\documentclass[reqno]{amsart} 
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\AtBeginDocument{{\noindent\small 
{\em Electronic Journal of Differential Equations},
Vol. 2001(2001), No. 31, pp. 1--20.\newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or 
http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu (login: ftp)}
\thanks{\copyright 2001 Southwest Texas State University.} 
\vspace{1cm}}

\begin{document} 

\title[\hfilneg EJDE--2001/31\hfil Potential theory for quasiliniear 
elliptic equations]
{Potential theory for quasiliniear elliptic equations} 

\author[A. Baalal \& A. Boukricha\hfil EJDE--2001/31\hfilneg]
{A. Baalal \& A. Boukricha  \break\quad\break
Dedicated to Prof. Wolfhard Hansen on his 60th birthday}


\address{Azeddine Baalal \hfill\break\noindent 
D\'{e}partement de Math\'{e}matiques et 
d'Informatique, Facult\'{e} des
Sciences A\"{i}n Chock\\
Km 8 Route El Jadida B.P. 5366 M\^{a}arif, Casablanca - Maroc}
\email{baalal@facsc-achok.ac.ma}

\address{Abderahman Boukricha \hfill\break\noindent
D\'{e}partement de Math\'{e}matiques, Facult\'{e} des Sciences de 
Tunis,\\
Campus Universitaire 1060 Tunis - Tunisie.}
\email{aboukricha@fst.rnu.tn}

\thanks{Submitted October 24, 2000. Published May 7, 2001.}
\thanks{Supported by Grant E02/C15 from the Tunisian Ministry of Higher 
Education}
\subjclass{31C15, 35J60}
\keywords{Quasilinear elliptic equation, Convergence property,
\hfill\break\indent  Keller-Osserman property, Evans functions }


\begin{abstract}
We discuss the potential theory associated with the quasilinear 
elliptic equation 
$$
-\mathop{\rm div}( \mathcal{A}(x,\nabla u))+\mathcal{B}(x,u)=0.
$$ 
We study the validity of Bauer convergence property, the Brelot 
convergence property. We discuss the validity of the 
Keller-Osserman property and the existence of Evans functions.
\end{abstract}

\maketitle


\makeatletter
\numberwithin{equation}{section}
\makeatother


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

\section{Introduction}

This paper is devoted to a study of the quasilinear elliptic equation 
\begin{equation}
-\mathop{\rm div}\left( \mathcal{A}(x,\nabla u)\right) 
+\mathcal{B}(x,u)=0\,,
\label{eq2}
\end{equation}
where $\mathcal{A}:\mathbb{R}^d\times \mathbb{R}^d\to 
\mathbb{R}^d$ and $\mathcal{B}:\mathbb{R}^d\times \mathbb{R}
\to \mathbb{R}$ are Carath\'{e}odory functions satisfying the
structure conditions given in Assumptions (I), (A1), 
(A2), (A3), and (M) below. 
In particular we are interested in the potential theory, the degeneracy 
of the sheaf of continuous solutions and the existence of Evans 
functions for the equation (\ref{eq2}).

Equation of the same type as (\ref{eq2}) were investigated in earlier
years in many interesting papers, \cite{Se,Tr,LU,MZ}.
An axiomatic potential theory associated with the equation 
$\mathop{\rm div}( \mathcal{A}(x,\nabla u)) =0$ was recently 
introduced and discussed in \cite{HKM}. These axiomatic setting are 
illustrated by the
study of the $p$-Laplace equation 
$\Delta_{p}u=\mathop{\rm div}( \left| \nabla
u\right| ^{p-2}\nabla u) $ obtained by $\mathcal{A}(x,\xi )=| \xi
|^{p-2}\xi $ for every $x\in \mathbb{R}^d$ and $\xi \in \mathbb{R}^d$.
 We have $\Delta_2=\Delta $ where $\Delta $, the Laplace operator
on $\mathbb{R}^d$.

Our paper is organized as follows: In the second section we introduce the basic notation. In the third section we present the
structure conditions needed for the mappings $\mathcal{A}$ and 
$\mathcal{B}$
in order to consider the equation (\ref{eq2}). We then use the 
variational
inequality to prove the solvability of the variational Dirichlet 
problem
related to (\ref{eq2}). In section 4 we prove a comparison principle 
for
supersolutions and subsolutions, existence and uniqueness of the 
Dirichlet
problem related to the sheaf $\mathcal{H}$ of continuous solutions of 
(\ref
{eq2}), as well as the existence of a basis of regular sets stable by
intersection. In the fifth section we discuss the potential theory 
associated
with equation (\ref{eq2}), prove that the harmonic sheaf $\mathcal{H}$
of solutions of (\ref{eq2}) satisfies the Bauer convergence property,
then introduce the presheaves of hyper-harmonic functions $^{\ast 
}\mathcal{H}
$ and of hypoharmonic functions $_{\ast }\mathcal{H}$ and prove a 
comparison
principle. In the sixth section we prove, using the obstacle problem, 
that 
$^{\ast }\mathcal{H}$ and $_{\ast }\mathcal{H}$ are sheaves. In the 
seventh
section we study the degeneracy of the sheaf $\mathcal{H}$; we are not 
able
to prove that the sheaf $\mathcal{H}$ is non degenerate even if we have 
the following Harnack inequality \cite{Se,Tr,MZ,B-h}:

\emph{For every open domain $U$ in $\mathbb{R}^d$ and
every compact subset $K$ of $U$ there exists two non-negative constants 
$c_1$ and $c_{2}$ such that
for every $h\in \mathcal{H}^{+}(U)$,}
\begin{equation*}
\sup_{K}h\leqslant c_1\inf_{K}h+c_{2}\,.
\end{equation*}

Let $U$ be an open subset of $\mathbb{R}^d$, $d\geqslant 1$ and $\alpha 
$
a positive real number, let $0<\epsilon <1$ and $b$ be a non-negative
function in $L_{\text{loc}}^{\frac{d}{p-\epsilon }}(\mathbb{R}^d)$. For 
every
open $U$ we consider the set $\mathcal{H}_\alpha (U)$ of all functions 
$u\in \mathcal{W}_{\text{loc}}^{1,p}(U)\cap \mathcal{C}(U)$ which are 
solutions of
the equation (\ref{eq2}) with $\mathcal{B}(x,\zeta )=b(x)\mathop{\rm 
sgn}(\zeta )\left|
\zeta \right| ^{\alpha }$, then $(\mathbb{R}^d,\mathcal{H}_\alpha )$ is
a nonlinear Bauer space. In particular $\mathcal{H}_\alpha $ is non
degenerate on $\mathbb{R}^d$. For $\alpha <p-1$, the Harnack inequality
and the Brelot convergence property are valid, but in contrast to the 
linear
and quasilinear theory (see e.g. \cite{HKM}) $(\mathbb{R}^d,\mathcal{H}
_\alpha )$ is not elliptic in the sense of Definition \ref{defell}. In 
the
eighth section, we define, as in \cite{B-ko}, regular Evans functions 
$u$
tending to the infinity (or exploding) at the regular boundary points 
of $U
$. We assume that $\mathcal{A}$ satisfies the following supplementary
derivability and homogeneity conditions:

\begin{itemize}
\item  For every $x_0\in \mathbb{R}^d$, the function $F$ from 
$\mathbb{R}^d$ to $\mathbb{R}^d$ defined by $F(x)=\mathcal{A}(x,x-x_0)$
is differentiable and $\mathop{\rm div}F$ is locally (essentially) 
bounded.

\item  $\mathcal{A}(x,\lambda \xi )=\lambda \left| \lambda \right|
^{p-2}\mathcal{A}(x,\xi )$ for every $\lambda \in \mathbb{R}$ and every 
$x$, 
$\xi \in \mathbb{R}^d$.
\end{itemize}

These conditions are satisfied in the particular case of the 
$p$-Laplace
operator with $p\geqslant 2$. We then prove that for every $\alpha 
>p-1$, 
the Keller-Osserman property in $(\mathbb{R}^d, \mathcal{H}_\alpha )$ 
is valid;  i. e., every open ball
admits a regular Evans function, which yields the validity of the 
Brelot
convergence property. Among others, we prove for $\alpha >p-1$ a 
theorem of the Liouville type in the form 
$\mathcal{H}_\alpha (\mathbb{R}^d)=\{ 0\} $. 
Finally in the ninth section , we consider some applications of the previous results to the 
case of
 the $p$-Laplace
operator, where we also prove the uniqueness of the regular Evans 
function
for star domain and strict positive $b$ and $\mathcal{H}_\alpha $ for 
$\alpha >p-1$.

Note that our methods are applicable to broader class of weighted
equations (see \cite{HKM}). The use of the constant weight $\equiv 1$ 
is
only for sake of simplicity.

\section{Notation}

We introduce the basic notation which will be observed throughout this 
paper.
$\mathbb{R}^d$ is the real Euclidean $d$-space, $d\geq 2$. For an open 
set 
$U$ of $\mathbb{R}^d$ and an positive integer $k$, $\mathcal{C}^k(U)$ 
is
the set of all $k$ times continuously differentiable functions on an 
open
set $U$. $\mathcal{C}^\infty (U):=\bigcap_{k\geq 1}\mathcal{C}^k(U)$ 
and 
$\mathcal{C}_{c}^\infty (U)$ the set of all functions in $\mathcal{C}
^\infty (U)$ compactly supported by $U$. For a measurable set $X$, 
$\mathcal{B}(X)$ denotes the set of all Borel numerical functions on 
$X$ and
for $q\geq 1$, $L^{q}(X)$ is the $q^{th}-$power Lebesgue space defined
on $X$. Given any set $\mathcal{Y}$ of functions $\mathcal{Y}_{b}$ (
$\mathcal{Y}^{+}$ resp.) denote the set of all functions in 
$\mathcal{Y}$
which are bounded (positive resp.). $\mathcal{W}^{1,q}(U)$ is the 
$(1,q)
$-Sobolev space on $U$. $\mathcal{W}_0^{1,q}(U)$ the closure of 
$\mathcal{C}
_{c}^\infty (U)$ in $\mathcal{W}^{1,q}(U)$, relatively to its norm. 
$\mathcal{W}^{-1,q'}(U)$ is the dual of $\mathcal{W}_0^{1,q}(U)$, 
$q'=q(q-1)^{-1}$. $u\wedge v$ (resp. $u\vee v$ ) is the infinimum
(resp. the maximum ) of $u$ and $v$; $u^{+}=u\vee 0$ and $u^{-}=u\wedge 
0$.

\section{Existence and Uniqueness of Solutions\label{ses}}

Let $\Omega $ be a bounded open subset of $\mathbb{R}^d$ ($d\geqslant 
1$).
We will investigate the existence of solutions 
$u\in \mathcal{W}^{1,p}(\Omega )$, $1<p\leqslant d$, of the 
variational Dirichlet problem associated with the quasilinear elliptic 
equation
\begin{equation*}
-\mathop{\rm div}\left( \mathcal{A}(x,\nabla u)\right) 
+\mathcal{B}(x,u)=0.
\end{equation*}

In this paper we suppose that the functions $\mathcal{A}:\mathbb{R}
^d\times \mathbb{R}^d\to \mathbb{R}^d$ and $\mathcal{B}:
\mathbb{R}^d\times \mathbb{R}\to \mathbb{R}$ are given
Carath\'{e}odory functions and the following structure conditions are
satisfied:
\begin{enumerate}
\item[\bf(I)]  $\zeta \to \mathcal{B}(x,\zeta )$ is increasing
and $\mathcal{B}(x,0)=0$ for every $x\in \mathbb{R}^d$.

\item[\bf (A1)]  There exists $0<\epsilon <1$ such that 
for any $u\in L^\infty (\mathbb{R}^d)$,
\begin{equation*}
\mathcal{B}(.,u(.))\in L_{\text{loc}}^{\frac{d}{p-\epsilon }}
(\mathbb{R}^d)\,.
\end{equation*}

\item[\bf (A2)]  There exists $\nu >0$ such that 
for every $\xi \in \mathbb{R}^d$,
\begin{equation*}
\left| \mathcal{A}(x,\xi )\right| \leqslant \nu \left| \xi \right| 
^{p-1}\,.
\end{equation*}

\item[\bf(A3)]  There exists $\mu >0$ such that 
for every $\xi \in \mathbb{R}^d$,
\begin{equation*}
\mathcal{A}(x,\xi ).\xi \geqslant \mu \left| \xi \right| ^{p}\,.
\end{equation*}

\item[\bf(M)]  For all $\xi $, $\xi '\in \mathbb{R}^d$ with $\xi \neq 
\xi'$,
\begin{equation*}
\left[ \mathcal{A}(x,\xi )-\mathcal{A}(x,\xi ')\right] \cdot \left(
\xi -\xi '\right) >0\,.
\end{equation*}
\end{enumerate}

We recall that assumptions (A2), (A3) and (M) are satisfied in
the framework of \cite{HKM} when the admissible weight is
 $\omega \equiv 1$.

