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\markboth{\hfil Dirichlet problem for quasi-linear elliptic equations 
\hfil EJDE--2002/82}
{EJDE--2002/82\hfil Azeddine Baalal \& Nedra BelHaj Rhouma \hfil}

\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2002}(2002), No. 82, pp. 1--18. \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} \\
 %
  Dirichlet problem for quasi-linear elliptic equations
 %
\thanks{ {\em Mathematics Subject Classifications:} 31C15, 35B65, 35J60.
\hfil\break\indent
{\em Key words:} Supersolution, Dirichlet problem, obstacle problem,
nonlinear potential theory.
\hfil\break\indent
\copyright 2002 Southwest Texas State University. \hfil\break\indent
Submitted April 9, 2002. Published October 2, 2002. \hfil\break\indent
Supported by Grant DGRST-E02/C15 from Tunisian Ministry of
Higher Education.} }
\date{}
%
\author{Azeddine Baalal \& Nedra BelHaj Rhouma}
\maketitle

\begin{abstract}
  We study  the Dirichlet Problem associated to the quasilinear
  elliptic problem
  \begin{equation*}
  -\sum_{i=1}^{n}\frac{\partial }{\partial x_i}\mathcal{A}_i(x,u(x),
  \nabla u(x))+\mathcal{B}(x,u(x),\nabla u(x))=0.
  \end{equation*}
  Then we define a potential theory related to this problem and we
  show that the sheaf of continuous solutions satisfies the Bauer
  axiomatic theory.
\end{abstract}

\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}{Remark}[section]

\section{Introduction}

The objective of this paper is to  study the weak solutions of
the following quasi-linear elliptic equation in $\mathbb{R}^{d}$,
($d\geq 2$):
\begin{equation}
-\sum_{i=1}^{n}\frac{\partial }{\partial x_i}\mathcal{A}_i(x,u(x),
\nabla u(x))+\mathcal{B}(x,u(x),\nabla u(x))=0\quad  \label{eq1}
\end{equation}
where $\mathcal{A}_i:\mathbb{R}^{d}\times \mathbb{R}\times \mathbb{R}
^{d}\to \mathbb{R}$ and $\mathcal{B}:\mathbb{R}^{d}\times \mathbb{R}\times
\mathbb{R}^{d}\to \mathbb{R}$ are given Carath\'{e}odory functions
satisfying the conditions introduced in section 2.

An example of equation (\ref{eq1}) is the perturbed $p$-Laplace
equation
\begin{equation}
-\mathop{\rm div}(|\nabla u| ^{p-2}\nabla u)+\mathcal{B}(.,u,\nabla
u)=0, \quad 1<p<d. \label{eq2}
\end{equation}
When $p=2$, equation (\ref{eq2}) reduces to the perturbed Laplace equation
\begin{equation}
-\Delta u+\mathcal{B}(.,u,\nabla u)=0.  \label{eq3}
\end{equation}
Another example included in this study is the  linear equation
$$
\mathcal{L}(u)=-\sum_{j}\Big(\sum_ia_{ij}\frac{
\partial u}{\partial x_i}+d_{j}u\Big)+\sum_{j}b_{j}\frac{\partial u}{
\partial x_{j}}+cu=0,
$$
where $\mathcal{L}$ is assumed to satisfy  conditions stated in
\cite{ST} (see also \cite{Her}).

Equation (\ref{eq1}) have been investigated in many interesting
papers \cite{Ser,Tr, Hei,Ma,BB}. Several papers have introduced an
axiomatic potential theory for the nonlinear equation (\ref{eq2})
when $\mathcal{B}=0$; see for example \cite{Hei}. For equations of
type (\ref{eq3}), see \cite{Baa,BB,BBM,Bou}.

The existence of weak  solutions of (\ref{eq1}) in variational
forms was treated by means of the sub-supersolution argument \cite{Deu, Deu1}.
Later on, Dancers/Sweers \cite{Dan},
Kura \cite{Kur}, Carl \cite{Car},
Lakshmikantham \cite{Lak}, Papageorgiou \cite{Pap}, Le/Schmitt \cite{LeS},
and others treated the existence of weak extremal solutions of nonlinear
equations of type (\ref{eq1}) by means of the sub-supersolution
method. Le \cite{Le} studied the existence of
extremal solutions of the problem
\begin{equation}
\int _{\Omega}A(x,\nabla u(x))(\nabla v-\nabla u)dx\geq
\int_{\Omega}\mathcal{B}(x,u(x))(v(x)-u(x))dx, \label{le}
\end{equation}
for all $v\in K$, $u\in K$, where $K$ is a closed convex subset of
$W_{0}^{1,p}(\Omega)$.

Note that the solutions of
(\ref{le}) correspond to the obstacle problem treated in section
5 of this paper. Remark that in the references cited above, often
$\mathcal{B}=\mathcal{B}(x,u(x))$ and the growth of $\mathcal{B}$
in $u$ is less then $p-1$  and when
$\mathcal{B}=\mathcal{B}(x,u,\nabla u)$, the growth of
$\mathcal{B}$ in $u$ and $\nabla u$ is less then $p-1$, but in our
case the growth of $\mathcal{B}$ in $\nabla u$ is is allowed to go
until $p-1+\frac{p}{n}$  and there is no condition on the growth
of $\mathcal{B}$ in $u$.

Our aim in this paper is to solve the   Dirichlet
 problem for (\ref{eq1}) with  a continuous data boundary and to give
an axiomatic of potential theory related to the associated
problem.

 This paper consists of four sections. First, we recall some
definitions for the (weak) subsolutions, supersolutions and
solutions of the equation (\ref {eq1}). In particular, we prove that
the supremum of two subsolutions is a subsolution and that the
infinimum of two supersolutions is also a supersolution.
In section 3, we give some conditions that allow us to have the
comparison principle for sub and supersolutions. After this
preparation we are able in section 4 to solve the Dirichlet
problem related to the equation (\ref{eq1}).
So at first we prove the existence of solutions to the
associated variational problem, after what we solve the Dirichlet
problem for continuous data boundary.
In the last section, we define a potential theory related to the
equation (\ref{eq1}), so we obtain that the sheaf of  continuous
solutions of (\ref{eq1}) satisfies the Bauer axiomatic theory
\cite{Bou}. We prove also that the set of all hyperharmonic
functions and the set of all hypoharmonic functions are sheaves.

\paragraph{Notation}
Throughout this paper we will use the following notation:
$\mathbb{R}^{d}$ is the real Euclidean $d$-space, $d\geq 2$. For
an open set $U$ of $\mathbb{R}^{d}$, we denote by $C^{k}(U)$ the
set of functions which $k$-th derivative is continuous for $k$
positive integer, $C^{\infty }(U)={\cap_{k\geq 1} }C^{k}(U)$ and by
$C_{0}^{\infty }(U)$ the set of all functions in $C^{\infty }(U)$
with compact support. $L^{q}(E)$ is the space of all
$q^{th}$-power Lebesgue
integrable functions defined on measurable set $E$. $W^{1,q}(U)$ is the
$(1,q)$-Sobolev space on $U$. $W_{0}^{1,q}(U)$ is the closure  of
$C_{0}^{\infty }(U)$ in $W^{1,q}(U)$ relatively to its norm.
$W^{-1,q'}(U)$ denotes the dual of $W_{0}^{1,q}(U)$, $q'=\frac{q}{q-1}$.
For a Lebesgue measurable set $E$, $|E| $ denotes the Lebesgue
measure of $E$. $u\vee v$ and $u\wedge v$ design respectively the supremum
and the infinimum of $u$ and $v$. $u^{+}=$ $u\vee 0$ and $u^{-}=$ $u\wedge 0$.
We write $\rightharpoonup $(resp. $\to $) to design the weak
(resp. strong) convergence.

