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\markboth{\hfil On the Keldys-Fichera boundary-value problem \hfil
EJDE--2002/87}
{EJDE--2002/87\hfil Zu-Chi Chen \& Ben-Jin Xuan \hfil}

\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2002}(2002), No. 87, pp. 1--13. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or
http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu  (login: ftp)}
 \vspace{\bigskipamount} \\
 %
  On the Keldys-Fichera boundary-value problem for degenerate
  quasilinear elliptic equations
 %
\thanks{ {\em Mathematics Subject Classifications:} 35J65, 35J30.
\hfil\break\indent
{\em Key words:} Keldys-Fichera, existence, uniqueness, maximum
principle,
\hfil\break\indent comparison principle.
\hfil\break\indent
\copyright 2002 Southwest Texas State University. \hfil\break\indent
Submitted April 27, 2002. Published October 10, 2002.
\hfil\break\indent
Supported by grants 10071080 and 10101024 from the  NNSF of China } }
\date{}
%
\author{Zu-Chi Chen \& Ben-Jin Xuan}
\maketitle

\begin{abstract}
  We prove existence and uniqueness theorems for the
  Keldys-Fichera  boundary-value problem using
  pseudo-monotone operators.
  The equation studied here is quasilinear, elliptic, and
  its set of degenerate points may be of non-zero measure.
  We also obtain  comparison and  maximum  principles
  for this problem.
\end{abstract}
\numberwithin{equation}{section}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}

\section{Introduction}

This article studies  the Keldys-Fichera boundary-value problems
(KFBVP)
for degenerate quasilinear elliptic equations.
For linear elliptic equations with nonnegative characteristic
form of second order, the KFBVP is well known and has been summarized
in detail by Oleinik and Radkevich \cite{[7]}.
However little information is known about this problem for
nonlinear equations. Ma and Yu \cite{[6]} discussed the KFBVP for the
degenerate quasilinear elliptic equation
\begin{equation}
\label{e1.1} Lu=D_i[a_{ij}(x,u)D_ju+b_i(x)u]-c(x,u)=f(x), \quad x\in
\Omega .
\end{equation}
In this article, the summation from $1$ to $n$ over repeated indices is
understood. They obtained an existence theorem by using the acute angle
principle for weakly continuous operators. In this paper we consider
the
more general degenerate quasilinear elliptic equation
\begin{equation}
\label{e1.2}
 Qu=-D_ia_i(x,u,Du)+a(x,u)=f(x), \quad x\in \Omega ,
\end{equation}
in a bounded domain $\Omega  \subset R^n$, $n\ge 2 $, with piecewise
$C^1$-smooth boundary $\partial \Omega $.
We also obtain existence results, different from \cite{[6]},
using the pseudo-monotone operator method.
Moreover, in this paper the set of degenerate points may be of
non-zero measure, which is different from \cite{[6]}.
Also, the comparison principle, the maximum
principle and a uniqueness theorem of the solution to the KFBVP for
(\ref{e1.2}) are discussed.

The article is organized as follows. Section 2 formulates the existence
theorem, the main result of this paper, and the preliminaries.
Section 3 contains the proof of the main result.
In section 4, we prove the comparison principle, the maximum principle
and the uniqueness theorem.

\section{Main Result}
 Let $\Omega  $ be a bounded domain in $R^n$ ($n\ge 2$) with piecewise
 $C^1$-boundary  $\partial \Omega  $ and the Sobolev imbedding theorems
 are valid for this domain. Assume the following hypotheses:
\begin{enumerate}
\item[(A1)] $a_i(x,z,p)$, $i=1,2,\dots ,n$, satisfy Carath\'eodory
conditions,
 i.e., they  are continuous in $(z,p)\in R\times R^n$ for $x$  a.e. in
 $\bar \Omega  $ and measurable in  $x\in \bar \Omega  $ for every
 $(z,p) \in \ R\times R^n$. Moreover, $a_i(x,z,p)$ and
 $a(x,z)$ possess integrable continuous derivatives in $p_i$ and $z$

\item[(A2)]  $a_i(x,0,0)\in W^{1/m, m'}(\partial \Omega  )$ for $m\ge
2$,
 \begin{gather*}
|a_i(x,z,p)|\le c(|p|^{m-1}+|z|^{l/m'}+\varphi _1(x)),
        \quad \varphi _1(x)\in W^{1/m, m'}(\partial \Omega ), \\
|a(x,z)|\le c(|z|^{l-1}+\varphi _2(x)), \quad
 \varphi _2(x)\in L_{l'}(\Omega  ),
\end{gather*}
for $(x,z,p)\in \bar \Omega  \times R \times R^n$, where $c>0$,
$2\le l<\bar m=m(n-1)/(n-m)$
if $m<n$ and $2\le l<\infty $ if $m=n$

\item[(A3)] There exist constants $\alpha >0 $, $\beta >0$ and a
continuous
function $\lambda (x)\ge 0$ such that for all
$(x,z,p)\in \bar \Omega  \times R \times R^n$ and any $\xi \in R^n$
it holds that
\begin{gather}
 \alpha ^{-1}\frac {\partial a_i}{\partial p_j}(x,0,0)\xi _i \xi _j \le
\frac {\partial a_i}
{\partial p_j}(x,z,p)\xi _i \xi _j \le \alpha \frac {\partial
a_i}{\partial p_j}(x,0,0)
\xi _i\xi _j, \label {e2.1} \\
 \beta ^{-1}\frac {\partial a_i}{\partial z}(x,0,0)\xi _i\le \frac
{\partial a_i}
{\partial z}(x,z,p)\xi _i \le \beta \frac {\partial a_i}{\partial
z}(x,0,0)\xi _i,
\label{e2.2} \\
 \frac {\partial a_i}{\partial p_j}(x,0,0)\xi _i \xi _j \ge \lambda
(x)|\xi |^2;\label{e2.3}
\end{gather}