Recall that $u\in \mathcal{W}_{\text{loc}}^{1,p}(\Omega )$ is a 
\emph{solution} of (\ref{eq2}) in $\Omega $ provided that
for all $\varphi \in \mathcal{W}_0^{1,p}(\Omega )$ and 
$\mathcal{B}(.,u)\in L_{\text{loc}}^{p^{\ast '}}(\Omega )$,
\begin{equation}
\int_{\Omega }\mathcal{A}(x,\nabla u)\cdot\nabla \varphi 
dx+\int_{\Omega }
\mathcal{B}(x,u)\varphi dx=0\,.  \label{eq22}
\end{equation}
A function $u\in \mathcal{W}_{\text{loc}}^{1,p}(\Omega )$ is termed 
\emph{subsolutions} (resp. \emph{supersolutions}) of (\ref{eq2}) if 
for all non-negative functions $\varphi \in \mathcal{W}_0^{1,p}(\Omega 
)$
and $\mathcal{B}(.,u)\in L_{\text{loc}}^{p^{\ast '}}(\Omega )$,
\begin{equation*}
\int_{\Omega }\mathcal{A}(x,\nabla u)\cdot\nabla \varphi 
dx+\int_{\Omega }
\mathcal{B}(x,u)\varphi dx\leqslant 0\quad \text{(resp. }\geqslant 0)\,.
\end{equation*}
If $u$ is a bounded subsolution (resp. bounded supersolution), then
for every $k\geqslant 0$, $u-k$ (resp. $u+k$) is also subsolution 
(resp. supersolution) for (\ref{eq2}).

For a positive constant $M$  and $u\in L^{p}(\Omega )$, we define 
the truncated function 
\begin{equation*}
\tau_{M}(u)(x)=\left\{ 
\begin{array}{ll}
-M & u(x)\leqslant -M \\ 
u(x) & -M<u(x)<M \\ 
M, & M\leqslant u(x)
\end{array}
\right.
\end{equation*}
(a.e. $x\in \Omega $). It is clear that the truncation mapping 
$\tau_{M}$
is bounded and continuous from $L^{p}(\Omega )$ to itself.

For $u\in \mathcal{W}^{1,p}(\Omega )$  and 
$\mathcal{B}(x,\tau_{M}(u))\in L_{\text{loc}}^{p^{\ast '}}(\Omega )$,
 we define
$\mathcal{L}_{M}:\mathcal{W}^{1,p}(\Omega )\to \mathcal{W}
^{-1,p^{'}}(\Omega )$ as 
\begin{equation*}
\left\langle \mathcal{L}_{M}(u),\varphi \right\rangle :=\int_{\Omega }
\mathcal{A}(x,\nabla u)\cdot\nabla \varphi dx
+\int_{\Omega }\mathcal{B}(x,\tau_{M}(u))\varphi dx\,,\quad
\varphi \in \mathcal{W}_0^{1,p}(\Omega )
\end{equation*}
here $\left\langle .,.\right\rangle $ is the pairing between 
$\mathcal{W}
^{-1,p^{'}}(\Omega )$ and $\mathcal{W}^{1,p}(\Omega )$. It follows
from Assumptions (A1), (A2), (A3), and the
carath\'{e}odory conditions that $\mathcal{L}_{M}$ is well defined. We
consider the variational inequality 
\begin{equation}
\left\langle \mathcal{L}_{M}(u),v-u\right\rangle \geqslant 0, 
\quad \forall v\in \mathcal{K}, u\in \mathcal{K},  \label{var1}
\end{equation}
where $\mathcal{K}$ is a given closed convex set in $\mathcal{W}^{1,p}(
\mathbb{\Omega })$ such that for given $f\in \mathcal{W}^{1,p}(\mathbb{
\Omega })$, 
\begin{equation*}
\mathcal{K}\subset f+\mathcal{W}_0^{1,p}(\mathbb{\Omega }).
\end{equation*}
Typical examples of closed convex sets $\mathcal{K}$ are as follows: 
for $f\in \mathcal{W}^{1,p}(\mathbb{\Omega })$ and $\psi
_1,\psi_2$ $:\Omega \to \left[ -\infty ,+\infty \right] $
let the convex set is 
\begin{equation}
\mathcal{K}_{\psi_1,\psi_2}^{f}=\mathcal{K}_{\psi_1,\psi
_{2}}^{f}(\Omega )=\left\{ u\in \mathcal{W}^{1,p}(\Omega ):\psi_1\leq
u\leqslant \psi_2\text{ a.e. in }\Omega,\, u-f\in \mathcal{W}
_0^{1,p}(\Omega )\right\}.  \label{con1}
\end{equation}
We write $\mathcal{K}_{\psi_1}^{f}=\mathcal{K}_{\psi_1,+\infty 
}^{f}(\Omega )$ and, if $f=\psi_1\in \mathcal{W}^{1,p}(\Omega )$, 
$\mathcal{K}_{f}=\mathcal{K}_{f}^{f}$. A function $u$ satisfying 
(\ref{var1}) with $M=+\infty $ and the closed convex sets 
$\mathcal{K}_{\psi_1}^{f}$ is called a \emph{solution to the obstacle 
problem} in $\mathcal{K}_{\psi_1}^{f}$. For the notion of obstacle 
problem, the reader is referred to monograph \cite[p. 60]{HKM} or 
\cite[Chap. 5]{MZ}.
We observe that any solution of the obstacle problem in 
$\mathcal{K}_{\psi_1}^{f}(\Omega )$ is always a supersolution of the 
equation (\ref{eq2}) in $\Omega $. Conversely, a supersolution $u$ is 
always a solution to the obstacle problem in 
$\mathcal{K}_{u}^{u}(\omega )$ for all open 
$\omega \subset \overline{\omega }\subset \Omega $. Furthermore a 
solution $u$ to equation (\ref{eq2}) in an open set $\Omega $ is a 
solution to the obstacle problem in 
$\mathcal{K}_{-\infty }^{u}(\omega )$ for all open 
$\omega \subset \overline{\omega }\subset \Omega $. Similarly, a 
solution to the obstacle problem in 
$\mathcal{K}_{-\infty }^{u}(\Omega )$ is a solution to (\ref{eq2}).

For the uniqueness of a solution to the obstacle problem we have 
following
lemma \cite[Lemma 3.22]{HKM}:

\begin{lemma} \label{lem1}
Suppose that $u$ is a solution to the obstacle problem in 
$\mathcal{K}_{g}^{f}(\Omega )$. If $v\in \mathcal{W}^{1,p}(\Omega )$ is 
a supersolution of (\ref{eq2}) in $\Omega $ such that 
$u\wedge v\in \mathcal{K}_{g}^{f}(\Omega )$, then a.e. $u\leqslant v$ 
in $\Omega $.
\end{lemma}

\begin{theorem}
\label{theo2}Let $\psi_1$ and $\psi_2$ in $L^\infty (\mathbb{\Omega 
})$, $f\in \mathcal{W}^{1,p}(\mathbb{\Omega })$ and $\mathcal{K}_{\psi
_1,\psi_2}^{f}$ as above assume that $\mathcal{K}_{\psi_1,\psi
_{2}}^{f}$ is non empty. Then for every positive constant $M$, $\left\| 
\psi
_1\right\|_{\infty }\vee \left\| \psi_2\right\|_{\infty }$ $\leqslant
M<+\infty $ the variational inequality \emph{(\ref{var1})} has a unique
solution. Moreover, if $w\in \mathcal{W}^{1,p}(\Omega )$ is a 
supersolution 
(resp. subsolution) to the equation $\emph{(\ref{eq2})}$ such
that $w\wedge u$ (resp. $w\vee u$) $\in \mathcal{K}_{\psi
_1,\psi_2}^{f}$, then $u\leqslant w$ (resp. $w\leqslant u$).
\end{theorem}

\begin{proof}
Let $\left\| \psi_1\right\|_{\infty }\vee \left\| \psi_2\right\|
_{\infty }$ $\leqslant M<+\infty $. If $u$, $v\in \mathcal{K}_{\psi
_1,\psi_2}^{f}$ are solutions of (\ref{var1}), it follows from 
$(\mathbf{I})$ and $(\mathbf{M})$ that 
\begin{eqnarray*}
0 &\geqslant &\int_{\Omega }\left[ \mathcal{A}(x,\nabla u)-\mathcal{A}
(x,\nabla v)\right] \cdot\nabla (v-u)dx \\
&&+\int_{\Omega }\left[ \mathcal{B}(x,\tau_{M}(u))-\mathcal{B}(x,\tau
_{M}(v))\right] (v-u)dx \\
&=&\left\langle \mathcal{L}_{M}(u)-\mathcal{L}_{M}(v),v-u\right\rangle
\geqslant 0,
\end{eqnarray*}
then $v-u$ is constant on connected components of $\Omega $. This, on 
the
other hand, since $v-u\in \mathcal{W}_0^{1,p}(\Omega )$, implies that 
$v=u$.

To prove the existence we will use \cite[Corollary III.1.8, p. 87]{KS}.
Since $\mathcal{K}_{\psi_1,\psi_2}^{f}$ is a non empty closed convex
subset of $\mathcal{W}^{1,p}(\Omega )$, it is enough to prove that 
$\mathcal{
L}_{M}$ is monotone, coercive and weakly continuous on 
$\mathcal{K}_{\psi
_1,\psi_2}^{f}$. We have 
\begin{eqnarray*}
\left\langle \mathcal{L}_{M}(u)-\mathcal{L}_{M}(v),u-v\right\rangle
&=&\int_{\Omega }\left[ \mathcal{A}(x,\nabla u)-\mathcal{A}(x,\nabla v)
\right] \cdot\nabla \left( u-v\right) dx+ \\
&&+\int_{\Omega }\left[ \mathcal{B}(x,\tau_{M}(u))-\mathcal{B}(x,\tau
_{M}(v))\right] .\left( u-v\right) dx\text{ }
\end{eqnarray*}
for all $v$, $u\in \mathcal{K}_{\psi_1,\psi_2}^{f}$ and the structure
conditions on $\mathcal{A}$ and $\mathcal{B}$ yield that 
$\mathcal{L}_{M}$
is monotone and coercive (for the definition of monotone or coercive
operator the reader is referred to \cite{Li,KS}).

To show that $\mathcal{L}_{M}$ is weakly continuous on 
$\mathcal{K}_{\psi
_1,\psi_2}^{f}$, let $(u_n)_n\subset \mathcal{K}_{\psi_1,\psi
_{2}}^{f}$ be a sequence that converges to $u\in $ $\mathcal{K}_{\psi
_1,\psi_2}^{f}$. There is a subsequence $(u_{n_{k}})_{k}$ such that 
$u_{n_{k}}\to u$ and $\nabla u_{n_{k}}\to \nabla u$ pointwise
a.e. in $\Omega $. Since $\mathcal{A}$ and $\mathcal{B}$ are
Carath\'{e}odory functions, $\mathcal{A}(.,\nabla u_{n_{k}})$ and 
$\mathcal{B
}(.,\tau_{M}(u_{n_{k}}))$ converges in measure to $\mathcal{A}(.,\nabla 
u)$
and $\mathcal{B}(x,\tau_{M}(u))$ respectively \cite{Kr}. Pick a
subsequence, indexed also by $n_{k}$, such that $\mathcal{A}(.,\nabla
u_{n_{k}})$ and $\mathcal{B}(.,\tau_{M}(u_{n_{k}}))$ converges 
pointwise
a.e. in $\Omega $ to $\mathcal{A}(.,\nabla u)$ and $\mathcal{B}(x,\tau
_{M}(u))$ respectively. Because $(u_{n_{k}})_{n_{k}}$ is bounded in 
$\mathcal{W}^{1,p}(\Omega )$, it follow that $\left( 
\mathcal{A}(.,\nabla
u_{n_{k}})\right)_{k}$ is bounded in $\left( L^{\frac{p}{p-1}}(\Omega
)\right) ^d$ and that $\mathcal{A}(.,\nabla u_{n_{k}})\rightharpoonup 
\mathcal{A}(.,\nabla u)$ weakly in $\left( L^{\frac{p}{p-1}}(\Omega 
)\right)
^d$. We have also $\mathcal{B}(.,\tau_{M}(u_{n_{k}}))\rightharpoonup 
\mathcal{B}(.,\tau_{M}(u))$ weakly in $L^{p^{\ast '}}(\Omega )$.
Since the weak limits are independent of the choice of the subsequence, 
we
have for all $\varphi \in \mathcal{W}_0^{1,p}(\Omega )$ 
\begin{equation*}
\left\langle \mathcal{L}_{M}(u_n),\varphi \right\rangle \to
\left\langle \mathcal{L}_{M}(u),\varphi \right\rangle
\end{equation*}
and hence $\mathcal{L}_{M}$ is weakly continuous on 
$\mathcal{K}_{\psi_1,\psi_2}^{f}$.