\section{Supersolutions of (\ref{eq1})}

Let $\Omega $ be a bounded domain in $\mathbb{R}^{d}$ $(d\geq 2)$
with smooth boundary $\partial \Omega $ and let $\mathcal{L}$ be a
quasi-linear elliptic differential operator in divergence form
$$
\quad \mathcal{L(}u)(x)=-\sum_{i=1}^{d}\frac{\partial }{\partial x_i
}\mathcal{A}_i(x,u(x),\nabla u(x))+\mathcal{B}(x,u(x),\nabla
u(x))\quad \mbox{ a.e. }x\in \Omega
$$
where
$\mathcal{A}_i:\mathbb{R}^{d}\times \mathbb{R}\times
\mathbb{R}^{d}\to \mathbb{R}$ and
$\mathcal{B} :\mathbb{R}^{d}\times \mathbb{R}\times
\mathbb{R}^{d}\to \mathbb{R}$ are given
Carath\'{e}odory functions. Let
$\mathcal{A}=(\mathcal{A}_{1},\dots,\mathcal{A}_{d})$ and $1<p<d$.
We suppose that the following conditions are
fulfilled: for a.e. $x\in \Omega ,\forall \zeta \in \mathbb{R}$
and $\xi,\xi '\in \mathbb{R}^{d}$:
\begin{enumerate}
\item[(P1)] $|\mathcal{A}(x,\zeta ,\xi )| \leq k_{0}(x)
+b_{0}(x)| \zeta | ^{p-1}+a| \xi |^{p-1}$

\item[(P2)] $ (\mathcal{A}(x,\zeta,\xi )
-\mathcal{A}(x,\zeta,\xi '))(\xi -\xi ')>0$,
if $\xi \neq \xi '$.

\item[(P3)] $ \mathcal{A}(x,\zeta,\xi )\xi \geq \alpha | \xi|^p-d_0(x)
| \zeta | ^p-e(x)$

\item[(P4)] $ | \mathcal{B}(x,\zeta,\xi )| \leq  k(x)+b(x)|
\zeta | ^{\alpha }+c| \xi | ^{r}$,
$0<r<\frac{p}{(p^\ast)'}$, $\alpha \geq 0$.
\end{enumerate}
Here $a$, $c$ and $\alpha $ are positive constants,
$p'=\frac{p}{p-1}$, $p^{\ast }=\frac{dp}{d-p}$, while $k_{0}$, $b_{0}$,
$d_{0}$, $e$, $k$ and $b$ are measurable functions on $\Omega $
satisfying:
$k_{0}\in L^{p'}$, $b_{0}\in L^{\frac{d}{p-1}}$, $k\in L^{q}$,
$(p^{\ast })'<q<(\frac{d}{p-\epsilon }\wedge \frac{p}{r})$ and
 $d_{0}$, $e$, $b\in L^{\frac{d}{p-\epsilon }}$, $(0<\epsilon <1)$.

We can easily show that if $u\in W^{1,p}(\Omega )$, then
$A(.,u,\nabla u)\in L^{p'}$ and that
$\mathcal{B}(.,u,\nabla u)\in L^{(p^{\ast })'}$ when
$\alpha \leq p-1$.

\paragraph{Definition} %\label{def1}
We say that a function $u\in W_{{\rm loc}}^{1,p}(\Omega )$ is a (weak)
 solution of (\ref{eq1}), if
\begin{equation}
\begin{gathered}
\mathcal{B}(.,u,\nabla u)\in L^{(p^{\ast })'} \\
\int_{\Omega }\mathcal{A}(.,u,\nabla u)\nabla \varphi
+\int_{\Omega }\mathcal{B}(.,u,\nabla u)\varphi =0,
\end{gathered} \label{S1}
\end{equation}
for all $\varphi \in W_{0}^{1,p}(\Omega )$.

We say that $u\in W_{{\rm loc}}^{1,p}(\Omega )$ is a
supersolution (resp. subsolution) of
(\ref{eq1}) if
\begin{gather*}
\mathcal{B}(.,u,\nabla u)\in L^{(p^{\ast })'} \\
\int_{\Omega }\mathcal{A}(.,u,\nabla u)\nabla \varphi
+\int_{\Omega }\mathcal{B}(.,u,\nabla u)\varphi \geq 0
\quad \mbox{(resp. $\leq 0$)}
\end{gather*}
for every nonnegative function $\varphi \in W_{0}^{1,p}(\Omega )$.
\smallskip

Note that if $u$ is a supersolution of (\ref{eq1}) then $-u$ is a
subsolution of the equation
$$
-\mathop{\rm div}\widehat{\mathcal{A}}+\widehat{\mathcal{B}}=0
$$
where $\widehat{\mathcal{A}}(x,\zeta ,\xi )=-\mathcal{A}(x,-\zeta ,-\xi
)$ and $\widehat{\mathcal{B}}(x,\zeta ,\xi )=-\mathcal{B}(x,-\zeta
,-\xi )$.
Furthermore, the structure of $\widehat{\mathcal{A}}$ and $\widehat{\mathcal{B%
}}$ are similar to that of $\mathcal{A}$ and $\mathcal{B}$.

We recall that if $u$ is a bounded supersolution (resp.
subsolution), then $u$ is upper (resp. lower) semicontinuous in
$\Omega $ \cite[Corollary 4.10] {Ma}.

\begin{proposition} \label{prop1.1}
Let $u$ and $v$ be two subsolutions of \eqref{eq1} in
$\Omega $ such that
$$ (\mathcal{A}(.,v,\nabla u)-\mathcal{A}(.,u,\nabla u))\nabla(v-u)\geq 0,
\quad \mbox{a.e.}x\in \Omega.
$$
Then, $\max (u,v)$ is also a subsolution. A similar
statement holds for the minimum of two supersolutions.
\end{proposition}

\paragraph{Proof.}
Fix $\varphi$ in $C_{0}^{\infty }(\Omega )$, $\varphi \geq 0$.
Let $\Omega_{1}=\{ x\in \Omega :u>v\}$,
$\Omega _{2}=\{ x\in \Omega :u\leq v\}$ and put
$I=\int_{\Omega }\mathcal{A}(.,u\vee v,\nabla (u\vee v))\nabla \varphi
=I_{1}+I_{2}$ where
$$
I_{1}=\int_{\Omega _{1}}\mathcal{A}(.,u,\nabla u)\nabla
\varphi \mbox{ \ and \ }I_{2}=\int_{\Omega
_{2}}\mathcal{A}(.,v,\nabla v)\nabla \varphi .
$$
Let $\rho _{n}:\mathbb{R}\to \mathbb{R}$ be such that
$\rho _{n}\in \mathcal{C}^{1}(\mathbb{R})$,
$$
\rho _{n}(t)=\begin{cases}
1 & \mbox{if }  t  \geq  1/n \\
0 & \mbox{if }  t  \leq  0 \end{cases}
$$
and $\rho _{n}'>0$ on $] 0,1/n[$. For each $x\in \Omega$ define
$q_{n}(x)=\rho _{n}((u-v)(x))$. We see that
$q_{n}\in W_{{\rm loc}}^{1,p}(\Omega )$, $q_{n}\to 1_{\Omega _{1}}$ and
$\| q_{n}\|_{\infty }\leq 1$. It follows by Lebesgue's Theorem of dominated
convergence that $I_{1}={\lim_{{n\to \infty }} }\int_{\Omega _{1}}q_{n}
\mathcal{A}(.,u,\nabla u).\nabla \varphi $ and
$I_{2}={\lim_{n\to \infty }}
\int_{\Omega_{2}}(1-q_{n})\mathcal{A}(.,v,\nabla v).\nabla \varphi $. Hence
\begin{eqnarray*}
\int_{\Omega }q_{n}\mathcal{A}(.,u,\nabla u).\nabla \varphi
&=&\int_{\Omega }\mathcal{A}(.,u,\nabla u)\nabla
.(q_{n}\varphi )-\int_{\Omega }\mathcal{A}(.,u,\nabla
u)\varphi .\nabla (q_{n}) \\ \ &\leq &-\int_{\Omega
}\mathcal{B}(.,u,\nabla u)(q_{n}\varphi )-\int_{\Omega
_{n}}\mathcal{A}(.,u,\nabla u)\varphi .\nabla (q_{n}),
\end{eqnarray*}
where $\Omega _{n}=\{ x\in \Omega :v<u<v+\frac{1}{n}\} $.