\item[(A4)] There exists a positive function $ \varphi _3(x)\in
L^1(\Omega  )$
such that
\begin{equation}
\label{e2.4}
a_i(x,z,p)p_i+a(x,z)z \ge h(|p|)|p|^m-\varphi _3(x)
\end{equation}
for $(x,z,p)\in \bar \Omega  \times R \times R^n$, where $h(t)$ with
$h(0)=0$ is a bounded, non-decreasing and continuous function on
$[0, \infty )$. Without loss
of generality, we assume that $\sup h(t)>1$.
\end{enumerate}

\begin{remark}\label{r2.1} \rm
From (\ref{e2.1}) and (\ref{e2.3}) we know that (\ref{e1.2}) may
degenerate
at the set $\{x\ |\  \lambda (x)=0\}$ which may have positive measure,
while in \cite{[6]} the set is
of zero measure.
\end{remark}

\begin{remark} \label{r2.2} \rm
Condition (A2) shows the growth orders of $a_i(x,z,p)$ and $a(x,z)$ in
$|z|$,
$|p|$ and $|z|$ respectively which often used in \cite{[2],[3]}
and other related papers. Condition (A3) is an
extension of the one in \cite{[6]}. Condition (A4) is a version of the
one in \cite{[1]}.
\end{remark}

We divide $\partial \Omega$ into the parts of the Keldys-Fichera type
as follows
\begin{equation} \begin{gathered}
\Sigma ^0=\big\{x \in \partial \Omega : \frac {\partial a_i}{\partial
p_j}
(x,0,0)\nu _i \nu _j=0\big\},\\
\Sigma _1=\big\{x \in \Sigma ^0 : \frac {\partial a_i}{\partial
z}(x,0,0)
\nu _i \le 0\big\}, \\
\Sigma _2=\Sigma ^0 \backslash \Sigma _1 , \quad
\Sigma _3=\partial \Omega  \backslash \Sigma ^0,
\end{gathered} \label{e2.5}
\end{equation}
where $\vec \nu =(\nu _1, \nu _2,\cdots ,\nu _n)$ is the unit outward
normal
to $\partial \Omega $.
For $m \ge 2$, we denote by $W^{1,m}(\Omega)$ the Sobolev space with
the norm
$$ \|u\|=(\|u\|_m^m + \|Du\|_m^m)^{1/m}, $$
where $\| \cdot \|_m $ is the $L^m(\Omega )$ norm. For $2\le m<n$, we
define that
$\tilde W^{1,m}(\Omega) $, the subspace of $W^{1,m}(\Omega) $, is the
closure of the set
$\{u\in C^1(\bar \Omega): u=0 \mbox { on } \Sigma _3\}$
in the norm $\| u\|$. Attaching the Keldys-Fichera boundary
condition that $u(x)=0$ on $\Sigma _2\cup \Sigma _3$ to (1.2) we then
get the
Keldys-Fichera boundary-value problem
\begin{equation} \begin{gathered}
-D_ia_i(x,u,Du)+a(x,u)=f(x), \quad x \in \Omega \\
       u=0, \quad x \in \Sigma _2\cup \Sigma _3.
\end{gathered}\label{e2.6}
\end{equation}

\begin{definition} \label{d2.1} \rm
A weak solution of (\ref{e2.6}) is defined to be an element
$u\in \tilde W^{1,m}(\Omega) $ such that
\begin{eqnarray}
&&\int _{\Omega}[a_i(x,u,Du)D_iv+a(x,u)v]dx- \int _{\Sigma
_1}a_i(x,u,0)\nu _ivds
\nonumber \\
&&=\int_{\Sigma _2}a_i(x,0,0)\nu _ivds+\int _{\Omega  }f(x)vdx,
\label{e2.7}
\end{eqnarray}
for any $v\in C^1(\bar \Omega  )$ vanishing on $\Sigma _3$.
\end{definition}

\begin{remark} \label{r2.3} \rm
One can find out that this definition is a proper counterpart
of the weak solution of the KFBVP for linear equations in \cite{[7]}.
\end{remark}

\begin{remark}\label{r2.4}\rm
(2.1) and (2.2) mean that in the definitions of $\Sigma  ^0$
and $\Sigma _1$ we can replace $\displaystyle \frac {\partial
a_i}{\partial p_j}(x,0,0)$ and
$\displaystyle \frac {\partial a_i}{\partial z}(x,0,0)$ by
$\displaystyle \frac {\partial a_i}{\partial p_j}(x,z,p)$ and
$\displaystyle \frac {\partial a_i}{\partial z}(x,z,p)$ respectively.
\end{remark}

\begin{remark} \label{r2.5} \rm
Multiplying (2.6) by $v\in C^1(\bar \Omega  )$ vanishing on $\Sigma _3$
and then integrating by parts, we can obtain (2.7).
Hence, the classical solution of (2.6) must be a weak solution.
\end{remark}

\begin{theorem}[Existence theorem] \label{th2.1}
 Assume that (A1)-(A4) hold and that $f(x)\in W^{-1,m'}(\Omega  )$.
 Then  (2.6) has a weak solution in $\tilde W^{1,m}(\Omega) $.
\end{theorem}

\section{Proof of Theorem \ref{th2.1}}
We confine ourselves to the case of $m<n$, and refer the reader to
Remark
\ref{r3.1} for the case of $m\ge n$.
For $\delta \in (0,\ 1]$ and $i=1,2,\cdots ,n$, define
$$a_{i \delta}(x,z,p)=a_i(x,z,p)+\delta p_i|p_i|^{m-2}\,.
$$

\begin{lemma} \label{le3.1}
For any $\delta \in (0,\ 1]$ the integral equation
\begin{eqnarray}
&&\int_\Omega  [a_{i \delta}(x,u,Du)D_iv+a(x,u)v]dx-\int_{\Sigma_1}
a_i(x,u,0)\nu _ivds \nonumber \\
 &&=\int_\Omega  f(x)vdx+\int_{\Sigma_2} a_i(x,0,0)\nu _ivds, \forall
v\in \tilde W^{1,m}(\Omega) \label{e3.1}
\end{eqnarray}
has a solution in $\tilde W^{1,m}(\Omega) $. This solution is denoted ,
$u_{\delta}$.
\end{lemma}