Let now $w\in \mathcal{W}^{1,p}(\Omega )$ be a supersolution of the 
equation 
(\ref{eq2}) such that $u\wedge w\in \mathcal{K}_{\psi_1,\psi_2}^{f}
$, then $u-\left( u\wedge w\right) \in \mathcal{W}_0^{1,p}(\Omega )$ 
and we
have 
\begin{eqnarray*}
0 &\leqslant &\int_{\Omega }\left[ \mathcal{A}(x,\nabla w)-\mathcal{A}
(x,\nabla u)\right] \cdot\nabla \left( u-\left( u\wedge w\right) 
\right) dx+ \\
&&+\int_{\Omega }\left[ \mathcal{B}(x,\tau_{M}(w))-\mathcal{B}(x,\tau
_{M}(u))\right] .\left( u-\left( u\wedge w\right) \right) dx \\
&=&\int_{\left\{ u>w\right\} }\left[ \mathcal{A}(x,\nabla \left( 
u\wedge
w\right) )-\mathcal{A}(x,\nabla u)\right] \cdot\nabla \left( u-\left( 
u\wedge
w\right) \right) dx+ \\
&&+\int_{\left\{ u>w\right\} }\left[ \mathcal{B}(x,\tau_{M}\left( 
u\wedge
w\right) )-\mathcal{B}(x,\tau_{M}(u))\right] .\left( u-\left( u\wedge
w\right) \right) dx \\
&\leqslant &0.
\end{eqnarray*}

It follow, by $(\mathbf{I})$ and $(\mathbf{M})$ , that $\nabla \left(
u-\left( u\wedge w\right) \right) =0$ a.e. in $\Omega $ and hence 
$u\leqslant w$ a.e. in $\Omega $. The same proof is valid if $w$ is a
subsolution.
\end{proof}

As an application of Theorem \ref{theo2}, we have the following two
theorems.

\begin{theorem}
\label{thdop}Let $f\in \mathcal{W}^{1,p}(\mathbb{\Omega })\cap $ 
$L^{\infty
}(\mathbb{\Omega })$  and 
\begin{equation*}
\mathcal{K}=\left\{ u\in \mathcal{W}^{1,p}(\Omega ):f\leq u\leqslant 
\text{ }
\left\| f\right\|_{\infty }\text{ a. e., }u-f\in \mathcal{W}
_0^{1,p}(\Omega )\right\} .
\end{equation*}
Then there exists $u\in \mathcal{K}$ such that 
\begin{equation*}
\left\langle \mathcal{L}(u),v-u\right\rangle \geqslant 0
\quad \text{for all } v\in \mathcal{K}\,.
\end{equation*}
Moreover, $u$ is a supersolution of (\ref{eq2}) in $\Omega$.
\end{theorem}

\begin{proof}
For $m>0$, by Theorem \ref{theo2} there exists a unique function 
$u_{m}$ in 
\begin{equation*}
\mathcal{K}_{f,\left\| f\right\|_{\infty }+m}^{f}=\left\{ u\in 
\mathcal{W}
^{1,p}(\Omega ):f\leqslant u\leqslant \left\| f\right\|_{\infty 
}+m\text{\
a. e., }u-f\in \mathcal{W}_0^{1,p}(\Omega )\right\}
\end{equation*}
such that 
\begin{equation*}
\left\langle \mathcal{L}_{\left\| f\right\|_{\infty
}+m}(u_{m}),v-u_{m}\right\rangle \geqslant 0
\end{equation*}
for all $v\in \mathcal{K}_{f,\left\| f\right\|_{\infty }+m}^{f}$. Since 
$u_{m}-\left\| f\right\|_{\infty }=u_{m}-f+f-\left\| f\right\|_{\infty
}\leqslant u_{m}-f$ and $\left( u_{m}-f\right) ^{+}\geqslant \left(
u_{m}-\left\| f\right\|_{\infty }\right) ^{+}$, we have $\eta :=\left(
u_{m}-\left\| f\right\|_{\infty }\right) ^{+}\in \mathcal{W}
_0^{1,p}(\Omega )$ (see e. g. \cite[Lemma1.25]{HKM} ). Moreover, since 
$u_{m}-\eta \in \mathcal{K}_{f,\left\| f\right\|_{\infty }+m}^{f}$ and 
$\left\| f\right\|_{\infty }$ is a supersolution of (\ref{eq2}), we 
have 
\begin{eqnarray*}
0 &\leqslant &-\int_{\Omega }\mathcal{A}(x,\nabla u_{m})\cdot\nabla 
\eta
dx-\int_{\Omega }\left[ \mathcal{B}(x,u_{m})-\mathcal{B}(x,\left\| 
f\right\|
_{\infty })\right] \eta dx \\
&=&-\int_{\left\{ u_{m}>\left\| f\right\|_{\infty }\right\} 
}\mathcal{A}
(x,\nabla u_{m})\cdot\nabla u_{m}dx+ \\
&&-\int_{\left\{ u_{m}>\left\| f\right\|_{\infty }\right\} }\left[ 
\mathcal{
B}(x,u_{m})-\mathcal{B}(x,\left\| f\right\|_{\infty })\right] \left(
u_{m}-\left\| f\right\|_{\infty }\right) dx \\
&\leqslant &0,
\end{eqnarray*}
then $\nabla \eta =0$ a.e. in $\Omega $ by $(\mathbf{M})$. Because 
$\eta \in 
\mathcal{W}_0^{1,p}(\Omega )$, $\eta =0$ a.e. in $\Omega $. It follows
that $u_{m}\leqslant \left\| f\right\|_{\infty }$ a.e. in $\Omega $. It
follows that $u_{m}\leqslant \left\| f\right\|_{\infty }$ a.e. in 
$\Omega 
$, and therefore $f\leqslant u_{m}<\left\| f\right\|_{\infty }+m$ a.e. 
in 
$\Omega $. Given a non-negative $\varphi \in \mathcal{C}_{c}^\infty 
(\Omega
)$ and $\varepsilon >0$ sufficiently small such that $u_{m}+\varepsilon
\varphi \in \mathcal{K}_{f,\left\| f\right\|_{\infty }+m}^{f}$,
consequently 
\begin{equation*}
\left\langle \mathcal{L}(u_{m}),\varphi \right\rangle \geqslant 0
\end{equation*}
which means that $u_{m}$ is a supersolution of (\ref{eq2}) in $\Omega 
$.
\end{proof}

\begin{theorem}
\label{thDir} Let $\Omega $ be a bounded open set of $\mathbb{R}^d$, 
$f\in 
\mathcal{W}^{1,p}(\Omega )\cap L^\infty (\Omega )$. Then there is a 
unique
function $u\in \mathcal{W}^{1,p}(\Omega )$ with $u-f\in \mathcal{W}
_0^{1,p}(\Omega )$ such that 
\begin{equation*}
\int_{\Omega }\mathcal{A}(x,\nabla u)\cdot\nabla \varphi 
dx+\int_{\Omega }
\mathcal{B}(x,u)\varphi dx=0,
\end{equation*}
whenever $\varphi \in \mathcal{W}_0^{1,p}(\Omega )$.
\end{theorem}

\begin{proof}
For $m>0$, by Theorem \ref{theo2}, there exists a unique $u_{m}$ in 
\begin{equation*}
\mathcal{K}_{f,m}:=\left\{ u\in \mathcal{W}^{1,p}(\Omega ):\left| 
u\right|
\leqslant \left\| f\right\|_{\infty }+m\text{ a. e., }u-f\in 
\mathcal{W}
_0^{1,p}(\Omega )\right\} ,
\end{equation*}
such that 
\begin{equation*}
\left\langle \mathcal{L}_{\left\| f\right\|_{\infty
}+m}(u_{m}),v-u_{m}\right\rangle \geqslant 0,
\end{equation*}
for all $v\in \mathcal{K}_{f,m}$. Since $u_{m}+\left\| 
f\right\|_{\infty
}=u_{m}-f+f+\left\| f\right\|_{\infty }\geqslant u_{m}-f$ and $\left(
u_{m}-f\right) ^{-}\leqslant \left( u_{m}+\left\| f\right\|_{\infty
}\right) \wedge 0$, we have $\eta :=\left( u_{m}+\left\| 
f\right\|_{\infty}\right) \wedge 0\in \mathcal{W}_0^{1,p}(\Omega )$ 
(see e. g. \cite[Lemma1.25]{HKM}). Moreover, since $\eta +u_{m}\in 
\mathcal{K}_{f,m}$
and $-\left\| f\right\|_{\infty }$ is a subsolution of (\ref{eq2}), we
have 
\begin{eqnarray*}
0 &\leqslant &\int_{\Omega }\mathcal{A}(x,\nabla u_{m})\cdot\nabla \eta
dx+\int_{\Omega }\left[ \mathcal{B}(x,u_{m})-\mathcal{B}(x,-\left\|
f\right\|_{\infty })\right] \eta dx \\
&=&-\int_{\left\{ u_{m}<-\left\| f\right\|_{\infty }\right\} 
}\mathcal{A}
(x,\nabla u_{m})\cdot\nabla u_{m}dx+ \\
&&-\int_{\left\{ u_{m}<-\left\| f\right\|_{\infty }\right\} }\left[ 
\mathcal{B}(x,u_{m})-\mathcal{B}(x,-\left\| f\right\|_{\infty })\right]
\left( u_{m}+\left\| f\right\|_{\infty }\right) dx \\
&\leqslant &0,
\end{eqnarray*}
then $\nabla \eta =0$ a.e. in $\Omega $ by $(\mathbf{M})$. Because 
$\eta \in 
\mathcal{W}_0^{1,p}(\Omega )$, $\eta =0$ a.e. in $\Omega $. It follows
that $-\left\| f\right\|_{\infty }\leqslant u_{m}$ a.e. in $\Omega $. 
Note
that $-u_{m}$ is also a solution in $\mathcal{K}_{-f,m}$ of the 
following
variational inequality 
\begin{eqnarray*}
\left\langle \widetilde{\mathcal{L}}_{\left\| f\right\|_{\infty
}+m}(u),v-u\right\rangle &= &
\int_{\Omega }\widetilde{\mathcal{A}}(x,\nabla u)\cdot\nabla 
\left( v-u\right) dx \\ 
&& +\int_{\Omega }\widetilde{\mathcal{B}}(x,\tau_{\left\| 
f\right\|_{\infty}+m}(u))\left( v-u\right) dx\geqslant 0,
\end{eqnarray*}
where $\widetilde{\mathcal{A}}(.,\xi )=-\mathcal{A}(.,-\xi )$ and 
$\widetilde{\mathcal{B}}(.,\zeta )=-\mathcal{B}(.,-\zeta )$ which 
satisfy the
same assumptions as $\mathcal{A}$ and $\mathcal{B}$. It follows that 
$u_{m}\leqslant \left\| f\right\|_{\infty }$ a.e. in $\Omega $, and
therefore $\left| u_{m}\right| <\left\| f\right\|_{\infty }+m$ a.e. in 
$\Omega $. Given $\varphi \in \mathcal{C}_{c}^\infty (\Omega )$ and 
$\varepsilon >0$ sufficiently small such that $u_{m}\pm \varepsilon 
\varphi
\in \mathcal{K}_{f,m}$, consequently 
\begin{equation*}
\left\langle \mathcal{L}(u_{m}),\varphi \right\rangle =0
\end{equation*}
which means that $u_{m}$ is a desired function.
\end{proof}

By regularity theory (e.g. \cite[Corollary 4.10]{MZ}), any bounded 
solution
of (\ref{eq2}) can be redefined in a set of measure zero so that it
becomes continuous.

\begin{definition}\rm
\label{defpr} A relatively compact open set $U$ is called 
$p\!-\!regularity$
if, for each function $f\in \mathcal{W}^{1,p}(U)\cap 
\mathcal{C}(\overline{U}
)$, the continuous solution $u$ of (\ref{eq2}) in $U$  with $u-f\in 
\mathcal{W}^{1,p}(U)$ satisfies $\lim_{x\to y}u(x)=f(y)$ for
all $y\in \partial U$.

A relatively compact open set $U$ is called regular, if for every 
continuous
function $f$ on $\partial U$, there exists a unique continuous solution 
$u$
of (\ref{eq2}) on $U$ such that $\lim_{x\to y}u(x)=f(y)$
for all $y\in \partial U$.
\end{definition}

If $U$ is $p$-regular and $f\in \mathcal{W}^{1,p}(U)\cap \mathcal{C}(
\overline{U})$, then the solution $u$ given by Theorem~\ref{thDir} 
satisfies 
\begin{equation*}
\lim_{x\in U, x\to z}u(x)=f(z)
\end{equation*}
for all $z\in \partial U$ \cite[Corollary 4.18]{MZ}.