Put $I_{n}=\int_{\Omega }q_{n}\mathcal{A}(.,u,\nabla
u).\nabla \varphi $ and $J_{n}=\int_{\Omega
}(1-q_{n})\mathcal{A}(.,v,\nabla v).\nabla \varphi $. Then,
similarly we have
$$
\int_{\Omega}(1-q_{n})\mathcal{A}(.,v,\nabla v).\nabla \varphi
\leq -\int_{\Omega }(1-q_{n})\mathcal{B}(.,v,\nabla v)\varphi
+\int_{\Omega _{n}}\mathcal{A}(.,v,\nabla v)\varphi .\nabla (q_{n}).
$$
So, we get
\begin{eqnarray*}
I_{n}+J_{n} &\leq &-\int_{\Omega }\mathcal{B}(.,u,\nabla
u)(q_{n}\varphi )-\int_{\Omega
}(1-q_{n})\mathcal{B}(.,v,\nabla v)\varphi \\
&&+\int_{\Omega _{n}}(\mathcal{A}(.,v,\nabla
v)-\mathcal{A}(.,u,\nabla u))\varphi .\nabla (q_{n}).
\end{eqnarray*}
Using that $\nabla (q_{n})=\rho _{n}'(u-v)\nabla
(u-v)$, we get
\begin{eqnarray*}
I_{n}+J_{n} &\leq &-\int_{\Omega }\mathcal{B}(.,u,\nabla
u)(q_{n}\varphi )-\int_{\Omega
}(1-q_{n})\mathcal{B}(.,v,\nabla v)\varphi \\
&&-\int_{\Omega _{n}}\rho _{n}'(u-v)(\mathcal{A}(.,v,\nabla v)-\mathcal{A}(.,u,\nabla u))\varphi
.\nabla (v-u) \\ \ &\leq &-\int_{\Omega
}\mathcal{B}(.,u,\nabla u)(q_{n}\varphi )-\int_{\Omega
}(1-q_{n})\mathcal{B}(.,v,\nabla v)\varphi .
\end{eqnarray*}
Finally, we have
$$ \int_{\Omega }\mathcal{A}(.,u\vee
v,\nabla (u\vee v)).\nabla \varphi +\int_{\Omega
}\mathcal{B}(.,u\vee v,\nabla (u\vee v))\varphi \leq 0
$$
which completes the proof. \hfill $\square$\smallskip

We say that $\mathcal{L}$ satisfies the property ($\pm $) , if for every
$k>0 $ and every supersolution (resp. subsolution) $u$ of
(\ref{eq1}), the function
$u+k$ (resp. $u-k$) is also a supersolution (resp. subsolution) of
(\ref{eq1})


\begin{remark} \label{rm1.1}
1) Suppose that for each $u\in W_{{\rm loc}}^{1,p}(\Omega )$ and each
$k>0$,
\begin{equation}
\int (\mathcal{A}(.,u+k,\nabla u)-\mathcal{A}(.,u,\nabla u)).\nabla \varphi
+\int (\mathcal{B}(.,u+k,\nabla u)-\mathcal{B}(.,u,\nabla u))\varphi \geq 0
\label{SH}
\end{equation}
for every nonnegative function $\varphi \in W_{0}^{1,p}(\Omega )$.
Then $\mathcal{L}$ satisfies the property ($\pm $).

\noindent 2) Note that if $\mathcal{L}(u)=-\sum_{j}\frac{\partial}{\partial
x_{j}}(\sum_ia_{ij}\frac{\partial u}{\partial x_i}
+d_{j}u)+(\sum_ib_i\frac{\partial u}{\partial x_i}+cu)$ is a
linear elliptic operator of second order satisfying the conditions of
\cite{Her}, then (\ref{SH}) is equivalent to
$(-\sum_{j}(d_{j})+c)\geq 0$ in the distributional sense.

\noindent 3) Suppose that $\mathcal{A}(x,\zeta ,\xi )=\mathcal{A}(x,\xi )$
and for a.e. $x\in \Omega $ and $\xi \in \mathbb{R}^{d}$ the map:
$\zeta \to \mathcal{B}(x,\zeta ,\xi )$ is increasing. Then the property
($\pm $) holds.
\end{remark}

\section{Comparison principle} \label{cp}

In this section, we will give some conditions needed for the comparison
principle. This principle makes it possible to solve the Dirichlet problem
and to develop a potential theory in our case.

We say that the \emph{comparison principle} holds for $\mathcal{L}$, if for
every supersolution $u$ and every subsolution $v$ of (\ref{eq1}) on $\Omega $%
, such that
$$ \limsup_{x\to y} v(x)\leq \liminf_{x\to y} u(x)
$$
for all $y\in \partial \Omega $ and both sides of the inequality are not
simultaneously $+\infty $ or $-\infty $, we have $v\leq u$ \ a.e. in
$\Omega $.

\begin{theorem} \label{thm1}
Suppose that the operator $\mathcal{L}$ satisfies
either one of the property \emph{(}$\pm $\emph{)} and the following
strict monotony condition (see \cite{Ne}):
$$
(\mathcal{A}(x,\zeta ,\xi )-\mathcal{A}(x,\zeta ',\xi
')).(\xi -\xi ')+(\mathcal{B}(x,\zeta ,\xi
)-\mathcal{B}(x,\zeta ',\xi '))(\zeta -\zeta
')>0 $$ for $(\zeta ,\xi )\neq (\zeta ',\xi
')$. Let $u$ be a supersolution and $v$ be a subsolution
of \emph{(\ref{eq1})}, on $\Omega $, such that $$
{\limsup_{x\to y} }v(x)\leq { \liminf_{x\to y}
}u(x)
$$
for all $y\in \partial \Omega $ and both sides of the inequality are not
simultaneously $+\infty $ or $-\infty $, then $v\leq u$ \ a.e. in $%
\Omega .$
\end{theorem}

\paragraph{Proof.}
Let $\varepsilon >0$ and $K$ be a compact subset of $\Omega $ such that $%
v-u\leq \varepsilon $ on $\Omega \backslash K$, then the function $%
\varphi =(v-u-\varepsilon )^{+}$ $\in W_{0}^{1,p}(\Omega )$. Testing by $%
\varphi $, we obtain that
\begin{eqnarray*}
0 & \leq & \int_{ v>u+\varepsilon}(\mathcal{A}
(.,u+\varepsilon ,\nabla u)-\mathcal{A}(.,v,\nabla
v))\nabla (v-u-\varepsilon ) \\
&& +  \int_{v>u+\varepsilon} (\mathcal{B} (.,u+\varepsilon ,\nabla
u)-\mathcal{B}(.,v,\nabla v))(v-u-\varepsilon )   \leq  0\,.
\end{eqnarray*}
Hence $\nabla (v-u-\varepsilon )^{+}=0$ and $(v-u-\varepsilon )^{+}$ $=0$
a.e. in $\Omega $. It follows that $v\leq u+\varepsilon $ a.e. in
$\Omega $ and therefore $v\leq u$ a.e. in $\Omega $
\hfill$\square$

\begin{corollary} \label{cor1}
we suppose that $\mathcal{A}(x,\zeta ,\xi )=\mathcal{A}(x,\xi )$
and $\mathcal{B}(x,\zeta ,\xi )=\mathcal{B}(\zeta )$ such that the map $%
\zeta \to \mathcal{B}(x,\zeta )$ is increasing \ for a.e. $x$ in $%
\Omega $. Then, the comparison principle holds.
\end{corollary}

\begin{theorem}
Suppose that
\begin{description}
\item[i)]  $[ \mathcal{A}(x,\zeta ,\xi )-\mathcal{A}(x,\zeta
',\xi ')].(\xi -\xi ')\geq \gamma | \xi -\xi '| ^p$ for all
$\zeta ,\zeta '$ in $\mathbb{R}$, for all  $\xi ,\xi '\in \mathbb{R}^{d}$,
a.e. $x$ in $ \Omega $ and for some $\gamma>0$ .