\paragraph{Proof}  Denote
\begin{eqnarray*}
\lefteqn{B_{\delta}(u,v)}\\
&=&\int_\Omega  [a_{i\delta}(x,u,Du)D_iv+a(x,u)v]dx-\int_{\Sigma_1}
a_i(x,u,0)\nu _ivds
\quad \forall v\in \tilde W^{1,m}(\Omega) .
\end{eqnarray*}
Then, by (A2), the H\"older's inequality and the Sobolev imbedding
theorem,
$W^{1,m}(\Omega) \hookrightarrow L^l(\Omega  )$, we obtain
\begin{eqnarray}
\lefteqn{\Big| \int_\Omega  [a_{i\delta}(x,u,Du)D_iv+a(x,u)v]dx \Big|}
\nonumber \\
&\le& c(\| Du\| _m^{m/m'}+
\| u\| _l^{l/m'}+n\| Du\| _m^{m/m'}+\| \varphi _1\| _{m'})\| v\|
\nonumber\\
&&+(\| u\|_l^{l/l'}+\| \varphi _2\| _{l'})\| v\| \label{e3.2} \\
&\le& c\| v\|,  \nonumber
\end{eqnarray}
where $c=c(\| u\| ,\| \varphi _1\| _{m'}, \| \varphi _2\| _{l'}, l, l',
m, m', n)$. Here
and in the sequel, the constant $c$ may vary in the context, and if
necessary
we will then indicate its dependence on other known quantities.

Because $\Sigma _1$ and $\Sigma _2$ are of $C^1$, the trace imbedding
$W^{1,m}(\Omega) \hookrightarrow L^{\bar m}(\Sigma _i), i=1,2,$ is
valid
(see $\Sigma _1$ of Chapter 3 in \cite{[4]}). Notice
that $l<\bar m, \ m<\bar m$, so the following integrals are well
defined,
and then by (A2) and the H$\ddot o$lder's inequality, it follows that
\begin{equation} \begin{aligned}
\Big| \int_{\Sigma_1} a_i(x,u,0)\nu _ivds|
\le& c\int_{\Sigma_1} (|u|^{l/m'}+|\varphi _1|)|v|ds \nonumber \\
\le&  c(\| u\| _{L^l(\Sigma _1)}^{l/m'}
+\| \varphi _1\| _{m'})\| v\| _{L^m(\Sigma _1)}
\le c\| v\|,
\end{aligned} \label{e3.3}
\end{equation}
where $c=c(\| u\| , \| \varphi _1\| _{m'}, l, m, m', |\Omega  |)$, and
the last inequality
is based on the trace imbedding theorem mentioned above. From
(\ref{e3.2}) and (\ref{e3.3})
we then know that there exists a continuous mapping $B(u): \tilde
W^{1,m}(\Omega) \to W^{-1,m'}(\Omega  )$
such that $(B(u),v)=B_{\delta}(u,v)$. Hence, for solving (\ref{e3.1})
it is
sufficient to find an element, say $u_{\delta}$, in $\tilde
W^{1,m}(\Omega) $ such that
\begin{equation}
\label{e3.4}
B(u_{\delta})=F,
\end{equation}
where $F$ defined by
$$ (F,v)=\int_\Omega  f(x)vdx+\int_{\Sigma_2} a_i(x,0,0)\nu _ivds. \
\forall v\in \tilde W^{1,m}(\Omega).
 $$
It is obvious that $F\in W^{-1,m'}$. In the following we employ the
pseudo-monotone operator method to solve (3.4). By (A2), it follows
that
\begin{equation}
\label{e3.5}
|a_{i\delta}(x,z,p)|\le (c+n\delta)\big(|p|^{m-1}+|z|^{l/m'}+\varphi
_1(x)\big).
\end{equation}
Let $\xi , \zeta \in R^n$, one can deduce, by (A3), that
\begin{eqnarray}
\lefteqn{[a_{i \delta}(x,z,\xi )-a_{i \delta}(x,z,\zeta )](\xi _i-\zeta
_i)}
 \nonumber \\
&=&[a_i(x,z,\xi )-a_i(x,z,\zeta )](\xi _i-\zeta _i)+\delta(\xi _i|\xi
_i|^{m-2}-\zeta
_i|\zeta _i|^{m-2})(\xi _i-\zeta _i) \nonumber \\
&=&\int _0^1\frac {\partial a_i}{\partial p_j}(x,z,\zeta
+t(\xi -\zeta ))(\xi _i-\zeta _i)(\xi _j-\zeta _j)dt \label{e3.6}\\
&&+\delta(\xi _i|\xi _i|^{m-2} -\zeta _i|\zeta _i|^{m-2})(\xi _i-\zeta
_i) \nonumber\\
&\ge& \delta(|\xi _i|^{m-1}-|\zeta |^{m-1})(|\xi _i|-|\zeta _i|)
>0,  \mbox{ for }  \xi \neq \zeta . \nonumber
\end{eqnarray}
Using (A4), we have
\begin{equation}
a_{i\delta}(x,z,p)p_i+a(x,z)z \ge  \delta\sum _{i=1}^n|p_i|^m-\varphi
_3(x)
 \ge  \delta 2^{(1-m)n}|p|^m-\varphi _3(x). \label{e3.7}
\end{equation}
Inequalities (\ref{e3.5})-(\ref{e3.7}) show that the operator $B$ in
$B(u)=F$ is pseudo-monotone(see Theorem 1 in \cite{[1]}).
Moreover, $B$ is coercive. In fact, by (\ref{e3.7}) and the
Poincar\'e inequality, we obtain
\begin{eqnarray*}
\int_\Omega  [a_{i \delta}(x,u,Du)D_iu+a(x,u)u]dx &\ge&
  \delta 2^{(1-m)n}\int_\Omega  |Du|^mdx-\int_\Omega  \varphi _3(x)dx
\\
&\ge& c\| u\| ^m-\int_\Omega  \varphi _3(x)dx,
\end{eqnarray*}
for $u\in \tilde W^{1,m}(\Omega) $, where $c=c(m,n,|\Omega  |,
\delta)$. Hence,
$$(B(u),u)\| u\| ^{-1}\ge c\| u\| ^{m-1}-\| u\| ^{-1}
\int_\Omega  \varphi _3(x)dx\to +\infty ,
 \ as \  \| u\| \to +\infty,
 $$
 which says that the pseudo-monotone operator $B$ is coercive.
Therefore, the
 equation $B(u)=F$ has a solution, say $u_{\delta}$, in $\tilde
W^{1,m}(\Omega) $.
This completes the proof of Lemma \ref{le3.1}. \hfill$\square$