\section{Comparison Principle and Dirichlet Problem\label{scp}}

The following \emph{comparison principle} is useful for the potential 
theory
associated with equation (\ref{eq2}):

\begin{lemma}
\label{lem3} Suppose that $u$ is a supersolution and $v$ is a 
subsolution on 
$\Omega $ such that 
\begin{equation*}
\limsup_{x\to y}v(x)\leqslant \liminf_{x\to
y}u(x)
\end{equation*}
for all $y\in \partial \Omega $ and if both sides of the inequality are 
not
simultaneously $+\infty $ or $-\infty $, then $v\leqslant u$ in $\Omega 
$.
\end{lemma}

\begin{proof}
By the regularity theory (see e.g. \cite[Corollary 4.10]{MZ}), we may 
assume
that $u$ is lower semicontinuous and $v$ is upper semicontinuous on 
$\Omega 
$. For fixed $\varepsilon >0$, the set $K_{\varepsilon }=\left\{ x\in 
\Omega
:v(x)\geqslant u(x)+\varepsilon \right\} $ is a compact subset of 
$\Omega $
and therefore $\varphi =(v-u-\varepsilon )^{+}\in \mathcal{W}_0^{1,p}(
\mathbb{R}^d)$. Testing by $\varphi $, we obtain
\begin{multline}
\int_{\left\{ v>u+\varepsilon \right\} }\left[ \mathcal{A}(x,\nabla
(u+\varepsilon ))-\mathcal{A}(x,\nabla v)\right] \cdot\nabla \varphi 
dx\\
+\int_{\left\{ v>u+\varepsilon \right\} }\left[ \mathcal{B}
(x,u+\varepsilon )-\mathcal{B}(x,v)\right] \varphi dx\geqslant 0
\end{multline}
Using Assumptions (I) and (M) we have
\begin{equation*}
\int_{\left\{ v>u+\varepsilon \right\} }\left[ \mathcal{A}(x,\nabla
u+\varepsilon )-\mathcal{A}(x,\nabla v)\right] \cdot\nabla 
(v-u-\varepsilon )dx=0
\end{equation*}
and again by M we infer that $v\leqslant u+\varepsilon $ on $\Omega $.
Letting $\varepsilon \to 0$ we have $v\leqslant u$ on $\Omega $.
\end{proof}

\begin{theorem}
\label{threg} Every $p$-regular set is regular in the sense of 
definition 
\ref{defpr}.
\end{theorem}

\begin{proof}
Let $\Omega $ be a $p$-regular set in $\mathbb{R}^d$ and $f$ be a
continuous function on $\partial \Omega .$ We shall prove that there 
exists
a unique continuous solution $u$ of (\ref{eq2}) on $\Omega $ such that 
$\lim_{x\to y}u(x)=f(y)$ for all $y\in \partial \Omega $. The
uniqueness is given by Lemma \ref{lem3}. By \cite[Theorem 4.11]{MZ}
we have the continuity of $u$. For the existence, we may suppose that 
$f\in 
\mathcal{C}_{c}(\mathbb{R}^d)$ (Tietze's extension theorem). Let $f_i$
be a sequence of functions from $\mathcal{C}_{c}^{1}(\mathbb{R}^d)$ 
such
that $\left| f_i-f\right| \leqslant 2^{-i}$ and $\left| f_i\right|
+\left| f\right| \leqslant M$ on $\overline{\Omega }$ for the same 
constant 
$M$ and for all $i$. Let $u_i$ $\in \mathcal{W}^{1,p}({\Omega })\cap 
\mathcal{C}(\overline{\Omega })$ be the unique solution for the 
Dirichlet
problem with boundary data $f_i$ (Theorem \ref{thDir}). Then from Lemma 
\ref{lem3} we deduce that $\left| u_i-u_{j}\right| \leqslant 
2^{-i}+2^{-j}$
and $\left| u_i\right| \leqslant M$ on $\Omega $ for all $i$ and $j$. 
We
denote by $u$ the limit of the sequence $(u_i)_i$. We will show that 
$u$
is a local solution of the equation. For this, we prove that the 
sequence 
$(\nabla u_i)_i$ is locally uniformly bounded in $\left( L^{p}(\Omega
)\right) ^d$. Let $\varphi =-\eta ^{p}u_i$, $\eta \in \mathcal{C}
_{c}^\infty ({\Omega })$, $0\leqslant \eta \leqslant 1$ and $\eta =1$ 
on 
$\omega \subset \overline{\omega }\subset \Omega $. Since $\varphi \in 
\mathcal{W}_0^{1,p}({\Omega })$, we have 
\begin{eqnarray*}
0 &=&\int_{\Omega }\mathcal{A}(x,\nabla u_i)\cdot\nabla \varphi
dx+\int_{\Omega }\mathcal{B}(x,u_i)\varphi dx \\
&=&\int_{\Omega }\mathcal{A}(x,\nabla u_i).(-\eta ^{p}\nabla
u_i-pu_i\eta ^{p-1}\nabla \eta )dx-\int_{\Omega }\eta ^{p}\mathcal{B}
(x,u_i)u_idx \\
&\leqslant &-\mu \int_{\Omega }\eta ^{p}\left| \nabla u_i\right|
^{p}dx+p\nu \int_{\Omega }\eta ^{p-1}\left| \nabla u_i\right| 
^{p-1}\left|
u_i\right| \left| \nabla \eta \right| dx+C(M,\left\| \eta
\right\|_{\infty },\left| \Omega \right| ), 
\end{eqnarray*}
and therefore, using the Young inequality, we obtain 
\begin{eqnarray*}
\lefteqn{\int_{\Omega }\eta ^{p}\left| \nabla u_i\right| ^{p}dx }\\
&\leqslant &p\frac{\varepsilon ^{p'}\nu }{\mu }\int_{\Omega }\eta 
^{p}\left| 
\nabla
u_i\right| ^{p}dx+p\frac{\nu }{\varepsilon ^{p}\mu }\int_{\Omega 
}\left|
u_i\right| ^{p}\left| \nabla \eta \right| ^{p}dx+C(M,\left\|
\eta \right\|_{\infty },\left| \Omega \right| )
\\
&\leqslant &p\frac{\varepsilon ^{p'}\nu }{\mu }\int_{\Omega }\eta
^{p}\left| \nabla u_i\right| ^{p}dx+C(M,\left\| \eta \right\|_{\infty
},\left| \Omega \right| ,\left\| \nabla \eta \right\|_{\infty 
},\varepsilon
).
\end{eqnarray*}
If $0<\varepsilon <\left( \frac{c_1}{pa_1}\right) ^{\frac{p-1}{p}}$,
then 
\begin{equation*}
\int_{\omega }\left| \nabla u_i\right| ^{p}dx\leqslant \frac{\mu
C(M,\left\| \eta \right\|_{\infty },\left| \Omega \right| ,\left\| 
\nabla
\eta \right\|_{\infty },\varepsilon )}{\mu -p\varepsilon ^{p'}\nu }
\text{ for all }i.
\end{equation*}
It follows that the sequence $(u_i)_i$ is locally uniformly bounded in 
$\mathcal{W}^{1,p}(\Omega )$. Fix $D\Subset G\Subset \Omega $. Since 
$(u_i)_i$ converges pointwise to $u$ and by \cite[Theorem 1.32]{HKM}, 
we
obtain that $u\in \mathcal{W}^{1,p}(D)$ and $(u_i)_i$ converges weakly,
in $\mathcal{W}^{1,p}(D)$, to $u$. Let $\eta \in \mathcal{C}_0^{\infty
}(G) $ such that $0\leqslant \eta \leqslant 1$, $\eta =1$ in $D$ and 
testing
by $\varphi =\eta (u-u_i)$ for the solution $u_i$, we have 
\begin{eqnarray*}
\lefteqn{-\int_{G}\eta \mathcal{A}(x,\nabla u_i)\cdot\nabla 
(u-u_i)dx}\\
&=&\int_{G}(u-u_i)\mathcal{A}(x,\nabla u_i)\cdot\nabla \eta dx 
+\int_{G}\eta \mathcal{B}(x,u_i)(u-u_i)dx \\
&\leqslant &\Big( \int_{G}\left| u-u_i\right| ^{p}dx\Big) ^{1/p}
\Big[ C+\nu \Big( \int_{G}\left| \nabla u_i\right| ^{p}dx\Big)
 ^{\frac{p-1}{p}}\Big] \\
&\leqslant &C\Big( \int_{G}\left| u-u_i\right| ^{p}dx\Big)^{1/p}.
\end{eqnarray*}
Since 
\begin{eqnarray*}
0 &\leqslant &\int_{D}\left[ \mathcal{A}(x,\nabla 
u)-\mathcal{A}(x,\nabla
u_i)\right] \cdot\nabla (u-u_i)dx \\
&\leqslant &\int_{G}\eta \mathcal{A}(x,\nabla u)\cdot\nabla (u-u_i)dx
+C\Big(\int_{G}\left| u-u_i\right| ^{p}dx\Big) ^{1/p}
\end{eqnarray*}
and the weak convergence of $(\nabla u_i)_i$ to $\nabla u$ implies that 
\begin{equation*}
\lim_{i\to \infty }\int_{G}\eta \mathcal{A}(x,\nabla u)\cdot\nabla
(u-u_i)dx=0,
\end{equation*}
we conclude 
\begin{equation*}
\lim_{i\to \infty }\int_{D}\left[ \mathcal{A}(x,\nabla u)-\mathcal{A}
(x,\nabla u_i)\right] \cdot\nabla (u-u_i)dx=0.
\end{equation*}
Now \cite[Lemma 3.73]{HKM} implies that $\mathcal{A}(x,\nabla u_i)$
converges to $\mathcal{A}(x,\nabla u)$ weakly in $\left( L^{p^{\prime
}}(D)\right) ^{n}$.

Let $\psi \in \mathcal{C}_0^\infty (G)$. By the continuity in measure 
of
the Carath\'{e}odory function $\mathcal{B}(x,z)$ \cite{Kr} and by using 
the
domination convergence theorem (in measure), we have 
\begin{equation*}
\lim_{i\to \infty }\int_{\Omega }\mathcal{B}(x,u_i)\psi
dx=\int_{\Omega }\mathcal{B}(x,u)\psi dx.
\end{equation*}
Finally we obtain 
\begin{eqnarray*}
0 &=&\lim_{i\to \infty }\left[ \int_{\Omega }\mathcal{A}
(x,\nabla u_i)\cdot\nabla \psi dx+\int_{\Omega }\mathcal{B}(x,u_i)\psi 
dx
\right] \\
&=&\int_{\Omega }\mathcal{A}(x,\nabla u)\cdot\nabla \psi 
dx+\int_{\Omega }
\mathcal{B}(x,u)\psi dx.
\end{eqnarray*}
By an application of \cite[Corollay 4.18]{MZ} for each $u_i$ we obtain 
\begin{equation*}
\lim_{x\in \Omega, x\to z}u_i(x)=f_i(z)
\end{equation*}
for all $z\in \partial \Omega $. From the following estimation, of $u$ 
on
all $\Omega $, 
\begin{equation*}
u_i-2^{-i}\leqslant u\leqslant u_i+2^{-i}\text{  for all }i
\end{equation*}
we deduce that for all $i
$\begin{equation*}
\text{ }f_i(z)-2^{-i}\leqslant \liminf_{\underset{x\in \Omega }{
x\to z}}u(z)\leqslant \limsup_{\underset{x\in \Omega }{x\to z
}}u(z)\leqslant f_i(z)+2^{-i}.
\end{equation*}
Letting $i\to \infty $ we obtain 
\begin{equation*}
\lim_{x\to z}u(x)=f(z)
\end{equation*}
for all $z\in \partial \Omega $ which finishes the proof.
\end{proof}

\begin{corollary}
There exists a basis $\mathcal{V}$ of regular sets which is stable by
intersection i.e. for every $U$ and $V$ in $\mathcal{V}$, we have 
$U\cap
V\in \mathcal{V}$.
\end{corollary}
The proof of this corollary can be found in 
Theorem \ref{threg} and \cite[Corollary 6.32]{HKM}.


For every open set $V$ and for every $f\in \mathcal{C}(\partial V)$ we 
shall
denote by $\textsc{H}_{V}f$  the \emph{solution of the Dirichlet 
problem}
for the equation (\ref{eq2}) on $V$ with the boundary data $f$.