\item[ii)]  For a.e. $x\in \Omega $ and for all $\xi \in \mathbb{R}^{d}$, the
map $\zeta \to \mathcal{B}(x,\zeta ,\xi )$ is increasing,

\item[iii)]  $| (\mathcal{B}(x,\zeta ,\xi )-\mathcal{B}(x,\zeta ,\xi
')| \leq b(x,\zeta )| \xi -\xi '| ^{p-1}$ for a.e. $x\in \Omega $,
for all
 $\zeta \in \mathbb{R}$ and for all  $\xi ,\xi '\in \mathbb{R}^{d}$.
Where $\sup_{| \zeta | \leq M}b(.,\zeta )\in
L_{{\rm loc}}^{s}(\Omega )$, $s>d$, for all $M>0$.
\end{description}
Then the comparison principle holds.
\end{theorem}

\paragraph{Proof.} The main idea in this proof comes from
Professor J. Maly'.
Let $\rho >0$, $M=\sup (v-u)$ and put $w=v-u-\rho $. Take $w^{+}$
as test function . Then, we get $$ \int_{\Omega }\left[
\mathcal{A}(.,u,\nabla u)-\mathcal{A}(.,v,\nabla v)\right] .\nabla
(w^{+})+\int_{\Omega }\left[ \mathcal{B}(.,u,\nabla
u)-\mathcal{B}(.,v,\nabla v)\right] (w^{+})\geq 0
$$
and by consequence
\begin{eqnarray*}
\gamma \int_{\Omega }| \nabla w^{+}| ^p &\leq
&\int_{\Omega }b(x,v)| \nabla w^{+}| ^{p-1}w^{+}
\\
&\leq &C\Big[ \int_{\Omega }| \nabla w^{+}| ^p%
\Big] ^{\frac{p-1}{p}}\Big[ \int_{\Omega }( w^{+})
^{p^{\ast }}\Big] ^{\frac{1}{p^{\ast }}}| A_{\rho }| ^{\frac{s-d}{sd}} \\
&\leq &C\| \nabla w^{+}\| _{p}^p\, | A_{\rho }| ^{\frac{s-d}{sd}}.
\end{eqnarray*}
where $A_{\rho }=\{ \rho <v-u<M\} $. Hence we get $| A_{\rho
}| \to 0$ when $\rho \to M$, which is impossible if $%
M>0$. Thus, $v\leq u$ on $\Omega $ \hfill$\square$

\section{Dirichlet Problem}

\subsection*{Existence of solutions for $0\leq \alpha \leq p-1$ and $0\leq r\leq p-1$}

\paragraph{Definition} %def3.1
Let $g\in W^{1,p}(\Omega )$. We say that $u$ is a solution of problem
$(P)$ if
\begin{gather*}
u-g\in W_{0}^{1,p}(\Omega ), \\
\int_{\Omega }\mathcal{A}(.,u,\nabla u).\nabla \varphi
+\int_{\Omega }\mathcal{B}(.,u,\nabla u)\varphi =0\quad \forall
\varphi \in W_{0}^{1,p}(\Omega ).
\end{gather*}

\begin{remark} %\label{rem1}
Put $v=u-g$, then $u$ is a solution of the above problem $(P)$ if
and only if $v$ is a solution of
\begin{equation}
\begin{gathered}
u\in W_{0}^{1,p}(\Omega ) \\
\int_{\Omega}{\mathcal{A}_{g}}(.,u,\nabla u)\nabla \varphi
+\int_{\Omega }{\mathcal{B}_{g}}(.,u,\nabla u)\varphi
=0,\quad \forall \varphi \in W_{0}^{1,p}(\Omega ),
\end{gathered} \label{P'}
\end{equation}
where ${\mathcal{A}_{g}}(.,u,\nabla u)=\mathcal{A}(.,u+g,\nabla (u+g))$ and
${\mathcal{B}_{g}}(.,u,\nabla u)=\mathcal{B}(.,u+g,\nabla (u+g))$.
\end{remark}

Let $T:  W_{0}^{1,p}(\Omega )  \to  W_{0}^{-1,p'}(\Omega )$
be  the operator defined by
$$ \langle T(u),v\rangle =\int {\mathcal{A}_{g}}(.,u,\nabla u)\nabla v
  +\int {\mathcal{B}_{g}}(.,u,\nabla u)v
\quad \forall v\in W_{0}^{1,p}(\Omega). $$ Next we will establish
the existence of solution of \eqref{P'} when $0\leq \alpha \leq
p-1$ and $0\leq r\leq p-1$. Let $C=C(d,p)$ be a constant such that
$\| u\|_{p*}\leq C \| u\|_{p}$ for every $u\in
W_{0}^{1,p}(\Omega)$. Then, we get the following result.

\begin{proposition} \label{prop0}
Suppose that $0\leq \alpha \leq p-1$ and $0\leq r\leq
p-1$. If $\Omega $ is small (i.e $\alpha >C(\| d_{0}\|
_{n/p}+\| b\| _{n/p})$), then the operator $T$ is coercive.
\end{proposition}

\paragraph{Proof.}
We have
\begin{eqnarray*}
\langle T(u),u\rangle & = &
\int \mathcal{A}(u+g,\nabla (u+g))\nabla u+\int \mathcal{B}(u+g,\nabla (u+g))u \\
& \geq & \big(\alpha -C\| d_{0}\| _{d/p}
-C\| b\| _{d/p}\big)\| \nabla u\| _{p}^p-H_{1}
(\|u\| ,\| \nabla u\| ,\| g\| ,\|
\nabla g\| )
\end{eqnarray*}
where $C=C(d,p)$ and the growth of $H_{1}$ in $\| u\| $ and $\| \nabla u\| $
is less then $p-1$. So, let $\Omega $ be small enough such that
$\alpha >C(\| d_{0}\|_{n/p}+\|b\| _{n/p})$.
Hence, $\frac{\langle T(u),u\rangle }{\| \nabla u\| _{p}}\to +\infty $ as
$\| \nabla u\| _{p}\to +\infty $ and therefore the operator $T$ is
coercive. \hfill$\square$

\begin{proposition} \label{lem1}
Suppose that $0\leq \alpha \leq p-1$ and $0\leq r\leq
p-1$. Then, the operator $T$ is pseudomonotone and satisfies the
well known property ($S_{+}$):\\
If $u_{n}\rightharpoonup u$ and
$\limsup_{n\to \infty }\langle T(u_{n})-T(u),u_{n}-u\rangle \leq 0 $,
then $ u_{n}\to u$.
\end{proposition}
The proof of this proposition is found in \cite{Ma}.

\begin{theorem}\label{thmoo}
Suppose that $T$ satisfies the coercive condition on $\Omega $.
Then  \eqref{P'} has at least one weak solution in
$W_{0}^{1,p}(\Omega )$.
\end{theorem}

\paragraph{Proof.}
The operator $T$ is pseudomonotone, bounded continuous and coercive. Hence,
by \cite{Ne} $T$ is surjective. \hfill$\square$

\subsection*{Existence of solutions for $\alpha\geq 0$ and $p-1<r<\frac{p}{(p^{\ast
})'}$}

\paragraph{Definition}
Let $g$ be an element of $W^{1-\frac 1p}(\partial \Omega )$.