Let $A_{\alpha  }(x,\eta,\xi),\ |\alpha  |\leq m$, be the functions
defined in
$\Omega  \times R^{N_1}\times R^{N_2}$ and satisfy the following
conditions: they
are continuous in $(\eta,\xi)$ for a.e. $x\in \Omega  $ and measurable
in $x$ for
every $(\eta,\xi)\in R^{N_1}\times R^{N_2}$ where $N_1$ is the number
of
 multi-index $\alpha  $ with  $|\alpha  |\leq m-1$ and $N_2$ is that of
$\alpha  $
with $|\alpha  |=m$. Moreover,
$$
|A_{\alpha  }(x,\eta,\xi)|\leq C(|\eta|^{p-1}+|\xi|^{p-1}+k(x)), \
k(x)\in
L^{p'}(\Omega  ), 1<p<\infty.
$$
We recall the following lemma.

\begin{lemma}[Lemma 2.1 of Chapter 2 in \cite{[5]}]
\label{le3.2}
Assume that $u_{\mu }\to u$  with $u\in W^{m-1,p}(\Omega  )$,  and
$v\in W^{m,p}(\Omega  )$. Then,
$$
A_{\alpha  }(x,\delta u_{\mu },D^mv)\to A_{\alpha  }(x, \delta u,D^mv)
\ \mbox {strongly
in} \ L^{p'}(\Omega  ),
$$
where $A_{\alpha  }\in L^{p'}(\Omega  )$ and
$ \delta u=\{u,Du,\cdots ,D^{m-1}u\}$.
\end{lemma}

\paragraph{Proof of Theorem \ref{th2.1}}
First we estimate a uniform bound for $u_{\delta},
0<\delta\le 1$. Let $u_{\delta}$ take the place of $u$ and $v$ in
(3.1),
then (3.1) reads
\begin{multline}
 \int_\Omega  [a_{i\delta}(x,u_\delta  ,Du_\delta  )D_iu_\delta
+a(x,u_\delta  )u_\delta  ]dx
-\int_{\Sigma_1} a_i(x,u_\delta ,0)\nu _iu_\delta  ds  \\
=\int_\Omega  f(x)u_\delta  dx+\int_{\Sigma_2} a_i(x,0,0)\nu _iu_\delta
ds. \label{e3.8}
\end{multline}
By condition (A4), it follows that
$$
a_{i\delta}(x,u_\delta  ,Du_\delta  )D_iu_\delta  +a(x,u_\delta
)u_\delta  \ge
\delta 2^{(1-m)n}|Du_\delta  |^m+h(|Du_\delta |)|Du_\delta  |^m-\varphi
_3(x),
$$
and then combining this with (\ref{e3.8}) yields
\begin{eqnarray}
\int_\Omega  h(|Du_\delta  |)|Du_\delta  |^mdx &\le& \int_\Omega
|Du_\delta  |^mdx+\int_\Omega  \varphi _3(x)dx
+\int_{\Sigma_1} a_i(x,u_\delta  ,0)\nu _iu_\delta  ds \nonumber \\
&&+\int_{\Sigma_2} a_i(x,0,0)\nu _iu_\delta  ds+\int_\Omega
f(x)u_\delta  dx.
\label{e3.9}
\end{eqnarray}
For the right-hand side of (\ref{e3.9}), we have the following
estimates
\begin{eqnarray*}
\lefteqn{\Big|\int_{\Sigma_1}a_i(x,u_\delta  ,0)\nu _iu_\delta
ds\Big|}\\
&\le& c\int_{\Sigma_1}(|u_\delta  |^{l/m'}+|\varphi _1|)|u_\delta
|ds \quad \mbox {by condition (A2)}  \\
&\le& c(\| u_\delta  \| _{L^l(\Sigma _1)}^{l/m'}+\|
\varphi _1\| _{m'})\| u_\delta  \| _{L^m(\Sigma _1)}
\quad \mbox{by H\"older's inequality} \\
&\le& c(\|Du_\delta  \| _m^{l/m'}+\| \varphi _1\| _{m'})\|Du_\delta  \|
_m \\
&& \hspace{2cm}\mbox {by the trace imbedding and Poincar\'e inequality}
\\
&\le& c(\frac 1{m'}+\frac 2m)\|Du_\delta  \| _m^m
+\frac 1{m'}\|\varphi _1\| _{m'}^{m'} \\
&& \hspace{2cm} \mbox{by Young's inequality and }
L^m(\Omega  )\hookrightarrow L^l (\Omega  ) \\
&\le& \varepsilon  \|Du_\delta  \| _m^m+M_1.\quad \mbox{by Young's
inequality}