\section{Nonlinear Potential Theory associated with the equation 
(\ref{eq2})} \label{snpt}

For every open set $U$ we shall denote by $\mathcal{U}(U)$ the set of 
all
relatively compact open, regular subset $V$ in $U$ with 
$\overline{V}\subset
U$.

By previous section and in order to obtain an axiomatic nonlinear 
potential
theory, we shall investigate the harmonic sheaf associated with 
(\ref{eq2}) and defined as follows: For every open subset $U$ of 
$\mathbb{R}^d$ ($d\geqslant 1$), we set
\begin{eqnarray*}
\mathcal{H}(U) &=&\big\{ u\in \mathcal{C}(U)\cap \mathcal{W}
_{\text{loc}}^{1,p}(U):u\text{ is a solution of }(\ref{eq2})\big\} \\
&=&\big\{ u\in \mathcal{C}(U):\text{\textsc{H}}_{V}u=u\text{ for every 
}
V\in \mathcal{U}(U)\big\} .
\end{eqnarray*}
Element in the set $\mathcal{H}(U)$ are called \emph{harmonic} on $U$.

We recall (see \cite{B-h}) that $(X,\mathcal{H})$ satisfies the 
\emph{Bauer
convergence property} if for every subset $U$ of $X$ and every monotone
sequence $(h_n)_n$ in $\mathcal{H}(U)$, we have $h=\underset{}{
\lim_{n\to \infty }}h_n\in \mathcal{H}(U)$ if it is locally
bounded.

\begin{proposition}
\label{pro2.1}Let be $U$ an open subset of $\mathbb{R}^d$. Then every
family $\mathcal{F}\subset \mathcal{H}(U)$ of locally uniformly bounded
harmonic functions is equicontinuous.
\end{proposition}

\begin{proof}
Let $V\subset \overline{V}\subset U$ and a family $\mathcal{F}$ 
$\subset 
\mathcal{H}(U)$ of locally uniformly bounded harmonic functions. Then 
$\sup
\left\{ \left| u(x)\right| :x\in \overline{V}\text{ and }u\in 
\mathcal{F}
\right\} <\infty $ and by \cite{MZ}, is equicontinuous on 
$\overline{V}$.
\end{proof}

\begin{corollary}
\label{corbc}We have the Bauer convergence properties and moreover 
every
locally bounded family of harmonic functions on an open set is 
relatively
compact.
\end{corollary}

\begin{proof}
Let $U$ be an open set and $\mathcal{F}$ a locally bounded subfamily of 
$\mathcal{H}(U)$. By Proposition \ref{pro2.1}, there exist a sequence 
$(u_n)_n$ in $\mathcal{F}$ which converge to $u$ on $U$ locally
uniformly. Let now $V\in \mathcal{U}(U)$. For every $\varepsilon >0$, 
there
exists $n_0\in \mathbb{N}$ such that $u-\varepsilon \leqslant
u_n\leqslant u+\varepsilon $ for every $n\geqslant n_0$. The comparison
principle yields therefore $\left( \text{\textsc{H}}_{V}u\right)
-\varepsilon \leqslant u_n\leqslant \left( 
\text{\textsc{H}}_{V}u\right)
+\varepsilon $, thus $\left( \text{\textsc{H}}_{V}u\right) -\varepsilon
\leqslant u\leqslant \left( \text{\textsc{H}}_{V}u\right) +\varepsilon 
$.
Letting $\varepsilon \to 0$, we get $u=$\textsc{H}$_{V}u$.
\end{proof}

\begin{proposition}
\cite{B-h} Let $V$ a regular subset of $\mathbb{R}^d$ and let 
$(f_n)_n$ and $f$ in $\mathcal{C}{(\partial V)}$ such that 
$(f_n)_n$ is a monotone sequence converging to $f$. Then $\sup_n
$\textsc{H}$_{V}f_n$ converge to \textsc{H}$_{V}f$.
\end{proposition}

\begin{proof}
Let $V$ a regular subset of $\mathbb{R}^d$ and let $(f_n)_n$ and 
$f$ in $\mathcal{C}{(\partial V)}$ such that $(f_n)_n$ is increasing
to $f$. Then, by Lemma \ref{lem3}, we have 
\begin{equation*}
\sup_n\text{\textsc{H}}_{V}f_n\leqslant \text{\textsc{H}}_{V}f\text{ }
\end{equation*}
and, by Corollary \ref{corbc} $\sup_n$\textsc{H}
$_{V}f_n\in \mathcal{H}(V)$.
Moreover, For every $n$ and every $z\in \partial V$ we have 
\begin{equation*}
f_n(z)\leq \liminf_{x\to z}(\sup_n\text{\textsc{H}}
_{V}f_n(x))\leq \limsup_{x\to z}(\sup_n\text{\textsc{H}}
_{V}f_n(x))\leqslant f(z).
\end{equation*}
Letting $n$ tend to infinity we obtain that 
\begin{equation*}
f(z)=\lim_{x\to z}(\sup_n\text{\textsc{H}}_{V}f_n)(x).
\end{equation*}
By Lemma \ref{lem3}, this shows that in fact \textsc{H}
$_{V}f=\sup_n\text{\textsc{H}}_{V}f_n$. An analogous proof can be given
if $(f_n)_n$ is decreasing. \newline
\end{proof}

\begin{corollary}
\cite{B-h} Let $V$ be a regular subset of $\mathbb{R}^d$ and $(f_n)_n$
and $(g_n)_n$ to sequences in $\mathcal{C}{(\partial V)}$ which
are monotone in the same sense such that $\lim_nf_n$ 
$=\lim_ng_n$. Then $\lim_n$\textsc{H}$_{V}f_n$ 
$=\lim_n\text{\textsc{H}}_{V}g_n$.
\end{corollary}

\begin{proof}
We assume without loss the generality that $(f_n)$ and $(g_n)$ are both
increasing. Obviously, \textsc{H}$_{V}(g_n\wedge f_{m})\leqslant 
$\textsc{H
}$_{V}g_n$ for every $n$ and $m$ in $\mathbb{N}$, hence $\sup_n
$\textsc{H}$_{V}(g_n\wedge f_{m})\leqslant \sup_n\text{\textsc{H}}
_{V}g_n$ for every $m$. Since the sequence $(g_n\wedge f_{m})_n$ is
increasing to $f_{m}$, the previous proposition implies that \textsc{H}
$_{V}f_{m}\leqslant \sup_n\text{\textsc{H}}_{V}g_n$. We then have 
$\sup_n$\textsc{H}$_{V}f_n\leqslant \sup_n\text{\textsc{H}
}_{V}g_n$. Permuting $(f_n)$ and $(g_n)$ we obtain the converse
inequality.
\end{proof}

Let $V$ be a regular subset of $\mathbb{R}^d$. For every lower bounded 
and
lower semicontinuous function $v$ on $\partial V$ we define the set
\begin{equation*}
\text{\textsc{H}}_{V}v=\sup_n\left\{ \text{\textsc{H}}_{V}f_n:(f_n)_n
\text{ in }\mathcal{C}{(\partial V)}\text{ and increasing to }v\right\} 
.
\end{equation*}
For every upper bounded and upper semicontinuous function $u$ on 
$\partial V$
we define 
\begin{equation*}
\text{\textsc{H}}_{V}u=\inf_n\left\{ \text{\textsc{H}}_{V}f_n:(f_n)_n
\text{ in }\mathcal{C}{(\partial V)}\text{ and decreasing to }u\right\} 
.
\end{equation*}

Let be $U$ an open set of $\mathbb{R}^d$. A lower semicontinuous and
locally lower bounded function $u$ from $U$  to $\overline{\mathbb{R}}$ 
is
termed \emph{hyperharmonic} on $U$ if \textsc{H}$_{V}u\leqslant u$ on 
$V$
for all $V$ in $\mathcal{U}(U)$. A upper semicontinuous and locally 
upper
bounded function $v$ from $U$ to $\overline{\mathbb{R}}$ is termed 
\emph{
hypoharmonic} on $U$ if \textsc{H}$_{V}u\geqslant u$ on $V$ for all $V$ 
in 
$\mathcal{U}(U)$. We will denote by $^{\ast \!}\mathcal{H}(U)$ 
(resp. $_{\ast\!}\mathcal{H}(U)$) the set of all hyperharmonic (resp. 
hypoharmonic)
functions on $U$.

For $u\in $ $^{\ast \!}\mathcal{H}(U)$, $v\in $ $_{\ast 
\!}\mathcal{H}(U)$
and $k\geqslant 0$ we have $u+k\in $ $^{\ast \!}\mathcal{H}(U)$ and 
$v-k\in $
$_{\ast \!}\mathcal{H}(U)$. Indeed, let $V$ $\in $ $\mathcal{U}(U)$ and 
a
continuous function such that $g\leqslant u+k$ on $\partial V$, then 
\textsc{
H}$_{V}(g-k)\leqslant $\textsc{H}$_{V}u\leqslant u$. Since $\left( 
\text{
\textsc{H}}_{V}g\right) -k\leqslant $\textsc{H}$_{V}(g-k)$, we 
therefore get 
\textsc{H}$_{V}g\leqslant u+k$ and thus $u+k\in $ $^{\ast 
\!}\mathcal{H}(U)$.

We have the following comparison principle:

\begin{lemma}
Suppose that $u$ is hyperharmonic and $v$ is hypoharmonic on an open 
set 
$U$. If 
\begin{equation*}
\underset{U\ni x\to y}{\limsup }v(x)\leqslant \underset{U\ni
x\to y}{\liminf }u(x)
\end{equation*}
for all $y\in \partial U$ and if both sides of the previous inequality 
are
not simultaneously $+\infty $ or $-\infty $, then $v\leqslant u$ in 
$U$.
\end{lemma}
The proof is the same as in \cite[p. 133]{HKM}.

\section{Sheaf Property for Hyperharmonic and Hypoharmonic Functions}
\label{ssp}

For  open subsets $U$ of $\mathbb{R}^d$, we denote by $\overline{
\mathcal{S}}(U)$ (resp. by $\underline{\mathcal{S}}(U)$) the set of all
supersolutions (resp. subsolutions) of the equation (\ref{eq2}) on $U$.

Recall that a map $\mathfrak{F}$ which to each open subset $U$ of 
$\mathbb{R}^d$ assigns a subset $\mathfrak{F}(U)$ of $\mathfrak{B}(U)$ 
is 
called sheaf if we have the following two properties: 
\\
(\emph{Presheaf Property}) For every two open subsets $U$, $V$ of 
$\mathbb{R}^d$ such that $U\subset $ $V$, 
$\mathfrak{F}(V)_{\left| U\right.}\subset \mathfrak{F}(U)$
\\
(\emph{Localization Property}) For any family 
$\left( U_i\right)_{i\in I}$ of open subsets and any numerical 
function $h$ on $U=\bigcup_{i\in I}U_i$, $h\in \mathfrak{F}(U)$ 
if $h_{\left| U_i\right. }\in \mathfrak{F}(U_i)$ for every $i\in 
I$. \smallskip

An easy verification gives that $\overline{\mathcal{S}}$ and 
$\underline{\mathcal{S}}$ are sheaves. Furthermore, we have the 
following
results which generalize many earlier \cite{Mae81,BBM,vG,HKM}.