We say that a function $u$ is a solution of \eqref{sD}
 with boundary value $g$ if
\begin{equation}
\begin{gathered}
u\in W^{1,p}(\Omega ), \mathcal{B}(.,u,\nabla u)\in L^{p*'}_{{\rm loc}}{\Omega}\\
u=g\mbox{ in }W^{1-\frac{1}{p}}(\partial\Omega ),\\
\int_{\Omega }\mathcal{A}(.,u,\nabla u)\nabla \varphi
+\int_{\Omega }\mathcal{B}(.,u,\nabla u)\varphi =0
\quad \forall \varphi \in W_{0}^{1,p}(\Omega ).
\end{gathered} \label{sD}
\end{equation}
(For the definition and properties of the space
$W^{1-\frac{1}{p}}(\partial \Omega )$ see e.g. \cite{Li}).

 We say that  $u$ is an upper supersolution of
(\ref{sD}) with boundary value $g$ if
\begin{gather*}
u\in W^{1,p}(\Omega ),  \mathcal{B}(.,u,\nabla u)\in L^{p*'}_{Loc}{\Omega}\\
u\geq g\mbox{ in }W^{1-\frac{1}{p}}(\partial \Omega ), \\
\int_{\Omega }\mathcal{A}(.,u,\nabla u)\nabla \varphi
+\int_{\Omega }\mathcal{B}(.,u,\nabla u)\varphi \geq 0
\end{gather*}
for all $\varphi \in W_{0}^{1,p}(\Omega )$ with $\varphi \geq 0$.\medskip

Similarly, a lower subsolution is characterized by the reverse
inequality signs in the above definition.


We recall the following result given in \cite[Theorem 2.2]{Leo}.

\begin{theorem}
Suppose that there exists an ordered pair $\varphi \leq \psi $ of
subsolution and supersolution of \emph{(\ref{sD})} satisfying the
following condition: There exists $k \in L^{q}(\Omega)$,
$q>p^*{}'$
such that for all
$\xi \in \mathbb{R}^{d}$ and all $\zeta$ with $ \varphi
(x)\leq \zeta \leq \psi (x)$,  $| \mathcal{B} (x,\zeta ,\xi
)| \leq k(x)+c| \xi | ^{r}$ a.e.$x\in \Omega$.
Then, \eqref{sD} has at least one solution $u\in
W_{0}^{1,p}(\Omega )$ such that $\varphi \leq u\leq \psi $.
\end{theorem}

\begin{proposition}
Suppose that \eqref{sD} admits a pair of bounded lower
subsolution $u$ and upper supersolution $v$ such that $u\leq v$,
then there exists a solution $w$ of \eqref {sD} such that
$u\leq w\leq v$.
\end{proposition}

\paragraph{Proof.}
Let $M$ be a positive real such that
$\| u\| _{\infty},\| v\| _{\infty },\| g\| _{\infty }\leq M$. Then, for each
$\zeta $ such that $u(x)-g(x)\leq \zeta \leq v(x)-g(x)$, we have
$| \mathcal{B}(x,\zeta ,\xi )| \leq k(x)+b(x)M^{\alpha}+2^{r}c| \nabla g| ^{r}
+c| \xi | ^{r}$ for a.e. $x\in \Omega $. In addition, $u$
(resp. $v$) is a lower subsolution (resp. upper supersolution) of
(\ref{sD}). Hence by the last Theorem, there exists a solution $w$ of
(\ref{sD}) such that $u\leq w\leq v$. \hfill$\square$


\begin{corollary}
Suppose that all positive constants are supersolutions and all negative
constants are subsolutions. Then for each $g\in W^{1,p}(\overline{\Omega }%
)\cap L^{\infty }(\Omega )$, there exists a bounded solution $w$
of (\ref{sD}) such that $\| w\| _{\infty }\leq \|
g\| _{\infty }$.
\end{corollary}

\paragraph{Proof.} We see that  $v=\| g\| _{\infty }$ is an upper
supersolution and $u=-\| g\| _{\infty }$ is a lower
subsolution. Hence by the Proposition given above, we get a solution
$u\leq w\leq v$ \hfill$\square$

\subsection{Dirichlet Problem}\label{sDP}

In this section, we assume that $\mathcal{A}(.,0,0)=0$ and
$\mathcal{B}(.,0,0)=0$ a.e. in $\Omega $, that the property ($\pm $)
is satisfied, and that the comparison principle holds.

Suppose that the open set $\Omega $ is regular ($p-$regular)
\cite{Ma,Hei}. Then it is known that if $u$ is a solution of
(\ref{eq1}) in $\Omega $ satisfying $u-f\in W_{0}^{1,p}(\Omega )$
with $f\in W^{1,p}(\Omega )\cap C(\overline{\Omega })$, then $$
\lim_{x\to z} u(x)=f(z) \quad \forall z\in \partial \Omega . $$

\paragraph{Definition} %def3.1
Let $f$ be a continuous function on $\partial \Omega $. We say
that $u$ $\in C(\overline{\Omega })\cap $ $W_{{\rm loc}}^{1,p}(\Omega )$
solves the Dirichlet problem with boundary value $f$ if $u$ is a
solution of \eqref{eq1} such that $\lim_{x\to z}u(x)=f(z)$,
for all  $z\in \partial \Omega $.

\begin{theorem} \label{thDP}
For each $f\in C(\partial \Omega )$, there exists $u$ in
$C(\overline{\Omega })\cap $ $W_{{\rm loc}}^{1,p}(\Omega )$
solving the Dirichlet problem with boundary value $f$.
\end{theorem}

\paragraph{Proof}
By the Tieze's extension Theorem, we can assume that
$f\in C_{c}^{\infty }(\mathbb{R}^{d})$. Let $(f_{n})_{n}$ be a sequence
of mollifiers of $f$ such that $\| f_{n}-f\| \leq 1/2^n$ on
$\overline{\Omega}$ .

let $u_{n}$ denote the continuous solution of
\begin{equation}
\begin{gathered}
u_{n}-f_{n}\in W_{0}^{1,p}(\Omega ), \\
\int_{\Omega}\mathcal{A}(.,u_{n},\nabla u_{n})\nabla \varphi
+\int_{\Omega }\mathcal{B}(.,u_{n},\nabla u_{n})\varphi =0,
\quad \forall \varphi \in W_{0}^{1,p}(\Omega ).
\end{gathered} \label{En}
\end{equation}
So, by the comparison principle, $|u_{n}-u_{m}| \leq \frac{1}{2^{n}}+\frac{1}{2^{m}}$. Hence,
the sequence $(u_{n})_{n}$  converges uniformly on
$\overline{\Omega }$ to a continuous function $u$. Let $M$ be a
positive real such that for all $n$: $| f_{n}|+| f| \leq M$ and
$| u_{n}| +| u|\leq M$ on $\Omega $.