\end{eqnarray*}
where $M_1=M_1(m, m', n, |\Omega  |, \| \varphi _1\| _{m'}, \varepsilon
)$
and $\varepsilon  >0 $ is any real number. Similarly,
$$
\Big|\int_\Omega  |Du_\delta  |^mdx+\int_{\Sigma_2}a_i(x,0,0)\nu
_iu_\delta  ds+\int_\Omega  f(x)u_\delta  dx
\Big|\le \varepsilon  \| u_\delta \| _m^m+M_2,
$$
where
$M_2=M_2(\| f\| _{m'}, \|\varphi _1\| _{m'}, m', |\Omega  |,
n,\varepsilon  )$.
Then, from (\ref{e3.9}) we get
\begin{equation}
\label{e3.10}
\int_\Omega  h(|Du_\delta  |)|Du_\delta  |^mdx\le 2\varepsilon  \|
Du_\delta  \| _m^m+M ,
\end{equation}
where $M=M_1+M_2+\| \varphi _3\| _{L^1(\Omega  )}$. Let \par
$\Omega  ( \delta ^{\ast })=\{x\in \Omega  \ |\  h(|Du_\delta  |)\le
\delta ^{\ast }\}, \ \alpha
( \delta ^{\ast })=\displaystyle \sup _{h(\eta )\le  \delta ^{\ast }}
\eta , \
0<\delta^{\ast }\le 1.$  \\
Obviously, $|Du_\delta  |\le \alpha  ( \delta ^{\ast })$ when $x\in
\Omega  ( \delta ^{\ast })$,
then
$$
\int _{\Omega  ( \delta ^{\ast })}|Du_\delta  |^mdx\le \alpha  ^m(
\delta ^{\ast })|
\Omega  |.
$$
With the aid of (\ref{e3.10}), we have
$$
\delta ^{\ast }\int _{\Omega  \backslash
\Omega  ( \delta ^{\ast })}|Du_\delta  |^mdx \le 2\varepsilon \|
Du_\delta  \| _m^m+M.
$$
Therefore,
$$
\delta ^{\ast }\int_\Omega  |Du_\delta  |^mdx\le 2\varepsilon \|
Du_\delta  \| _m^m+ \delta ^{\ast }\alpha^m
( \delta ^{\ast })|\Omega  |+M.
$$
Choosing $ \delta ^{\ast }=1, \ \varepsilon  =1/4$ in above inequality
yields $\| Du_\delta \| _m^m \le 2\alpha  ^m(1)|\Omega  |+2M$.
Then, by Poincar\'e inequality,
we finally obtain the uniform bound of $\{u_\delta  \}$ that
\begin{equation}
\label{e3.11}
\|u_\delta  \| \le c,
\end{equation}
where $c$ is independent of $ \delta $.
 Hence, there is a subsequence of $\{u_\delta  \}$,
denoted still by $\{u_\delta  \}$, converging weakly to an element
$u\in \tilde W^{1,m}(\Omega) $.
Replacing $u$ and $v$ in (\ref{e3.1}) by $u_\delta  $ and $u_\delta
-v$ respectively,
then (\ref{e3.1}) reads
\begin{multline*}
\int_\Omega  [a _{i \delta }(x,u_\delta  ,Du_\delta  )D_i(u_\delta
-v)+ a(x,u_\delta  )(u_\delta  -v)]dx
-\int_{\Sigma_1} a_i(x,u_\delta  ,0)\nu _i(u_\delta  -v)ds \\
= \int_\Omega  f(x)(u_\delta  -v)dx+\int_{\Sigma_2}a_i(x,0,0)\nu
_i(u_\delta  -v)ds.
\end{multline*}
Substituting (\ref{e3.6}) on the above equality yields
\begin{multline}
\int_\Omega  f(x)(u_\delta  -v)dx+\int_{\Sigma_1}a_i(x,u_\delta  ,0)\nu
_i(u_\delta  -v)ds
+\int_{\Sigma_2}a_i(x,0,0)\nu _i(u_\delta  -v)ds \\
-\int_\Omega  a_{i \delta }(x,u_\delta  ,Dv)D_i(u_\delta  -v)dx
-\int_\Omega  a(x,u_\delta  )(u_\delta  -v)dx \\
\ge  \delta \int_\Omega  (|D_iu_\delta  |^{m-1}-|D_iv|^{m-1})
(|D_iu_\delta  |-|D_iv|)dx. \label{e3.12}
\end{multline}
Next, we consider the convergence of the right hand side
of (3.12) as $ \delta \to 0$, in three steps.