\begin{theorem}
\label{theo3} Let $U$ be a non empty open subset in $\mathbb{R}^d$ and 
$u\in $ $^{\ast \!}\mathcal{H}(U)\cap \mathfrak{B}_{b}(U)$. Then $u$ is 
a
supersolution on $U$.
\end{theorem}

\begin{proof}
First, we shall prove that for every open $O\subset\overline{O}\subset 
U$,
there exists an increasing sequence $(u_i)_i$ in
in $O$ of supersolutions such that $u=\lim_{i\to \infty }u_i$ 
on $O$. Let $(\varphi_i)_i$ be an increasing sequence in 
$\mathcal{C}_{c}^\infty (U)$ such
that $u=\sup_i\varphi_i$ on $O$. Let $u_i$ be the solution of
the obstacle problem in the non empty convex set 
\begin{equation*}
\mathcal{K}_i:=\left\{ v\in \mathcal{W}^{1,p}(O):\varphi_i\leqslant
v\leqslant \left\| \varphi_i\right\|_{\infty }+\left\| \varphi
_{i+1}\right\|_{\infty }\text{ and }v-\varphi_i\in \mathcal{W}
_0^{1,p}(O)\right\} .
\end{equation*}
The existence and the uniqueness are given respectively by Theorem \ref
{theo2}; moreover is a supersolution (Theorem\ref{thdop}). Since 
$u_{i+1}$
is a supersolution and $u_i\wedge u_{i+1}\in \mathcal{K}_i$, we have 
$u_i\leqslant u_{i+1}$ in $O$.We have to prove that the sequence 
$(u_i)_i$ is increasing to $u$. Let $x_0$ be an element of the open
subset $G_i:=\left\{ x\in O:\varphi_i(x)<u_i(x)\right\} $ and $\omega 
$ be a domain such that $x_0\in \omega \subset $ $\overline{\omega }
\subset G_i$. Since for every $\psi \in \mathcal{C}_{c}^\infty (\omega 
)$
and for sufficiently small $\left| \varepsilon \right| $ $u_i\pm
\varepsilon \psi \in \mathcal{K}_i$, 
\begin{equation*}
\int_{\omega }\mathcal{A}(x,\nabla u_i)\cdot \nabla \psi dx
+\int_{\omega }\mathcal{B}(x,u_i)\psi dx=0\,.
\end{equation*}
Then $u_i$ is a solution of the equation (\ref{eq2}) on $\omega $ and 
by
the sheaf property of $\mathcal{H}$, $u_i$ is a solution of the 
equation $(
\ref{eq2})$ on $G_i$. Now the comparison principle implies that 
$u_i\leqslant u$ on $G_i$, hence $\varphi_i\leqslant u_i\leqslant u$
on $O$ and therefore $u=\sup_iu_i$. Finally, the boundedness of
the sequence $(u_i)_i$ and the same techniques in the proof of Theorem 
\ref
{threg} yield that $(u_i)_i$ is locally bounded in 
$\mathcal{W}^{1,p}(O)$
and that $u$ is a supersolution of the equation (\ref{eq2}) in $O$.
\end{proof}

\begin{corollary}
Let $U$ be a non empty open subset in $\mathbb{R}^d$ and $u\in $ 
$\mathcal{
W}_\text{loc}^{1,p}(U)\cap $ $^{\ast \!}\mathcal{H}(U)$. Then $u$ is a
supersolution on $U$. Moreover the infinimum of two supersolutions is 
also a
supersolution.
\end{corollary}

\begin{proof}
Let $u\in $ $\mathcal{W}_\text{loc}^{1,p}(U)\cap $ $^{\ast 
\!}\mathcal{H}(U)$.
The Theorem \ref{theo3} implies that $u\wedge n$ is a supersolution for 
all 
$n\in \mathbb{N}$, consequently we have for every positive $\varphi \in 
\mathcal{C}_{c}^\infty (U)$ 
\begin{eqnarray*}
0 &\leqslant &\int_U\mathcal{A}(x,\nabla \left( u\wedge n\right) 
)\cdot\nabla
\varphi dx+\int_U\mathcal{B}(x,u\wedge n)\varphi dx \\
&=&\int_{\left\{ u<n\right\} }\mathcal{A}(x,\nabla u)\cdot\nabla 
\varphi
dx+\int_U\mathcal{B}(x,u\wedge n)\varphi dx.
\end{eqnarray*}
Letting $n\to +\infty $ we obtain 
\begin{equation*}
0\leqslant \int_U\mathcal{A}(x,\nabla u)\cdot\nabla \varphi dx+\int_U
\mathcal{B}(x,u)\varphi dx
\end{equation*}
for all positive $\varphi \in \mathcal{C}_{c}^\infty (U)$, thus $u$ is 
a
supersolution.
Moreover, if $u$ and $v$ are two supersolutions then $u\wedge v$ $\in $ 
$\mathcal{W}_\text{loc}^{1,p}(U)\cap $ $^{\ast \!}\mathcal{H}(U)$ so 
$u\wedge v$
is a supersolution.
\end{proof}

\begin{theorem}
$^{\ast \!}\mathcal{H}$ is a sheaf.
\end{theorem}

\begin{proof}
Let $(U_i)_{i\in I}$ be a family of open subsets of $\mathbb{R}^d$, 
$U=\bigcup_{i\in I}U_i$ and $h\in $ $^{\ast \!}\mathcal{H}(U_i)$
for every $i\in I$. Then by the definition of hyperharmonic function, 
we
have $h\wedge n\in $ $^{\ast \!}\mathcal{H}(U_i)$ for every $(i,n)\in
I\times \mathbb{N}$ and by Theorem \ref{theo3}, $h\wedge n$ is a
supersolution on each $U_i$. Since $\overline{\mathcal{S}}$ is a sheaf, 
we
get $h\wedge n\in \overline{\mathcal{S}}(U)\subset $ $^{\ast 
\!}\mathcal{H}
(U)$. Thus $h=\sup_nh\wedge n\in $ $^{\ast \!}\mathcal{H}(U)$ and 
$^{\ast \!}\mathcal{H}$ is a sheaf.
\end{proof}

\begin{remark} \rm
For every open subset $U$ of $\mathbb{R}^d$, let 
$\widetilde{\mathcal{H}}
(U)$ denote the set of all $u\in \mathcal{W}^{1,p}(U)\cap 
\mathcal{C}(U)$
such that 
$\widetilde{\mathcal{B}}(x,u)\in L_\text{loc}^{p^{\ast '}}(U)$ and
\begin{equation*}
\int_U\mathcal{A}(x,\nabla u)\cdot\nabla \varphi dx+\int_U
\widetilde{\mathcal{B}}(x,u)\varphi dx=0
\end{equation*}
for every $\varphi \in \mathcal{W}_0^{1,p}(U)$, where 
$\widetilde{\mathcal{
B}}(x,\zeta )=-\widetilde{\mathcal{B}}(x,-\zeta )$. It is easy to see 
that
the mapping $\zeta \to \widetilde{\mathcal{B}}(x,\zeta )$ is
increasing and that $u\in \mathcal{H}(U)$ if and only if $-u\in 
\widetilde{
\mathcal{H}}(U)$. Furthermore $\mathcal{H}$ and 
$\widetilde{\mathcal{H}}$
have the same regular sets and for every $V\in \mathcal{U}(U)$ and 
$f\in 
\mathcal{C}(\partial V)$ we have 
\textsc{H}$_{V}f=-\widetilde{\text{\textsc{H
}}}_{V}(-f)$. It follows that $u\in $ $_{\ast \!}\mathcal{H}(U)$ if and 
only
if $-u\in $ $^{\ast \!}\widetilde{\mathcal{H}}(U)$ and therefore 
$_{\ast \!}
\mathcal{H}$ is a sheaf.
\end{remark}

\section{The degeneracy of the sheaf $\mathcal{H}$\label{shi}}

As in the previous section we consider the sheaf $\mathcal{H}$ defined 
by
(\ref{eq2}). Recall that the \emph{Harnack inequality} or
the \emph{Harnack principle} is satisfied by $\mathcal{H}$ if for every
domain $U$ of $\mathbb{R}^d$ and every compact subset $K$ in $U$, there
exists two constants $c_1\geqslant 0$ and $c_{2}\geqslant 0$ such that
for every $h\in \mathcal{H}^{+}(U)$,
\begin{equation}
\sup_{x\in K}h(x)\leqslant c_1\inf_{x\in K}h(x)+c_{2}  \tag{HI}
\label{hi}
\end{equation}


We remark that, if for every $\lambda >0$ and $h\in \mathcal{H}^{+}(U)$ 
we
have $\lambda h\in \mathcal{H}^{+}(U)$, then we can choose $c_{2}=0$ 
and we
obtain the classical Harnack inequality.

The Harnack inequality, for quasilinear elliptic equation,  is proved 
in
the fundamental tools of Serrin \cite{Se}, see also \cite{Tr,Le}.
For the linear case see \cite{H-h,BHH,AS,GT}.

In the rest of this section, we assume that $\mathcal{B}$ satisfy the
following supplementary condition.
\begin{itemize}
\item[$(\ast )$]  There exists $b\in L_\text{loc}^{\frac{d}{p-\epsilon 
}}(\mathbb{
R}^d)$, $0<\epsilon <1$, such that $\left| \mathcal{B}(x,\zeta )\right|
\leqslant b(x)\left| \zeta \right| ^{\alpha }$ for every $x\in 
\mathbb{R}
^d $ and $\zeta \in \mathbb{R}$.
\end{itemize}

\subsection*{Small powers $(0<\alpha <p-1)$}

We have the validity of Harnack principle given by the following 
proposition.

\begin{proposition}
Let $\mathcal{H}$ be the sheaf of the continuous solutions of the 
equation 
$\emph{(\ref{eq2})}$. Assume that the condition $(\ast )$ is satisfies 
with 
$0<\alpha <p-1$. Then the Harnack principle is satisfied by 
$\mathcal{H}$.
\end{proposition}

The proof of this proposition can be found in \cite[p. 178]{MZ} or 
\cite{Se}


\begin{definition}\rm
\label{defell}The sheaf $\mathcal{H}$ is called elliptic if for every
regular domain $V$ in $\mathbb{R}^d$, $x\in V$ and $f\in \mathcal{C}
^{+}(\partial V)$, \textsc{H}$_{V}f(x)=0$ if and only if $f=0$.
\end{definition}

In the following example, we have the Harnack inequality but not the
ellipticity. This is in contrast to the linear theory or quasilinear 
setting of nonlinear potential theory given by the 
$\mathcal{A}$-harmonic functions in \cite{HKM}.

\begin{example} \rm
We assume that $\mathcal{B}(x,\zeta )=\mathop{\rm sgn}(\zeta )\left| 
\zeta \right|
^{\alpha }$ with $0<\alpha <p-1$ and $\mathcal{A}(x,\xi )=\left| \xi 
\right|
^{p-2}\xi $. Let $u=cr^{\beta }$ with $\beta =p(p-1-\alpha )^{-1}$ and 
\begin{equation*}
c=p^{\frac{p-1}{p-1-\alpha }}(p-1-\alpha )^{\frac{p}{p-1-\alpha 
}}\left[
d(p-1-\alpha )+\alpha p\right] ^{\frac{1}{p-1-\alpha }}.
\end{equation*}
With an easy verification, we will find that for every $x_0\in 
\mathbb{R}
^d$ and ball \textsc{B}$(x_0,\rho )$, there exists a solution $u$ (in
the form $c\left\| x-x_0\right\| ^{\beta }$) on \textsc{B}$(x_0,\rho )$
such that $\Delta_{p}u=u^{\alpha }$ with $u(x_0)=0$ and $u(x)>0$ for
every $x\in $\textsc{B}$(x_0,\rho )\setminus \left\{ x_0\right\} $. We
therefore obtain that the sheaf $\mathcal{H}$ is not elliptic and 
curiously
we have the existence of a basis of regular set $\mathcal{V}$ such that 
for
every $V\in \mathcal{V}$, there exist $x_0$ $\in V$ and $f\in 
\mathcal{C}
(\partial V)$ with $f>0$ on $\partial V$ and \textsc{H}$_{V}f(x_0)=0$.
\end{example}

We will prove that the sheaf given in the previous example is 
non-degenerate in the following sense:

\begin{definition}\rm
A sheaf $\mathcal{H}$ is called non-degenerate on an open $U$ if for 
every $x\in U$, there exists a neighborhood $V$ of $x$ and 
$h\in \mathcal{H}(V)$ with $h(x)\neq 0$.
\end{definition}

\begin{proposition}
Assume that the condition $(\ast )$ is satisfies with $0<\alpha <p-1$ 
and 
$\mathcal{A}(x,\lambda \xi )=\lambda \left| \lambda \right| 
^{p-2}\mathcal{A}
(x,\xi )$ for all $x,\xi \in \mathbb{R}^d$ and for all $\lambda \in 
\mathbb{R}$. Then the sheaf $\mathcal{H}$ is non degenerate and more we
have: for every regular set $V$ and $x\in V$, $\sup_{h\in \mathcal{H}
(V)}h(x)=+\infty $.
\end{proposition}

\begin{proof}
It is sufficient to prove that for every $x_0\in \mathbb{R}^d$, $\rho 
>0$, 
$n\in \mathbb{N}$ and $u_n=$ \textsc{H}$_{\text{\textsc{B}}(x_0,\rho 
)}n$
we have $u_n$ converges to infinity at any point of 
\textsc{B}$(x_0,\rho
)$. The comparison principle yields that $0\leqslant u_n\leqslant n$ on 
\textsc{B}$(x_0,\rho )$. Put $u_n=nv_n$, we then obtain: 
\begin{equation*}
\int \mathcal{A}(x,\nabla v_n)\nabla \varphi dx+n^{1-p}\int \mathcal{B}
(x,nv_n)\varphi dx=0
\end{equation*}
for every $\varphi \in \mathcal{C}_{c}^\infty (\text{\textsc{B}}
(x_0,\rho ))$ and for every $n\in \mathbb{N}^{\ast }$.
The assumptions on $\mathcal{B}$ yields 
\begin{equation*}
\lim_{n\to \infty }\int \mathcal{A}(x,\nabla v_n)\nabla \varphi
dx=0\text{;}
\end{equation*}
since $0\leqslant v_n\leqslant 1$, we have 
\begin{equation*}
\left| n^{1-p}\mathcal{B}(x,nv_n)\right| \leqslant n^{\alpha
-p+1}b(x)\leqslant b(x)
\end{equation*}
and by \cite[Theorem 4.19]{MZ}, $v_n$ are equicontinuous on the closure 
$\overline{\text{\textsc{B}}}_{x_0,\rho }$ of the ball \textsc{B}
$(x_0,\rho )$, then by the Ascoli's theorem, $(v_n)_n$ admits a
subsequence which is uniformly convergent on 
$\overline{\text{\textsc{B}}}
_{x_0,\rho }$ to a continuous function $v$ on 
$\overline{\text{\textsc{B}}}
_{x_0,\rho }$. Further we can easily verify that $v\in \mathcal{W}
_\text{loc}^{1,p}(\text{\textsc{B}}(x_0,\rho ))$ and 
\begin{equation*}
\int \mathcal{A}(x,\nabla v)\nabla \varphi dx=0
\end{equation*}
for every $\varphi \in \mathcal{W}_0^{1,p}(\text{\textsc{B}}(x_0,\rho 
))$.
 Since $v=1$ on $\partial $\textsc{B}$(x_0,\rho )$, $v=1$ on 
$\overline{
\text{\textsc{B}}}_{x_0,\rho }$. The relation $u_n=nv_n$ yields the
desired result.
\end{proof}