Let $G\subset \overline{G}\subset \Omega $ , take $\varphi $ as a
test function in (\ref{En}) such that $\varphi =\eta
^pu_{n},\eta \in C_{c}^{\infty }(\Omega ),0\leq \eta \leq 1$ and
$\eta =1$ on $G$. Then
\begin{multline*}
 \int_{\Omega}\mathcal{A}(.,u_{n},\nabla u_{n})\eta ^p\nabla(u_{n})\\
=-p\int_{\Omega }\mathcal{A}(.,u_{n},\nabla
u_{n})u_{n}\eta ^{p-1}\nabla (\eta )-\int_{\Omega
}\mathcal{B}(.,u_{n},\nabla u_{n})u_{n}\eta ^p
\end{multline*}
Using the assumptions on $\mathcal{A}$ and $\mathcal{B}$, we get
\begin{eqnarray*}
\lefteqn{\alpha \int_{\Omega }\eta ^p| \nabla (u_{n})|^p }\\
&\leq &pM\int_{\Omega }k_{0}| \nabla \eta |
+pM^p\int_{\Omega }b_{0}| \nabla \eta | +
pM\int_{\Omega }a| \nabla u_{n}| ^{p-1}\eta
^{p-1}| \nabla \eta | \\ &&\ +cM\int_{\Omega
}| \nabla u_{n}| ^{r}\eta ^p+\int_{\Omega
}(M^pd_{0}+Mk+M^{\alpha +1}b+e) \\
 &\leq & a(p-1)^{-1}M\varepsilon ^{\frac{p}{p-1}}(\int_{\Omega
}| \nabla u_{n}| ^p\eta ^p)+crp^{-1}M\varepsilon ^{\frac{p}{r}}
(\int_{\Omega }| \nabla u_{n}| ^p\eta ^p)\\
&&+C(M,\Omega ,\eta ,\nabla \eta ).
\end{eqnarray*}
Thus, for $\varepsilon $ small enough, we obtain
 $$
\int_{G}| \nabla (u_{n})| ^p\leq C(M,\Omega
,\eta ,\nabla \eta ,\varepsilon ).
$$
So $(\nabla u_{n})_{n}$ is bounded in $L^p(G)$ and therefore
$(\nabla u_{n})_{n}$ converges weakly to $\nabla u$ in
$(L^p(G))^{d}$.

Fix $D$ an open subset of $G$ and let $\eta \in C_{0}^{\infty
}(G)$ such that $0\leq \eta \leq 1$ and $\eta =1$ on $D$. Take
$\psi =\eta (u_{n}-u)$ as test function, then
\begin{eqnarray*}
\lefteqn{-\int_{\Omega }\eta \mathcal{A}(.,u_{n},\nabla u_{n}).\nabla
(u_{n}-u)}\\
 &=&\int_{\Omega }(u_{n}-u)\mathcal{A}(.,u_{n},\nabla
u_{n}).\nabla \eta +
\int_{\Omega }\mathcal{B}(.,u_{n},\nabla u_{n})(u_{n}-u)\eta
\end{eqnarray*}
Since $\mathcal{A}(.,u_{n},\nabla u_{n})$ is bounded in
$L^{p'}(G)$ and $\mathcal{B}(.,u_{n},\nabla u_{n})$ is
bounded in $L^{q}(G)$,
\begin{gather*}
{\lim_{n\to \infty } }\int_{G}\mathcal{A}
(.,u_{n},\nabla u_{n})(u_{n}-u)\nabla \eta =0,\\
\lim_{n\to \infty } \int_{G}\mathcal{B}
(.,u_{n},\nabla u_{n})(u_{n}-u) \eta =0.
\end{gather*}
Consequently,
$\lim_{n\to \infty } \int_{G}\mathcal{A}(.,u_{n},\nabla u_{n})\eta\nabla
(u_{n}-u)=0$ and
$$
\lim_{n\to \infty } \int_{G} (\mathcal{A}(.,u_{n},\nabla u_{n})
-\mathcal{A}(.,u_{n},\nabla u))\nabla (u_{n}-u)=0.
$$
To complete the proof, we need to prove that $(\nabla u_{n})_{n}$
converges to $\nabla u$ a.e. in $\Omega $. That is the aim of the
following lemma.

\begin{lemma} \label{lweak}
Let $G\subset \Omega $ and suppose that the sequence
$(\nabla u_{n})_{n}$ is bounded in $L^p(G)$ and
$$
{\lim_{n\to \infty } }\int_{G}\left[ \mathcal{A}
(.,u_{n},\nabla u_{n})-\mathcal{A}(.,u,\nabla u)\right] .\nabla
(u_{n}-u)=0.
$$
Then $ \mathcal{A}(.,u_{n},\nabla u_{n})\to
\mathcal{A}(.,u,\nabla u)$  weakly in $L^{p'}(G)$.
\end{lemma}

\paragraph{Proof.}
Put $v_{n}=\left[ \mathcal{A}(.,u_{n},\nabla u_{n})-\mathcal{A}%
(.,u_{n},\nabla u)\right] .\nabla (u_{n}-u)$. Since
\begin{eqnarray*}
\int_{G}v_{n} &=&\int_{G}\left[ \mathcal{A}(.,u_{n},\nabla
u_{n})-\mathcal{A}(.,u,\nabla u)\right] .\nabla (u_{n}-u) \\
&&-\int_{G}\left[ \mathcal{A}(.,u_{n},\nabla u)-\mathcal{A}%
(.,u,\nabla u)\right] .\nabla (u_{n}-u),
\end{eqnarray*}
for a subsequence we get
$$ {\lim_{n\to \infty }
}\left[ \mathcal{A}(.,u_{n},\nabla
u_{n})-\mathcal{A}(.,u_{n},\nabla u)\right] .\nabla (u_{n}-u)=0 $$
a.e. $x\in G\setminus N$ with $| N| =0$. Let $x\in
G\setminus N$. By the assumptions on $\mathcal{A}$ we have
$$
 v_n(x)\geq \alpha | \nabla u_n(x)| ^p-F(|
\nabla u_n(x)| ^{p-1},| \nabla u(x)| ^{p-1}).
$$
Consequently, $(\nabla u_{n}(x))_{n}$ is bounded and converges to
some $\xi \in\mathbb{R}^{d}$. It follows that
$[\mathcal{A}(.,u,\xi )-\mathcal{A}(.,u,\nabla u)].(\xi -\nabla u)=0$
and hence $\xi =\nabla u$. Finally we conclude that
$\mathcal{A}(.,u_{n},\nabla u_{n})\to \mathcal{A}(.,u,\nabla u)$ a.e. in
$G$ and $\mathcal{A}(.,u_{n},\nabla u_{n})$ converge weakly to
$\mathcal{A}(.,u,\nabla u)$ in $L^{p'}(G)$. \hfill$\square$

Now we go back to the proof of Theorem \ref{thDP}.
Using Lemma \ref{lweak}, we conclude that
$\nabla u_{n}\to \nabla u$ a.e. in $\Omega $ and
$\mathcal{A}(.,u_{n},\nabla u_{n})\rightharpoonup \mathcal{A}(.,u,\nabla u)$
in $L^{p'}(D)$. Hence,
$$
\int_{D}\mathcal{A}(.,u,\nabla u)\nabla \varphi +\int_{D}%
\mathcal{B}(.,u,\nabla u)\varphi =0
\quad \forall \varphi \in C_{0}^{\infty }(\Omega ).
$$
Moreover, using the fact that
$$
-\frac{1}{2^{n}}-\frac{1}{2^{m}}\leq u_{m}-u_{n}\leq \frac{1}{2^{n}%
}+\frac{1}{2^{m}}\quad\forall n, m
$$ we obtain
$$
-\frac{1}{2^{n}}+u_{n}\leq u\leq \frac{1}{2^{n}}+u_{n},\quad\forall n.
$$
So, we deduce that for all $n$ and all $z\in \partial \Omega $,
$$
 -\frac{1}{2^{n}}+f_{n}(z)\leq
\liminf_{x\in \Omega, x\to z}u(z)\leq \limsup_{x\in \Omega,x\to z
}u(z)\leq \frac{1}{2^{n}}+f_{n}(z) $$ which implies ${\lim_{x\to
z} u(x)}=f(z)$ and completes the proof of Theorem \ref{thDP}.
\hfill$\square$

\begin{remark} \label{remsup} \rm
Using the same techniques as in the proof of Theorem \ref{thDP}
 we can show that every increasing and locally bounded
sequence $(u_{n})_{n}$ of supersolutions of \eqref{eq1}
in $\Omega $ is locally bounded in $W^{1,p}(\Omega )$ and that
$u=\lim_{n}u_{n}$ is a supersolution of \eqref{eq1} in
$\Omega $.
\end{remark}

\section{Sheaf property for Superharmonic functions}

\subsection*{The obstacle Problem}

\paragraph{Definition}
Let $f$, $h\in W^{1,p}(\Omega )$ and let
$$
K_{f,h}=\big\{ u\in W^{1,p}(\Omega ):h\leq u\mbox{ a.e. in }
\Omega , u-f\in W_{0}^{1,p}(\Omega )\big\}.
$$
If $f=h$, we denote $K_{f,h}=K_{f}$.

