n}(\partial\Omega)}
+ \|u\|^{q_n}_{L^{q_n}(\Omega)}))^{q_0/p},
$$
where
$$
T_n=\Big((\frac{1}{C_1}+\frac{1}{C_2})\frac{1}{B_n}\Big)^{q_0/p}.
$$
Obviously, $\lim_{n\to +\infty}B_n=0$, so we have
$\lim_{n\to +\infty}|D_n|=1$; so there exists $n_0\in \mathbf{N}^{+}$,
such that $|D_n|\leq2$ when $n>n_0$.
 Consequently,
$$
\|u\|^{q_{n+1}}_{L^{s'q_{n+1}}(\partial\Omega)}
+\|u\|^{q_{n+1}}_{L^{q_{n+1}}(\Omega)}
\leq \overline{C}(q_n)^{q_0}(\|u\|^{q_n}_{L^{s'q_n}(\partial\Omega)}+
\|u\|^{q_n}_{L^{q_n}(\Omega)})^{\frac{q_0}{p}},
$$
where
$$
\overline{C}=\frac{1}{p^{q_0}}
 \Big((\frac{1}{C_1}+\frac{1}{C_2})\max(R,\ 2)\Big)^{q_0/p}.
$$
Setting
$$
v_n=\big(\|u\|^{q_n}_{L^{s'q_n}(\partial\Omega)}
+\|u\|^{q_n}_{L^{q_n}(\Omega)}\big)^{1/q_n},
$$
we have $v^{q_{n+1}}_{n+1}\leq \overline{C}(q_n)^{q_0}(v^{q_n}_n)^{q_0/p}$
 for all $n\geq n_0$, and
$$
\ln(v_{n+1})\leq \frac{B}{q_{n+1}}+p\frac{\ln(q_n)}{q_n}+\ln(v_n)
\leq B\sum_{n_0+1\leq k\leq n+1}(\frac{1}{q_{k}})
+p\sum_{n_0\leq k\leq n}(\frac{\ln(q_{k})}{q_{k}})
+\ln(v_{n_0}),
$$
for all $n\geq n_0$, where
$B=\ln (\overline{C})$. By $0<\frac{p}{q_0}<1$, we have
$$
\sum_{n_0+1\leq k\leq n+1}(\frac{1}{q_{k}})\leq\frac{q_0}{q_0-p}.
$$
Since
\begin{align*}
     \sum_{n_0\leq k\leq n}\frac{\ln(q_{k})}{q_{k}}
& =\sum_{n_0\leq k\leq n}(\frac{\ln(q_0)}{q_0}
 +\frac{\ln(q_0)-\ln(p)}{q_0}k)(\frac{p}{q_0})^{k}
 :=\sum_{n_0\leq k\leq n}(\theta+\eta k)(\frac{p}{q_0})^{k}\\
&\leq\sum_{k\geq 0}(\theta+\eta k)(\frac{p}{q_0})^{k}
 =\frac{\theta q_0}{q_0-p}+\frac{\eta pq_0}{(q_0-p)^{2}},
\end{align*}
we have
$$
\ln(v_n)\leq \frac{q}{(q_0-p)}(B+\theta p)
 +\frac{\eta p^{2}q_0}{(q_0-p)^{2}}+\ln(v_{n_0}):=A,\quad \forall n\geq n_0.
$$
Thus 
$$v_n=(\|u\|^{q_n}_{L^{s'q_n}(\partial\Omega)}
+\|u\|^{q_n}_{L^{q_n}(\Omega)})\leq\exp^{A}:=C,\quad \forall n\geq n_0.
$$
\end{proof}

\begin{lemma} \label{lem4.2}
Let $\partial\Omega$ be $C^{2,\alpha}(\partial\Omega)$ with $\alpha\in (0,1)$
and $u$ be in $E\cap L^{\infty}(\Omega)$ such that
 $\operatorname{div}(a(x)|Du|^{p-2}Du)\in L^{\infty}(\Omega)$, then
$u\in C^{1,\beta}(\overline{\Omega})$  for some $\beta\in (0,1)$
and
$$
\|u\|_{C^{1,\beta}(\overline{\Omega})}
\leq K\big(N,\ p,\ \|u\|_{L^{\infty}(\Omega)},
 \|\operatorname{div}(a(x)|Du|^{p-2}Du)\|_{L^{\infty}(\Omega)} \big).
$$
\end{lemma}

The above  lemma is similar to \cite[Lemma 2.2]{4}, and  is also a
result in \cite{5}.

\begin{proof}[Proof of Theorem \ref{thm1.2}]
 (1) By  Lemma \ref{lem4.1} we know that
$$
\|u\|_{L^{q_n}(\Omega)}\leq C, \quad
\|u\|_{L^{s'q_n}(\partial\Omega)}\leq C,\quad \forall n\geq n_0.
$$
then we obtain
\begin{gather*}
\|u\|_{L^{\infty}(\Omega)}\leq\lim_{n\to +\infty}\sup \|u\|_{L^{q_n}(\Omega)}\leq C,
\\
\|u\|_{L^{\infty}(\partial\Omega)}\leq\lim_{n\to +\infty}
 \sup \|u\|_{L^{s'q_n}(\partial\Omega)}\leq C.
\end{gather*}
Hence, (1) of Theorem \ref{thm1.2} is proved.

(2) By (1) of Theorem \ref{thm1.2}, we obtain that the solution $u$ is
in $E\cap L^{\infty}(\Omega)$. Using
$\|\operatorname{div}(a(x)|Du|^{p-2}Du)\|_{L^{\infty}(\Omega)}
=\|u\|^{p-1}_{L^{\infty}(\Omega)}$, we have
$\operatorname{div}(a(x)|Du|^{p-2}Du)=|u|^{p-2}u\in L^{\infty}(\Omega)$.
 So $u$ is in $C^{1,\beta}(\overline{\Omega})$ for some
$\beta\in (0,1)$ and $\|u\|_{C^{1,\beta}
(\overline{\Omega})}\leq K(N,\ p,\ \|u\|_{L^{\infty}(\Omega)})$.
Indeed, we have $\|u\|_{L^{\infty}(\Omega)}\leq C$ for
$1<p<N$, where $C$ depends on $G$, $\|u\|_{L^{s'q_0}(\partial\Omega)}$, and
$\|V\|_{L^{s}(\partial\Omega)}$, then we have
$$
K=K\big(p,N, G,\|u\|_{L^{s'q_0}(\partial\Omega)},
\|V\|_{L^{s}(\partial\Omega)}\big).
$$
\end{proof}

\subsection*{Acknowledgments}
The authors express their gratitude to the anonymous referees for their
comments and remarks. This research was supported by project 11ZR1424500
from the  Shanghai Natural Science Foundation.

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\end{document}