\paragraph{step 1:}
Since $u_\delta  \to u$ weakly in $\tilde W^{1,m}(\Omega) $ and because
of the trace imbedding $W^{1,m}(\Omega)
\hookrightarrow L^m(\Sigma _1)$, we can assume, choose a subsequence if
necessary, that $u_\delta  \to u$
weakly in $L^m(\Sigma _1)$. Therefore, noticing (3.11), we have
\begin{gather*}
\int_\Omega  f(x)(u_\delta  -v)dx\to \int_\Omega  f(x)(u-v)dx, \\
\int_{\Sigma_2}a_{i \delta }(x,0,0)\nu _i(u_\delta  -v)ds\to
\int_{\Sigma_2}a_i(x,0,0)\nu _i(u-v)ds.
\end{gather*}
\paragraph{step 2:}
\begin{eqnarray}
\lefteqn{\int_\Omega  a_{i \delta }(x,u_\delta  ,Dv)D_i(u_\delta  -v)dx
}\nonumber \\
&=&\int_\Omega  [a_{i \delta }(x,u_\delta  ,Dv)-a_{i \delta
}(x,u,Dv)]D_i(u_\delta  -v)dx \nonumber \\
&&+\int_\Omega  a_{i \delta }(x,u,Dv)D_i(u_\delta  -v)dx=I_1+I_2.
\label{e3.13}
\end{eqnarray}
Because $W^{1,m}(\Omega) \hookrightarrow L^m(\Omega  )$ is compact,
then $u_\delta  \to u$ strongly in $L^m(\Omega  )$.
By lemma \ref{le3.2} and (\ref{e3.11}) we know that $a_{i \delta
}(x,u_\delta  ,Dv)$ tends to
$a_i(x,u,Dv)$ strongly in $L^{m'}(\Omega  )$. This and (\ref{e3.11})
show that
$$
|I_1|\le \| a_{i \delta }(x,u_\delta  ,Dv)-a_{i \delta }(x,u,Dv)\|
_{m'}\|
 D_i(u_\delta  -v)\| _m \to 0,
$$
then $I_1\to 0$. It is obvious that $I_2\to \int_\Omega
a_i(x,u,Dv)D_i(u-v)dx$
since
$u_\delta  \to u$ weakly in $\tilde W^{1,m}(\Omega) $ and
(\ref{e3.11}). Therefore, (\ref{e3.13}) yields
$$\int_\Omega  a_{i \delta }(x,u_\delta  ,Dv)D_i(u_\delta  -v)dx\to
\int_\Omega  a_i(x,u,Dv)D_i(u-v)dx.$$
For the integral
$$\int_\Omega  a(x,u_\delta  )(u_\delta  -v)dx=\int_\Omega
a(x,u_\delta  )(u_\delta  -u)dx+\int_\Omega  a(x,u_\delta  )(u-v)dx=I_3+I_4,$$
by the compact imbedding $W^{1,m}(\Omega) \hookrightarrow L^l(\Omega
)$ it holds that $u_\delta  \to u$ strongly
in $L^l(\Omega  )$, then, noticing (\ref{e3.11}), we get
\begin{eqnarray}
 |I_3|& \le & \| a(x,u_\delta  )\|  _{l'}\| u_\delta  -u\|_l \le (\|
u_\delta  \| _l^{l/l'}+
 \| \varphi _2\| _{l'})\| u_\delta  -u\| _l \nonumber \\
 & \le &(\| u_\delta  \|_l^{l/l'}+\| \varphi _2\| _{l'})\| u_\delta
-u\| _l\le c\| u_\delta  -u\| _l\to 0.
\label{e3.14}
\end{eqnarray}
Because $u_\delta  \to u$ strongly in $L^l(\Omega  )$, then $u_\delta
\to u$ almost everywhere
in $\Omega  $. On the other hand, we have already know that $\| a(x,
u_\delta  )\| _{l'}
\le c $, and hence $a(x,u_\delta  )\to A(x)$ weakly in $L^{l'}(\Omega
)$. Then, Based
on Lemma 1.3 of Chapter 1 in \cite{[5]}, we have $A(x)=a(x,u)$, and
hence
$$\int_\Omega  a(x,u_\delta  )(u-v)dx\to \int_\Omega  a(x,u)(u-v)dx. $$
Combining this with (\ref{e3.14}) yields
$$
\int_\Omega  a(x,u_\delta  )(u_\delta  -v)dx\to \int_\Omega
a(x,u)(u-v)dx.
$$
\paragraph{step 3:}
\begin{eqnarray}
\lefteqn{\int_{\Sigma_1}a_i(x,u_\delta  ,0)\nu _i(u_\delta  -v)ds
}\nonumber \\
&=&\int_{\Sigma_1}a_i(x,u_\delta
,0)\nu_i(u-v)ds+\int_{\Sigma_1}a_i(x,u_\delta  ,0)\nu _i(u_\delta  -u)ds.
\label{e3.15}
\end{eqnarray}
For the first integral in the above equation, based on the compact
trace imbedding
$W^{1,m}(\Omega) \hookrightarrow L^m(\Sigma _1)$ and $u_\delta  \to u$
weakly in $W^{1,m}(\Omega) $ we know that $u_\delta  \to u$
strongly in $L^m(\Sigma _1)$. Then, by Lemma \ref{e3.2} and noticing
that $a_i(x,u_\delta  ,0)
\in L^{m'}(\Sigma _1)$, it yields
\begin{equation}
\label{e3.16}
\int_{\Sigma_1}a_i(x,u_\delta  ,0)\nu _i(u-v)ds\to
\int_{\Sigma_1}a_i(x,u,0)\nu _i(u-v)ds.
\end{equation}
Now, consider the second integral of (\ref{e3.15}). Because the trace
imbedding $W^{1,m}(\Omega)
\hookrightarrow L^m(\Sigma _1)$ is compact, hence $\| u_\delta  -u\|
_{L^m(\Sigma _1)}\to 0$. Noticing
that $a_i(x,u_\delta  ,0)\in L^{m'}(\Sigma _1)$ and (\ref{e3.11}),
using H$\ddot o$lder's
inequality and the compact trace imbedding $W^{1,m}(\Omega)
\hookrightarrow L^l(\Sigma _1)$,
it holds that
\begin{eqnarray}
\lefteqn{\Big|\int_{\Sigma_1}a_i(x,u_\delta  ,0)\nu _i(u_\delta
-u)ds\Big|}\nonumber\\
&\le& \| a_i(x,u_\delta  ,0)\|_{L^{m'}(\Sigma _1)}
\| u_\delta  -u\| _{L^m(\Sigma _1)} \nonumber \\
&\le& c(\| u_\delta  \| _{L^l(\Sigma _1)}^{l/m'}
+\| \varphi \| _{m'})\| u_\delta  -u\| _{L^m(\Sigma _1)}
\label{e3.17}\\
&\le& c(\| u_\delta  \|^{l/m'}+\| \varphi _1\| _{m'})\|u_\delta  -u\|
_{L^m(\Sigma 1)}\to 0.
\nonumber
\end{eqnarray}
Returning to (\ref{e3.15}), by (\ref{e3.16}) and (\ref{e3.17}), we have
$$\int_{\Sigma_1}a_i(x,u_\delta  ,0)\nu _i (u_\delta  -v)ds\to
\int_{\Sigma_1}a_i(x,u,0)\nu _i(u-v)ds. $$
Now, let $ \delta \to 0$ in (3.12), by $\mbox {\it {1), 2), 3)}}$
 and (\ref{e3.11}), we obtain that
\begin{multline}
\lefteqn{\int_\Omega
[a_i(x,u,Dv)D_i(u-v)+a(x,u)(u-v)]dx-\int_{\Sigma_1}a_i(x,u,0)\nu_i(u-v)ds}
 \\
\le \int_\Omega  f(x)(u-v)dx+\int_{\Sigma_2}a_i(x,0,0)\nu _i(u-v)ds,
\quad \forall v \in \tilde W^{1,m}(\Omega)\,.
\label{e3.18}
\end{multline}
For any real number $\alpha  >0$ and any $\zeta (x)\in \tilde
W^{1,m}(\Omega) $, choosing $v=u-\alpha
\zeta (x)$ in (\ref{e3.18}) and then let $\alpha  \to 0$, it yields
\begin{multline}
\int_\Omega  [a_i(x,u,Du)D_i\zeta + a(x,u)\zeta
]dx-\int_{\Sigma_1}a_i(x,u,0)
\nu _i\zeta (x)ds  \\
 \le \int_\Omega  f(x)\zeta dx+\int_{\Sigma_2}a_i(x,0,0)\nu _i\zeta ds.
\label{e3.19}
\end{multline}
The inverse inequality of (\ref{e3.19}) holds if $\alpha  <0$. This
completes the proof of
Theorem \ref{th2.1}. \hfill$\square$