We say that a function $u\in K_{f,h}$ is a solution to the
obstacle problem in $K_{f,h}$ if $$ \int_{\Omega
}\mathcal{A}(.,u,\nabla u).\nabla (v-u)+\int_{\Omega
}\mathcal{B}(.,u,\nabla u)(v-u)\geq 0 $$ whenever $v\in $
$K_{f,h}$.
This function $u$ is called solution of the problem with obstacle $h$
and boundary value $f$.

\begin{remark} \label{rm4.1} \rm
Since $u+\varphi \in K_{f,h}$ for all nonnegative $\varphi \in
W_{0}^{1,p}(\Omega )$, the solution $u$ to the obstacle problem is
always a supersolution of \emph{(\ref{eq1})} in $\Omega $.
Conversely, a supersolution of \emph{(\ref{eq1})} is always a
solution to the obstacle problem in $K_{u}(D)$ for all open
$D\subset \overline{D}\subset \Omega$.
\end{remark}

\begin{theorem} \label{thop}
Let $h$ and $f\ $ be in $W^{1,p}(\Omega )\cap L^{\infty }(\Omega)$.
If $v$ is an upper bounded supersolution of \eqref{sD}
with boundary value $f$ such that $v\geq h$, then there exists
a solution $u$ to the obstacle problem in $K_{f,h}$ with $u\leq v$.
\end{theorem}

\paragraph{Proof.}
As in \cite{Leo}, we introduce the function
$$
g(x,\zeta ,\xi )=\begin{cases}
\widetilde{\mathcal{B}}(x,\zeta ,\xi ) & \mbox{if }  \zeta \leq v(x)\\
\widetilde{\mathcal{B}}(x,v,\nabla v)  & \mbox{if }  \zeta >v(x).
\end{cases}
$$
As in \cite{Hes}, we define the function
$$
\mbox{\bf {a}}(x,\zeta ,\xi )=\begin{cases}
\mathcal{A}(x,\zeta ,\xi ) & \mbox{if }  \zeta \leq v(x)\\
\mathcal{A}(x,v,\nabla v) & \mbox{if }  \zeta>v(x).
\end{cases}
$$ Note that $\bf {a}$ satisfies the conditions (P1), (P2), and
(P3).

A Lemma in \cite[p.52]{Deu} proves that the map $u\to g(x,u,\nabla
u) $ from $W^{1,p}(\Omega )$ to $L^{p'}(\Omega )$ is bounded and
continuous. Without loss of generality we can assume that $r\geq p-1$.
Let $l=\max \{ q',\frac{p}{p-r}\} -1$, and
define the following penalty term
$$ \gamma (x,s)=[(s-v(x))^{+}]^l \quad \forall x\in \Omega , s\in\mathbb{R}.
$$
Let $M>0$ and consider the map $T:K_{0,h}$ $\to
W^{-1,p'}(\Omega )$ defined by
$$
\langle T(u),w\rangle =\int_{\Omega }\mbox{\bf{a}}(.,u,\nabla
u)\nabla w+\int_{\Omega }g(.,u,\nabla
u)w+M\int_{\Omega }\gamma (.,u)w.
$$
Then for any $u,w\in K_{0,h}$, we have
\begin{gather*}
| \int_{\Omega }g(x,u,\nabla u)w| \leq
c_{1}\| w\| _{l+1}+c_{2}\| \nabla u\|_{p}^{r}\| w\| _{l+1}, \\
| \int_{\Omega}\gamma (x,u)w| \leq c_{3}\| w\|_{l+1}
+c_{4}\| u\| _{l+1}^{l}\| w\| _{l+1},
\end{gather*}
and for each $u\in K_{f,h}-f$ , we have
$$
\int_{\Omega}\gamma (.,u)u\geq c_{5}\| u\| _{l+1}^{l+1}-c_{6}.
$$
An easy computation shows that for $\varepsilon >0$,
\begin{eqnarray*}
(T(u),u) &\geq &(\alpha -c_{2}\varepsilon )\| \nabla u\|
_{p}^p-(c\| u\| _{p}^p+c_{1}\| u\|
_{l+1}^{l+1}+c_{2}c(\varepsilon )\| u\| _{l+1}^{l+1}) \\
&&+Mc_{5}\| u\| _{l+1}^{l+1}-Mc_{6}-c_{1}c_{7}.
\end{eqnarray*}
where $c(\varepsilon )$ is a constant which depends on $\varepsilon $ and
$c>0$. Now, we choose $M$ large to get the operator $T$ coercive.
Since $T$ is bounded , pseudomonotone and continuous, then by a Theorem
in \cite{Ne}, there exists $w\in $ $K_{0,h}$ such that $(T(w),u-w)\geq 0$ for
all $ u\in K_{0,h}$.

Next, we show that $w\leq v$. Since $w-((w-v)\vee 0)\in K_{0,h}$
and since $v$ is a supersolution of (\ref{sD}), it follows that $$
\int_{\{ w>v\} }[\mathcal{A}(.,w,\nabla w)- \mathcal{A}(.,v,\nabla
v)]\nabla (w-v) \leq M\int_{\{ w>v\} }\gamma (.,w)(v-w). $$ Thus
by (P2), $(w-v)^{+}=0$ a.e. in $\Omega $ and hence, $w\leq v$ on
$\Omega $. Finally, if we take $w_{1}=w+f$, we obtain a
supersolution of the obstacle problem $K_{f,h}$. \hfill$\square$

\subsection*{Nonlinear Harmonic Space}

\paragraph{Definition}
Let $V$ be a regular set. For every $f\in C(\partial V)$, we denote by
$H_{V}f$ the solution of the Dirichlet problem with the boundary data $f$.

\begin{proposition}
Let $f$ and $g$ in $C(\partial V)$ be such that $f\leq g$. Then
\begin{description}
\item[i)]  $H_{V}f\leq H_{V}g$
\item[ii)]  For every $k\geq 0$, we have $H_{V}(k+f)\leq H_{V}(f)+k$
and $H_{V}(f)-k\leq H_{V}(f-k)$.
\end{description}
\end{proposition}

\paragraph{Definition}
Let $U$ be an open set. We denote by $\mathcal{U}(U)$ the set of
all open, regular subsets of $U$ which are relatively compact in
$U$.

We say that a function $u$ is harmonic on $U$, if $u\in C(U)$ and
$u$ is a solution of (\ref{eq1}). We denote by $\mathcal{H}(U)\
$the set of all harmonic functions on $U$. Then,
$$
\mathcal{H}(U)=\big\{ u\in C(U):H_{V}u=u\mbox{ for every }V\in \mathcal{U}
(U)\big\}.
$$

A lower semicontinuous function $u$ is said to be hyperharmonic on
$U$, if
\begin{itemize}
\item  $-\infty <u$

\item  $u\neq \infty $ in each component of $U$

\item  For each regular set $V\subset \overline{V}\subset \Omega $ and for
every $f\in \mathcal{H}(V)\cap C(\overline{V})$, the inequality
$f\leq u $ on $\partial V$ implies $f\leq u$ in $V$ .
\end{itemize}
We denote by $^{*}\mathcal{H}(U)$ the set of all hyperharmonic
functions on $U$.