\subsection*{Big Powers $(\protect\alpha \geqslant p-1)$}

We shall investigate (\ref{eq2}) in the case $\alpha
\geqslant p-1$. Let $\mathcal{H}$ be the sheaf of the continuous 
solutions
of (\ref{eq2}). In \cite{MZ} or \cite{Se}, we find the following form
 of the Harnack inequality.

\begin{theorem}
\label{thbp} Assume that the condition $(\ast )$ is satisfies with 
$\alpha
\geqslant p-1$. Then 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 $C=C(K,M)>0$ such that for every $u\in \mathcal{H}^{+}(U)$ 
with 
$u\leqslant M$,
\begin{equation*}
\sup_{K}u\leqslant C\inf_{K}u \,.
\end{equation*}
\end{theorem}

\begin{corollary}
If the condition $(\ast )$ is satisfies with $\alpha \geqslant p-1$, 
then $\mathcal{H}$ is non-degenerate and elliptic. Moreover,
for every domain $U$ in $\mathbb{R}^d$ and $u\in \mathcal{H}^{+}(U)$, 
we have either $u>0$ on $U$ or $u=0$ on $U$.
\end{corollary}

\begin{remark}\rm
If $\alpha =p-1$, the constant in \emph{Theorem \ref{thbp} }does not 
depend
on $M$ and we have the classical form of the Harnack inequality.
\end{remark}

We recall that a sheaf $\mathcal{H}$ satisfies the \emph{Brelot
convergence property}  if for every domain $U$ in $\mathbb{R}^d$ and 
for
every monotone sequence $(h_n)_n\subset \mathcal{H}(U)$ we have 
$\lim_nh_n\in \mathcal{H}(U)$ if it is not identically $+\infty $
on $U$.

Using the same proof as in \cite{B-h}, we have the following 
proposition.

\begin{proposition}
\label{pBr} If the Harnack inequality is satisfied by $\mathcal{H}$, 
then
the convergence property of Brelot is fulfilled by $\mathcal{H}$.
\end{proposition}

\begin{remark} \rm
In contrast to the linear case \emph{(see \cite{LW})} the converse of 
\emph{
Proposition \ref{pBr}} is not true \emph{(see \cite{B-ko})} and hence 
the
validity of the convergence property of Brelot does not imply the 
validity
of the Harnack inequality.
\end{remark}

\subsection*{An Application}
Let $\mathcal{H}_\alpha $ be the sheaf of all continuous solution of 
the equation
\begin{equation*}
-\mathop{\rm div}\mathcal{A}(x,\nabla u)+b(x)\mathop{\rm sgn}(u)\left| 
u\right| ^{\alpha }=0
\end{equation*}
where $b\in L_\text{loc}^{\frac{d}{d-\epsilon }}(\mathbb{R}^d)$, 
$b\geqslant 0$
and $0<\epsilon <1$.

\begin{theorem}
a)  For each $0<\alpha <p-1$, $(\mathbb{R}^d,\mathcal{H}_{\alpha
})$ is a Bauer harmonic space satisfying the Brelot convergence 
property,
but it is not elliptic in the sense of Definition \ref{defell}.
\\
b)  For each $\alpha \geqslant p-1$, $(\mathbb{R}^d,\mathcal{H}_\alpha 
)$ 
is a Bauer harmonic space elliptic in the sense of Definition 
\ref{defell} and the convergence property of Brelot is fulfilled by 
$\mathcal{H}_{p-1}$.
\end{theorem}

\section{Keller-Osserman Property\label{sko}}
Let $\mathcal{H}$ be the sheaf of continuous solutions related to the
equation (\ref{eq2}).

\begin{definition} \rm
Let $U$ be a relatively compact open subset of $\mathbb{R}^d$. A 
function 
$u\in \mathcal{H}^{+}(U)$ is called regular Evans function for 
$\mathcal{H}$
and $U$ if $\underset{U\ni x\to z}{\lim }u(x)=+\infty $ for every
regular point $z$ in the boundary of $U$.
\end{definition}

For an investigation of regular Evans functions see \cite{B-ko}.

\begin{definition} \rm
\label{defko} We shall say that $\mathcal{H}$ satisfies the 
Keller-Osserman property, denoted (KO), if every ball admits a regular 
Evans function for $\mathcal{H}$.
\end{definition}

As in \cite[Proposition 1.3]{B-ko}, we have the following proposition.

\begin{proposition}
$\mathcal{H}$ satisfies the (KO) condition if and only if 
$\mathcal{H}^{+}$
is locally uniformly bounded (i.e. for every non empty open set $U$ in 
$\mathbb{R}^d$ and for every compact $K\subset U$, there exists a 
constant 
$C>0$ such that $\sup_{K}u\leqslant C$ for every $u\in \mathcal{H}
^{+}(U)$).
\end{proposition}

\begin{corollary}
If $\mathcal{H}$ fulfills the (KO) property, then $\mathcal{H}$ 
satisfies
the Brelot convergence property.
\end{corollary}

\begin{theorem}
\label{theo65} Assume that $\mathcal{A}$ and $\mathcal{B}$ satisfies 
the
following supplementary conditions

\begin{itemize}
\item[i)]  For every $x_0\in \mathbb{R}^d$, the function $F$ from 
$\mathbb{R}^d$ to $\mathbb{R}^d$ defined by

$F(x)=\mathcal{A}(x,x-x_0)$ is differentiable and $\mathop{\rm div}F$ 
is locally
(essentially) bounded.

\item[ii)]  $\mathcal{A}(x,\lambda \xi )=\lambda \left| \lambda \right|
^{p-2}\mathcal{A}(x,\xi )$ for every $\lambda \in \mathbb{R}$ and every 
$x$, 
$\xi \in \mathbb{R}^d$.

\item[iii)]  $\left| \mathcal{B}(x,\zeta )\right| \geqslant b(x)\left|
\zeta \right| ^{\alpha }$, $\alpha >p-1$ where $b\in 
L_\text{loc}^{\frac{d}{
d-\epsilon }}(\mathbb{R}^d)$, $0<\epsilon <1$, with 
$\underset{U}{\mathop{\rm ess\, inf}}b(x)>0$ for every relatively 
compact $U$ in $\mathbb{R}^d$.
\end{itemize}

Then the (KO) property is valid by $\mathcal{H}$.
\end{theorem}

\begin{proof}
Let $U$ be the ball with center $x_0\in \mathbb{R}^d$ and radius $R$.
Put $f(x)=R^{2}-\left\| x-x_0\right\| ^{2}$ and $g=cf^{-\beta }$, we
obtain the desired property if we find a constant $c>0$ such that $g$ 
is a
supersolution of the equation (\ref{eq2}). We have $\nabla
f(x)=-2(x-x_0) $ and $\nabla g(x)=2c\beta \left( f(x)\right) ^{-(\beta
+1)}(x-x_0)$ and then 
\begin{equation*}
\mathcal{A}(x,\nabla g(x))=(2c\beta )^{p-1}\left( f(x)\right) ^{-(\beta
+1)(p-1)}\mathcal{A}(x,x-x_0).
\end{equation*}
Let $\varphi \in \mathcal{C}_{c}^\infty (U)$, $\varphi \geqslant 0$ and 
we
set $I_{\varphi }=\int \mathcal{A}(x,\nabla g)\nabla \varphi dx+\int 
\mathcal{B}(x,g)\varphi dx$, then 
\begin{eqnarray*}
I_{\varphi } &=&-\int \mathop{\rm div}\mathcal{A}(x,\nabla g)\varphi 
dx+\int 
\mathcal{B}(x,g)\varphi dx \\
&=&-\int \Big[ 2(\beta +1)(p-1)(2c\beta )^{p-1}f^{-(\beta +1)(p-1)-1}
\mathcal{A}(x,x-x_0).(x-x_0) \\
&&+ (2c\beta )^{p-1}f^{-(\beta +1)(p-1)}\mathop{\rm div}
\mathcal{A}(x,x-x_0)-\mathcal{B}(x,g)\Big] \varphi dx \\
&\geqslant &-\int \Big[ 2(\beta +1)(p-1)(2c\beta )^{p-1}f^{-(\beta
+1)(p-1)-1}\mathcal{A}(x,x-x_0).(x-x_0) \\
&&+(2c\beta )^{p-1}f^{-(\beta +1)(p-1)}\mathop{\rm div}
\mathcal{A}(x,x-x_0)-c^{\alpha }bf^{-\alpha \beta }\Big] \varphi dx \\
&=&-\int \Big[ 2c^{p-1-\alpha }(2\beta )^{p-1}(\beta 
+1)(p-1)\mathcal{A}
(x,x-x_0).(x-x_0) \\
&&+c^{p-1-\alpha }(2\beta )^{p-1}f\mathop{\rm div}\mathcal{A}
(x,x-x_0)-bf^{\beta (p-1-\alpha )+p}\Big] c^{\alpha }f^{-(\beta
+1)(p-1)-1}\varphi dx.
\end{eqnarray*}
Putting $\beta =p(\alpha -p+1)^{-1}$ we obtain 
\begin{eqnarray*}
I_{\varphi } &\geqslant &-\int \Big[ 2(\tfrac{2p}{\alpha -p+1})^{p-1}(
\tfrac{\alpha +1}{\alpha -p+1})(p-1)\mathcal{A}(x,x-x_0).(x-x_0)
\\
&& +(\tfrac{2p}{\alpha -p+1})^{p-1}f\mathop{\rm div}\mathcal{A}
(x,x-x_0)-c^{\alpha -p+1}b\Big] c^{p-1}f^{\frac{\alpha p}{p-1-\alpha }
}\varphi dx.
\end{eqnarray*}
It follows from A2 that $\mathcal{A}(x,x-x_0).(x-x_0)$ is
locally bounded. Hence if we take $c$ so that$\tfrac{p-1}{\alpha -p+1}$ 
\begin{multline*}
c \geqslant \Big[ \sup_{x\in U}\Big\{ \tfrac{2(\alpha +1)(p-1)}{
\alpha -p+1}\frac{| \mathcal{A}(x,x-x_0).(x-x_0)| }{b(x)}
 +R^{2}\frac{| \mathop{\rm div}\mathcal{A}
(x,x-x_0)| }{b(x)}\Big\} \Big] ^{\frac{1}{\alpha -p+1}}\\
\times\Big( \frac{2p}{\alpha -p+1}\Big) ^{\frac{p-1}{\alpha -p+1}},
\end{multline*} 
then $I_{\varphi }\geqslant 0$ holds for every $\varphi \in \mathcal{C}
_{c}^\infty (U)$ with $\varphi \geqslant 0$. Thus the function 
$g(x)=c(R^{2}-\left\| x-x_0\right\| ^{2})^{p(p-1-\alpha )}$ is a
supersolution satisfying $\underset{x\to z}{\lim }g(x)=+\infty $
for every $z\in \partial U$. By the comparison principle we have 
\textsc{H}
$_Un\leqslant g$ for every $n\in \mathbb{N}$ and therefore, the 
increasing
sequence $($\textsc{H}$_Un)_n$ of harmonic functions is locally
uniformly bounded on $U$. The Bauer convergence property implies that 
$u=
\underset{n}{\sup }$\textsc{H}$_Un\in \mathcal{H}(U)$, therefore we 
have 
$\underset{x\to z}{\liminf }u(x)\geqslant n$ for every $z$ in 
$\partial U$, thus $\underset{x\to z}{\lim }u(x)=+\infty $ for every 
$z$ in $\partial U$ and $u$ is a regular Evans function. Since $U$ is 
an
arbitrary ball, we get the desired property.
\end{proof}