\begin{remark} \label{r3.1} \rm
If $m>n$ or $n=2$, Assumption (A2) is replaced by the following
assumption (A2)
$'$ which allows us to use the Sobolev imbedding $W^{1,m}(\Omega) $ to
bounded and continuous
function space $C_B(\omega)$ (see Chapter 7.7 in \cite{[2]}). So the
proof is easier than that of
case $m<n$.
\begin{enumerate}
\item[(A2)']
$ |a_i(x,z,p)|\leq h_1(z)(|p|^{m-1}+k_1(x))$,
$ k_1(x)\in L^{m'}(\omega)$, for $i=1,2,\cdots, n$.
$|a(x,z)|\leq h_2(x)k_2(x)$, $k_2(x)\in L^1(\omega)$,
where $h_i(z)(i=1,2)$ is a continuous function.
\end{enumerate}
\end{remark}

\section{Comparison Principle and Uniqueness Theorem}

We first consider the linear operator
\begin{equation}
\label{e4.1}
Lu=-D_i(a^{ij}(x)D_j u+b^i(x)u)+cu, \quad x\in \omega, \quad
(a^{ij}(x))\geq 0.
\end{equation}
Denote
\begin{gather*}
\Sigma ^0=\big\{x\in \partial \Omega  : a^{ij}(x)\nu_i\nu_j=0\big\},
\\
\Sigma _1=\big\{ x\in \Sigma ^0 : b^i(x)\nu_i\leq 0\big\},  \\
\Sigma _2=\Sigma ^0\setminus \Sigma _1, \quad \Sigma _3=\partial \Omega
\setminus \Sigma ^0,  \\
C^*=\big\{\varphi \in C^1(\bar \Omega  ) : \varphi \geq 0,
\varphi |_{\Sigma _3}=0\big\},
\end{gather*}
For  $u\in \tilde W^{1,m}(\Omega)$ and $v\in C^*$, let
\begin{equation}
(Lu,\ v)=\int_\Omega  [(a^{ij}D_ju+b^iu)D_iv+cuv]dx
-\int_{\Sigma _1}b^i\nu_i uvds,
\label{e4.2}
\end{equation}

\begin{definition}\label{de4.1} \rm
By a weak subsolution (supersolution)
$u\in W^{1,m}(\Omega  )(m\geq 2)$ of $Lu=0$ in $\Omega  $, we mean that
$(Lu,\ v)\geq 0$ ($\leq 0$) holds for all $\varphi \in C^*$.
\end{definition}

\begin{lemma}[Maximum Principle] \label{le4.1}
Suppose that the coefficients of $L$ satisfy
\begin{gather*}
a^{ij}, c\in C(\bar \Omega  ),\quad b^i\in C^1(\bar \Omega ),
\quad b^i \nu_i\big|_{\Sigma _1}\leq 0,\quad b^i \nu_i\big|_{\Sigma
_2}> 0,\\
D_i b^i \leq \min \{c,\ 2c\}, \quad \Sigma _1\in C^1,
\quad \Sigma _2\cup \Sigma _3\neq \emptyset.
\end{gather*}
If the minimum of the weak subsolution of $Lu=0$ is nonpositive
(or the maximum of the weak supersolution is nonnegative),
then it must be achieved on $\overline{\Sigma _2\cup\Sigma _3}$.
\end{lemma}

\paragraph{Proof}
 Suppose that $u$ is a weak subsolution of $Lu=0$, by definition
\ref{de4.1},
we have
\begin{equation}
\label{e4.3}
\int_\Omega  [(a^{ij}D_ju+b^iu)D_iv+cuv]dx-\int_{\Sigma _1}b^i\nu_i
uvds\geq 0,
 \quad \forall v\in C^*.
 \end{equation}
 Let $l=\displaystyle \inf_{\Sigma _2 \cup \Sigma _3}u \leq 0$,
 $$ w=(l-u)^+= \begin{cases}
 l-u, &\mbox{when } u<l, \\ 0 &  \mbox{otherwise}.
 \end{cases}
$$
Then $w$ belongs to the closure of $C^*$ in the norm of
Sobolev space $W^{1,m}(\omega)$, so we can choose $w$ as a test
function in (\ref{e4.3})
and obtain
\begin{equation}
\label{e4.4}
\int_\Omega  [(a^{ij}D_ju+b^iu)D_iw+cuw]\,dx-
\int_{\Sigma _1}b^i\nu_i uw\,ds \geq 0,
\end{equation}
Noting that $\Sigma _1\in C^1$ and $W^{1,m}(\Omega )$ can be imbedded
into
 $L^{mn/(n-m)}(\Omega )\subset L^2(\Omega )$, the left hand side of
(\ref{e4.4})
 is well defined. Integrating the term $b^iuD_jw$ in (\ref{e4.4}) by
parts
 yields
 \begin{eqnarray}
\lefteqn{ \int_{\Omega _+}[-a^{ij}D_iwD_jw
+ \frac12(D_ib^i-2c)w^2-(D_ib^i-c)lw]dx }\nonumber \\
& \geq
&-\frac12\int_{\Sigma_1}b^i\nu_iw^2ds+\int_{\Sigma_2}(\frac12w^2-lw)b^i
\nu_i\,ds  \label{e4.5} \\
& \geq &-\frac12\int_{\Sigma_1}b^i \nu_iw^2ds \geq 0, \nonumber
\end{eqnarray}
where $\Omega _+=\{x\in \Omega : u<l\}$. On the other hand, from the
assumption of
Lemma \ref{le4.1}, the integrand on $\Omega _+$ is negative which
implies
$|\Omega _+|=0$, and hence
$$ \inf_\Omega  u\geq \inf_{\Sigma_2\cup \Sigma_3}u.
$$
For the case of weak supersolution, let $u$ is a weak supersolution of
$Lu=0$,
replacing $u$ by $-u$ in the preceding arguments,  we obtain
$$ \sup_\Omega  u \leq \sup_{\Sigma_2 \cup \Sigma_3}u.
$$
Thus the proof of Lemma \ref{le4.1} is completed. \hfill$\square$

\begin{remark}\label{r4.1} \rm
Replacing $\overline{\Sigma_2\cup \Sigma _3}$ by $\overline{\Sigma_3}$
in the proof of Lemma \ref{le4.1}, we can deduce that
$\displaystyle \inf_\Omega  u \geq \displaystyle \inf_{\Sigma_3}u$ and
$\displaystyle
\sup_\Omega  u \leq \displaystyle \sup_{\Sigma_3}u$ for the subsolution
and the
supersolution of $Lu=0$ respectively. Hence for the weak solution of
$Lu=0$ with
$u|_{\Sigma_3}=0$, it must be that $u|_{\Sigma_2}=0$.
\end{remark}