An upper semicontinuous function $u$ is said to be hypoharmonic on
$U$, if
\begin{itemize}
\item  $u<+\infty $

\item  $u\neq \infty $ in each component of $U$

\item  For each regular set $V\subset \overline{V}\subset \Omega $ and each $%
f\in \mathcal{H}(\overline{V})$ $\cap C(\overline{V})$, the inequality $%
f\geq u$ on $\partial V$ implies $f\geq u$ in $V$ .
\end{itemize}
We denote by $\mathcal{H}_{*}(U)$ the set of all hypoharmonic
functions on $U$.


\begin{proposition}
Let $u\in {}^{\ast}\mathcal{H}(U)$ and $v\in \mathcal{H}_{\ast }(U)$, then
for each $k\geq 0$ we have $u+k\in {}^{\ast }\mathcal{H}(U)$ and $v-k\in
\mathcal{H}_{\ast }(U)$.
\end{proposition}

\begin{proposition}
Let $u$ be a superharmonic function and $v$ be a subharmonic
function on $U$ such that $$ {\lim \sup_{x\to z} }v(x)\leq
{ \lim \inf_{x\to z} }u(x) $$ for all $z\in \partial U$,
and both sides of the previous inequality are not simultaneously
$+\infty $ or $-\infty $, then $v\leq u$ in $U$.
\end{proposition}

\paragraph{Proof.}
Let $x\in U$ and $\varepsilon >0$. Choose a regular open set $V\subset
\overline{V}\subset U$ such that $x\in V$ and $v<u+\varepsilon $ on
$\partial V$. Let $(\varphi _i)$ $\in C^{\infty }(\Omega )$ be a decreasing
sequence converging to $v$ in $\overline{V}$. Then $\varphi
_i\leq u+\varepsilon $ on $\partial V$ for i large. Let $h=$
$H_{V}(\varphi _i)$, then $v\leq h\leq u+\varepsilon $ on $V$.
By letting $\varepsilon
\to 0$, we get $v(x)\leq u(x)$. \hfill$\square$

\begin{theorem}\label{thcB}
The space $(\mathbb{R}^{d},\mathcal{H})$ satisfies the
Bauer convergence property.
\end{theorem}

\paragraph{Proof.}
Let $(u_{n})_{n}$ be an increasing sequence in $\mathcal{H}(U)$
locally bounded. By Theorem 4.11 in \cite{Ma}, for every $V\subset
\overline{V}\subset U$, the set $\{ u_{n}(x),x\in
\overline{V,}n\in \mathbb{N}\} $ is equicontinuous . Then the
sequence converges locally and uniformly in $U$ to a continuous
function $u$. Take $\varepsilon >0$, since $u-\varepsilon \leq
u_{n}\leq u$ $+\varepsilon $ , we get $H_{V}(u)-\varepsilon \leq
u_{n}\leq H_{V}(u)+\varepsilon $  and therefore $H_{V}(u)=u$
\hfill$\square$


\begin{theorem}
Suppose that the conditions in subsection \ref{sDP} are satisfied,
$k_{0}=e=k=0$ and $\alpha \geq p-1$. Then
$(\mathbb{R}^{d},\mathcal{H})$ is a nonlinear Bauer harmonic
space.
\end{theorem}

\paragraph{Proof.}
It is clear that $\mathcal{H}$ is a sheaf of continuous functions
and by Theorem \ref{thDP} there exists a basis of regular sets
stable by intersection. The Bauer convergence property is
fulfilled by Theorem \ref {thcB}. Since $k_{0}=e=k=0$ and $\alpha
\geq p-1$, we have the following form of the Harnack inequality
(e.g. \cite{Ma},\cite{Tr} or \cite{Ser}): For every non empty open
set $U$ in $\mathbb{R}^{d}$, for every constant $M>0$ and every
compact $K$ in $U$, there exists a constant $C=C(K,M)$ such hat
$$
\sup_{K}u\leq C\inf_{K}u
$$
for every $u\in \mathcal{H}^{+}(U)$
with $u\leq M$. It follows that the sheaf $\mathcal{H}$ is non
degenerate. \hfill$\square$


\begin{theorem}
Suppose that the condition of strict monotony holds. Let $u$ $\in $ $%
\mathcal{H}^{\ast }(\Omega )\cap L^{\infty }(\Omega )$. Then $u$ is a
supersolution on $U.$
\end{theorem}

\paragraph{Proof.}
Let $V\subset \overline{V}\subset \Omega $. Let $(\varphi _i)_i$
be an increasing sequence in $C_{c}^{\infty }(\Omega )$ such that
$\ u={\sup_i} \varphi _i$ on $\overline{V}$. Let
$$
K_{\varphi _i}=\big\{ w\in W_{{\rm loc}}^{1,p}(\Omega ):\varphi _i\leq w%
\mbox{, \ }w-\varphi _i\in W_{0}^{1,p}(V)\big\} .
$$
We know by Theorem \ref{thop} that there exists a solution $u_i$ to the
obstacle problem $K_{\varphi _i}$ such that
$\|u_i\|_{\infty }\leq \| \varphi _i\| _{\infty }$. We claim that
$(u_i)_i$ is increasing. In fact $u_i\wedge u_{i+1}\in K_{\varphi _i}$,
 then
\begin{eqnarray*}
\int_{\{ u_i>u_{i+1}\}
}(\mathcal{A}(.,u_i,\nabla u_i)-\mathcal{A}(.,u_{i+1},\nabla
u_{i+1}))\nabla (u_{i+1}-u_i) &  \\ +\int_{\{
u_i>u_{i+1}\} }(\mathcal{B}(.,u_i,\nabla
u_i)-\mathcal{B}(.,u_{i+1},\nabla u_{i+1}))(u_{i+1}-u_i) &
\geq 0.
\end{eqnarray*}
Hence $\nabla (u_{i+1}-u_i)^{+}=0$ a.e. which yields that
$u_i\leq u_{i+1}$ a.e. in $V$ .

On the other hand, for each $i$ the function $u_i$ is a solution
of (\ref {eq1}) in $D_i:=\{ \varphi _i<u_i\} $.
Indeed, let $\psi \in C_{c}^{\infty }(W)$, $W \subset \overline{W
}\subset D_i $, and $\varepsilon >0$ such that $\varepsilon
\| \psi \|
\leq \inf_{\overline{W}}(u_i-\varphi _i)$. Then, we get $%
u_i+\varepsilon \psi \in K_{\varphi _i}$ and
$$
\int_{W }\mathcal{A}(.,u_i,\nabla u_i).\nabla \psi
+\int_{W }\mathcal{B}(.,u_i,\nabla u_i)\psi =0.
$$
Since
$$
\lim \inf_{x\to y} u(x)\geq u(y)\geq \varphi
_i(y)={\lim_{x\to y} }u_i(x)
$$
for all $y\in \partial D_i$, it yields, by the comparison principle, that
$u\geq u_i$ in $D_i$. Hence $u\geq u_i$ in $D$. Thus
$u={\lim_{i\to \infty } }\varphi _i\leq
\lim_{i\to \infty } u_i\leq u$.
Finally, using Remark \ref{remsup} we complete the proof. \hfill$\square$

\begin{theorem} \label{thm4.45}
Suppose that the condition of strict monotonicity holds. Then
$^{\ast }\mathcal{H}$ is a sheaf.
\end{theorem}
The proof of this theorem is the same as in \cite[Theorem 4.2]{BB}.


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\noindent\textsc{Azeddine Baalal}\\
D\'{e}partement de Math\'{e}matiques et d'Informatique,\\
Facult\'{e} des Sciences A\"{i}n Chock, \\
Km 8 Route El Jadida BP 5366 M\^{a}arif, Casablanca, Maroc.\\
E-mail: baalal@facsc-achok.ac.ma \medskip

\noindent\textsc{Nedra BelHaj Rhouma}\\
Institut Pr\'{e}paratoire aux Etudes d'Ing\'{e}nieurs de Tunis,\\
2 Rue Jawaher Lel Nehru, 1008 Montfleury, Tunis, Tunisie.\\
E-mail: Nedra.BelHajRhouma@ipeit.rnu.tn


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