\begin{corollary}
Under the assumptions in Theorem \ref{theo65}, for every
ball \textsc{B}$=$\textsc{B}$(x_0,R)$ with center $x_0$ and radius $R$
and for every $u$ $\in \mathcal{H}(U)$,  
\begin{equation*}
\left| u(x_0)\right| \leqslant cR^{\frac{2p}{p-1-\alpha }}
\end{equation*}
where 
\begin{multline*}
c =\big[ \sup_{x\in \text{\textsc{B}}}\Big\{ \tfrac{2(\alpha
+1)(p-1)}{\alpha -p+1}\frac{\left| \mathcal{A}(x,x-x_0).(x-x_0)\right| 
}{
b(x)} +R^{2}\frac{\left| \mathop{\rm div}\mathcal{A}
(x,x-x_0)\right| }{b(x)}\Big\} \Big] ^{\frac{1}{\alpha -p+1}} \\
\times \Big( \frac{2p}{\alpha -p+1}\Big) ^{\frac{p-1}{\alpha -p+1}}.
\end{multline*}
\end{corollary}

\begin{proof}
From the proof of the previous theorem, if \textsc{B}$_n=$\textsc{B}
$(x_0,R(1-n^{-1}))$, $n\geqslant 2$, we have 
\begin{equation*}
u(x_0)\leqslant c_n\left( \tfrac{R(n-1)}{n}\right) ^{\frac{2p}{
p-1-\alpha }}
\end{equation*}
for every $n\geqslant 2$ and 
\begin{eqnarray*}
c_n &=&\left[ \sup_{x\in \text{\textsc{B}}_n}\left\{ \tfrac{2(\alpha
+1)(p-1)}{\alpha -p+1}\frac{\left| \mathcal{A}(x,-x_0).(x-x_0)\right| 
}{
b(x)}\right. \right. \\
&&\left. \left. +\left( \tfrac{R(n-1)}{n}\right) ^{2}\frac{\left| 
\mathop{\rm div}
\mathcal{A}(x,x-x_0)\right| }{b(x)}\right\} \right] ^{\frac{1}{\alpha 
-p+1}
}\left( \tfrac{2p}{\alpha -p+1}\right) ^{\frac{p-1}{\alpha -p+1}} \\
&\leqslant &\text{ }\left[ \text{ }\sup_{x\in \text{\textsc{B}}}\left\{ 
\tfrac{2(\alpha +1)(p-1)}{\alpha -p+1}\frac{\left| \mathcal{A}
(x,x-x_0).(x-x_0)\right| }{b(x)}\right. \right. \\
&&\hspace*{0.7cm}\hspace*{0.7cm}\left. \left. +R^{2}\frac{\left| 
\mathop{\rm div}
\mathcal{A}(x,x-x_0)\right| }{b(x)}\right\} \right] ^{\frac{1}{\alpha 
-p+1}
}\left( \tfrac{2p}{\alpha -p+1}\right) ^{\frac{p-1}{\alpha -p+1}}.
\end{eqnarray*}
Then we obtain the inequality 
\begin{equation*}
u(x_0)\leqslant cR^{\frac{2p}{p-1-\alpha }}.
\end{equation*}
Since $-u$ is a solution of similarly equation, we get 
\begin{equation*}
-u(x_0)\leqslant cR^{\frac{2p}{p-1-\alpha }}
\end{equation*}
with the same constant $c$ as before. Then we have the desired 
inequality.
\end{proof}

We now have a Liouville like theorem.

\begin{theorem}
Assume that the conditions in \emph{Theorem \ref{theo65}} are satisfied 
and
that 
\begin{equation*}
\liminf_{R\to \infty }\left( R^{-2p}M(R)\right) =0
\end{equation*}
where 
\begin{equation*}
M(R)=\sup_{\left\| x-x_0\right\| \leqslant R}\left\{ \tfrac{2(\alpha
+1)(p-1)}{\alpha -p+1}\frac{\left| \mathcal{A}(x,x-x_0).(x-x_0)\right| 
}{
b(x)}+R^{2}\frac{\left| \mathop{\rm div}\mathcal{A}(x,x-x_0)\right| 
}{b(x)}
\right\} .
\end{equation*}
Then $u\equiv 0$ is the unique solution of the equation (\ref{eq2}) on 
$\mathbb{R}^d$.
\end{theorem}

\begin{proof}
Let $u$ be a solution of the equation (\ref{eq2}) on $\mathbb{R}^d$. By
the previous corollary, we have for every $x_0\in \mathbb{R}^d$ and
every $R>0$
\begin{eqnarray*}
\left| u(x_0)\right| &\leqslant &\text{ }\left[ \sup_{\left\|
x-x_0\right\| \leqslant R}\left\{ \tfrac{2(\alpha +1)(p-1)}{\alpha 
-p+1}
\frac{\left| \mathcal{A}(x,x-x_0).(x-x_0)\right| }{b(x)}\right. \right.
\\
&&\hspace*{1cm}\left. \left. +R^{2}\frac{\left| \mathop{\rm 
div}\mathcal{A}
(x,x-x_0)\right| }{b(x)}\right\} R^{-2p}\right] ^{\frac{1}{\alpha -p+1}
}\left( \tfrac{2p}{\alpha -p+1}\right) ^{\frac{p-1}{\alpha -p+1}}.
\end{eqnarray*}
Hence $u(x_0)=0$ and $u\equiv 0$.
\end{proof}

\section{Applications}

We shall use the previous results for the investigation of the $p\!-\!
$Laplace $\Delta_{p}$, $p\geqslant 2$ which is the Laplace operator if 
$p=2
$. $\Delta_{p}$  is associated with $\mathcal{A}(x,\xi )=\left| \xi 
\right|
^{p-2}\xi $, an easy calculation gives 
$\mathop{\rm div}\mathcal{A}(x,x-x_0)=(d+p-2)
\left\| x-x_0\right\| ^{p-2}$. Let, for every $\alpha >0$, $\mathcal{H}
_\alpha $ denote the sheaf of all continuous solution of the equation
\begin{equation}
-\Delta_{p}u+b(x)\mathop{\rm sgn}(u)\left| u\right| ^{\alpha }=0  
\label{eqapp}
\end{equation}
where $b\in L_\text{loc}^{\frac{d}{d-\epsilon }}(\mathbb{R}^d)$, 
$b\geqslant 0$
and $0<\epsilon <1$.

\begin{theorem} \label{theoapp}
Assume that $p\geqslant 2$. For $\alpha >0$, let $\mathcal{H}
_\alpha $ denote the sheaf of all continuous solution of the equation 
\begin{equation*}
-\Delta_{p}u+b(x)\mathop{\rm sgn}(u)\left| u\right| ^{\alpha }=0\,.
\end{equation*}
where $b\in L_\text{loc}^{\frac{d}{d-\epsilon }}(\mathbb{R}^d)$, 
$b\geqslant 0$ and $0<\epsilon <1$. Then
\begin{enumerate}
\item For every $\alpha >0$, $(\mathbb{R}^d,\mathcal{H}_\alpha )$
is a nonlinear Bauer harmonic space with the Brelot convergence 
Property.

\item $\mathcal{H}_\alpha $ is elliptic for every $\alpha \geqslant 
p-1$.

\item If $\alpha >p-1$ and $\inf_U b>0$ for every
relatively compact open $U$ in $\mathbb{R}^d$, then the property (KO) 
is
satisfied by $\mathcal{H}_\alpha $.

\item If $\alpha >p-1$ and , $\inf_{\mathbb{R}^d} b>0$,
then $\mathcal{H}_\alpha (\mathbb{R}^d)=\{ 0\} $.
\end{enumerate}
\end{theorem}

\begin{theorem}
Let $U\subset \mathbb{R}^d$ be an bounded open set whose boundary, 
$\partial U$, can be  represented locally as a graph of function with 
H\"{o}lder
continuous derivatives. Assume that $\alpha >p-1$. Then $U$ admits a 
regular
Evans function for $\mathcal{H}$.
\end{theorem}

\begin{proof}
We first prove the existence of  a continuous supersolution $v$ on $U$ 
such
that $\lim_{x\to z}v(x)=+\infty $, for every $z\in \partial U$.

Let $f$ in $\mathcal{C}_{c}^\infty (U)$ be a positive function ($f\neq 
0$)
and $w\in \mathcal{W}_0^{1,p}(U)$ be the solution of the problem 
$$\begin{gathered}
\int_U\left| \nabla w\right| ^{p-2}\nabla w\cdot\nabla \varphi
dx=\int_Uf\varphi dx , \quad \varphi \in \mathcal{W}_0^{1,p}(U) \\ 
w=0 \quad \text{on }\partial U 
\end{gathered} $$


By the regularity theory, $w$ has a H\"{o}lder continuous gradient, $w$ 
is
continuous supersolution $w>0$ in $U$, $\lim_{x\to z}w(x)=0$
for every $z\in \partial U$ and $\left\| w\right\|_{\infty }+\left\| 
\nabla
w\right\|_{\infty }\to 0$ as $\left\| f\right\|_{\infty
}\to 0$.
Then we set $v=w^{-\beta }$ and look for $\beta >0$ and $f$ such that 
$$ 
\int_U\left| \nabla v\right| ^{p-2}\nabla v\cdot\nabla \varphi
dx+\int_Ub(x)v^{\alpha }\varphi dx\geqslant 0 \quad \varphi
\geqslant 0, \varphi\in \mathcal{W}_0^{1,p}(U).
$$
For every $\varphi \geqslant 0$,$\in \mathcal{W}_0^{1,p}(U)$, we have
\begin{align*}
\int_U| \nabla v| ^{p-2}\nabla v\cdot\nabla \varphi dx 
=&-\beta
^{p-1}\int_Uw^{-(\beta +1)(p-1)}| \nabla w| ^{p-2}\nabla
w\cdot\nabla \varphi dx \\
=&-\beta ^{p-1}\int_U| \nabla w| ^{p-2}\nabla w\cdot\nabla
(w^{-(\beta +1)(p-1)}\varphi )dx \\
&-\beta ^{p-1}(\beta +1)(p-1)\int_Uw^{-(\beta +1)(p-1)-1}\varphi |
\nabla w| ^{p}dx \\
=&-\beta ^{p-1}\int_Uw^{-(\beta +1)(p-1)-1}[ wf+(\beta
+1)(p-1)| \nabla w| ^{p}] \varphi dx;
\end{align*}
thus 
\begin{multline*}
\int_U\left| \nabla v\right| ^{p-2}\nabla v\cdot\nabla \varphi dx \\
+\beta^{p-1}\int_Ubv^{\frac{(\beta +1)(p-1)+1}{\beta }}
\left[ b^{-1}wf+(\beta
+1)(p-1)b^{-1}\left| \nabla w\right| ^{p}\right] \varphi dx=0\,.
\end{multline*}
Put $\beta =\frac{p}{\alpha -p+1}$ and choose $f$ such that $wf+(\beta
+1)(p-1)\left| \nabla w\right| ^{p}\leqslant b\beta ^{1-p}$. Then 
\begin{equation*}
\int_U\left| \nabla v\right| ^{p-2}\nabla v\cdot\nabla \varphi
dx+\int_Ubv^{\alpha }\varphi dx\geqslant 0,\text{ for every }\varphi
\geqslant 0, \varphi\in \mathcal{W}_0^{1,p}(U);
\end{equation*}
therefore, $v$ is a continuous supersolution of (\ref{eqapp}) such that 
$\lim_{x\to z}v(x)=+\infty $, for every $z\in \partial U$.

Let $u_n$ denote the continuous solution of the problem 
$$\begin{gathered}
\int_U\left| \nabla u\right| ^{p-2}\nabla u\cdot\nabla \varphi
dx+\int_Ubu^{\alpha }\varphi dx=0 , \quad  \varphi \in \mathcal{W}
_0^{1,p}(U) \\ 
u=n\in \mathbb{N} \quad \text{on }\partial U
\end{gathered} $$
By the comparison principle we have $0\leqslant u_n\leqslant v$ for all 
$n$
and by the convergence property, the function $u=\sup_nu_n$ is a
regular Evans function for $\mathcal{H}$ and $U$.
\end{proof}

\begin{theorem}
Let $\alpha >p-1$ and let $U$ be a star domain and $b$ continuous and
strictly positive function on $\mathbb{R}^d$. Assume that the 
conditions
in Theorem \ref{theoapp} are satisfied. If there exists a regular
Evans function $u$ associated with $U$ and $\mathcal{H}_\alpha $,
then $u$ is unique.
\end{theorem}

The proof is the same as in \cite{B-h} and \cite{Dy} when $b\equiv 1$.


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\end{document}