Let $u,\ v\in \tilde W^{1,m}(\Omega)$, we say that $Qu \leq Qv$ in the
sense of distributions, if
\begin{eqnarray*}
 \lefteqn{\int_{\Omega }[a_i(x,u,Du)D_i \varphi
+a(x,u)\varphi]dx-\int_{\Sigma_1}a_i(x,u,0)
\nu_i \varphi \,ds }\\
&\leq& \int_{\Omega }[a_i(x,v,Dv)D_i \varphi +a(x,v)
\varphi]dx-\int_{\Sigma_1}
a_i(x,v,0)\nu_i \varphi \,ds
\end{eqnarray*}
holds for any $\varphi \in C^*$. Denote
$u_t(x)=tu+(1-t)v(x), 0\leq t\leq 1$ and $a_{it}=a_i(x,\ u_t,\ Du_t)$
for
$i=1, 2,\cdots, n$. $a_t=a_i(x, u_t)$. Then, we have the following
comparison
principle.

\begin{theorem}[Comparison Principle] \label{th4.1}
 Suppose that
$a_i(x,z,p)\in C^1(\bar \Omega \times R \times R^n)\times C^2(\Omega
\times R \times R^n)$,
$a(x,z)\in C^1(\bar\Omega \times R)$,
$\nu_iD_za_{it}|_{\Sigma _1}\leq 0$,
$\nu_iD_za_{it}|_{\Sigma _2}>0$, $D_{x_iz}a_{it}\leq \min\{D_za_t,\
2D_za_t\}$
and (A3) hold.
If $u, v\in C^1(\bar \Omega  )\cap \tilde W^{1,m}(\Omega)$ satisfy
$Qu\leq Qv$ in $\Omega $ and $u\leq v$ on $\Sigma _2\cup\Sigma _3$,
then $u\leq v$ in $\Omega $.
\end{theorem}

\paragraph{Proof}
 By the condition $Qu\leq Qv$, for any $\varphi \in C^{*}$, we have
\begin{eqnarray}
0&\geq &\int_0^1dt\int_\Omega
\{[D_{p_j}a_{it}D_j(u-v)+D_za_{it}(u-v)]D_i \varphi
+D_za_t(u-v)] \varphi \}dx \nonumber \\
&-&\int_0^1dt\int_{\Sigma _1}D_za_i(x,u_t,0)(u-v)\nu_i \varphi ds
\nonumber \\
&=&\int_\Omega \{[a^{ij}D_j(u-v)+b^i(u-v)]D_i \varphi +c(u-v)\varphi
\}dx
\label{e4.6} \\
&-&\int_{\Sigma _1}b^i \nu_i(u-v)\varphi ds, \nonumber
\end{eqnarray}
where $a^{ij}=\displaystyle \int_0^1D_{p_j}a_{it}dt,\ b^i=\displaystyle
\int_0^1D_za_{it}dt,\ c=\displaystyle\int_0^1D_za_tdt$. Let $w=u-v$.
From
(\ref{e4.6}), we have
$$
\int_\Omega [(a^{ij}D_jw+b^iw)D_i \varphi + cw \varphi]dx-\int_{\Sigma
_1}b^i \nu_i w
\varphi ds \geq 0,
$$
i.e., $w$ is a supersolution of the liner equation
$$ Lu=-D_i(a^{ij}(x)D_j u+b^i(x)u)+cu=0.
$$
From the assumptions of this theorem, the assumptions of Lemma
\ref{le4.1}
are all satisfied, thus Lemma \ref{le4.1} implies that
$\sup_{\omega}w \leq \sup_{\Sigma _2\cup \Sigma _3}w\leq 0$,
hence $u \leq v$ in $\omega$ which proves Theorem \ref{th4.1}.
\hfill$\square$

\begin{theorem}[Uniqueness Theorem] \label{th4.2}
Suppose that the coefficients of $Q$ satisfy
the conditions in Theorem \ref{th4.1}, then the
$C^1(\bar \Omega  )\cap \tilde W^{1,m}(\Omega) $-weak
solution of problem (\ref{e2.6}) is unique.
\end{theorem}

\paragraph{Proof}
Suppose that $u$ and $v$ are two weak solutions of problem
(\ref{e2.6}).
By Remark \ref{r4.1}, it holds that $u=v=0$ on
$\overline{\Sigma _2\cup \Sigma _3}$. Using Theorem \ref{th4.1}, we
find
that $u\leq v$ as well as $v\leq u$, hence $u=v$ in $\Omega $.
This completes the proof. \hfill$\square$

\begin{thebibliography}{00} \frenchspacing

\bibitem{[1]} F. E. Browder, Pseudo-monotone operator and nonlinear
elliptic
          boundary value problem or unbounded domains, Proc. North.
Acad.
          Sci. U.S.A., {\bf 74}(1977), 2659-2661.

\bibitem{[2]} D. Gilbarg and N. S. Trudinger, Elliptic Partial
Differential
         Equations of Second Order, Springer-Verlag, New York, 1983.

\bibitem{[3]} O. A. Ladyzenskaja and N. N. Uralceva, Linear and
Quasilinear
         Equations of Elliptic Type (in Russian), Scince Press, Moscow,
1973.

\bibitem{[4]} J. L. Lions, Probl$\grave e$mes Aux Limites Dans Les
$\acute E$
          quations Aux D$\acute e$riv$\acute e$es Partielles, Les
Presses
          de l'Universit$\acute e$ de Montr$\acute e$al, Montr$\acute
e$al,
          1967.

\bibitem{[5]} J. L. Lions, Quelques m$\acute e$thodes de r$\acute
e$solution
         des probl$\grave e$mes aux limites nonl$\acute e$aires, Dunod
and
         Gauthier-Villars, Paris, 1969.

\bibitem{[6]} T. Ma and Y. Q. Yu, The Keldys-Fichera boundary value
problem
         for degenerate quasilinear elliptic equations of second order,
         Differential and Integral Equations,{\bf 2}(1989), 379-388.

\bibitem{[7]} O. A. Oleinik and E. V. Radkevich, Second order equations
with
         nonnegative characteristic form, AMS Rhode Island and Plenum
Press,
         New York, 1973.
\end{thebibliography}

\noindent\textsc{Zu-Chi Chen} \\
Department of Mathematics\\
University of Science and Technology of China\\
Hefei, Anhui 230026, P. R. China \\
Email: chenzc@ustc.edu.cn
 \smallskip


\noindent\textsc{Ben-Jin Xuan}\\
Department of Mathematics \\
University of Science and Technology of China\\
Hefei, Anhui 230026, P. R. China \\
and \\
Depart. de matematicas, Universidad Nacional\\
Bogota, Colombia\\
Email: wenyuanxbj@yahoo.com
\end{document}