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\markboth{ On the $L^{\infty}$-regularity}
{L. Aharouch, E. Azroul and A. Benkirane}

\begin{document}
\setcounter{page}{1}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
2002-Fez conference on Partial Differential Equations,\newline 
Electronic Journal of Differential Equations, 
Conference 09, 2002, pp 1--8. \newline
http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu (login: ftp)}
 \vspace{\bigskipamount} \\
%
  On the $L^{\infty}$-regularity of solutions of nonlinear
  elliptic equations in Orlicz spaces
%
\thanks{ {\em Mathematics Subject Classifications:} 35J60.
\hfil\break\indent
{\em Key words:} Orlicz Sobolev spaces, boundary-value problems.
\hfil\break\indent
\copyright 2002 Southwest Texas State University. \hfil\break\indent
Published December 28, 2002.} }

\date{}
\author{Lahsen Aharouch,  Elhoussine Azroul, \&  Abdelmoujib Benkirane} 
\maketitle

\begin{abstract} 
 Our main result is a maximum principle bounding the absolute
 values of the solution in terms of the supremum of the
 absolute values of the boundary data.
\end{abstract}

\numberwithin{equation}{section}
\newtheorem{lem}{Lemma}[section]
\newtheorem{thm}{Theorem}[section]
\newtheorem{Def}{Definition}[section]
\newtheorem{rem}{Remark}[section]


\section{Introduction}
Let $\Omega$ be a bounded Lipshitz domain in $\mathbb{R}^n$
($n\geq 1$), let $M(t)$ be an $N$-function i.e. continuous,
convex, with $M(t)>0 $ for $t>0$, $M(t)/t\to 0$ as $t\to 0$ and
$M(t)/t\to \infty$ as $t\to \infty$, and $m(t)$ be its right
derivatives. Consider the nonlinear boundary-value problem
$$
\begin{gathered}
Au = -\mathop{\rm div} a(\nabla u) = f \quad \mbox{in } \Omega \\
u = \theta \quad \mbox{in } \partial\Omega
\end{gathered}
\eqno{(1.1)}
$$
with prescribed boundary datum $\theta$, where $a = \{a_i , 1\leq
i\leq n\}$  is a vector of Carath\'eodory functions defined on
$\mathbb{R}^n$  satisfying the hypotheses
\begin{itemize}
\item[(H1)] $|a(\xi)| \leq \lambda_1 m(|\xi|)$ for all 
$\xi \in \mathbb{R}^n$ and some positive constant $\lambda_1$.

\item[(H2)] $ a(\xi).\xi \geq \lambda_2 |\xi|m(|\xi|)$
for all $\xi \in \mathbb{R}^n$ and some positive constant $\lambda_2$.
\end{itemize}

\begin{Def}\rm
Let $\theta\in W^1 L_M (\Omega)$ and $f\in L^1 (\Omega)$.
A function $u\in W^1 L_M (\Omega)$ is called a weak solution of the
 boundary-value problem (1.1) if
$u-{\theta} \in W_{0}^1 L_M (\Omega)$ and
$$\int_{\Omega}a(\nabla u).\nabla \varphi = \int_{\Omega}f. \varphi
\quad \mbox{for all } \varphi\in {\cal C}_{0}^{\infty}(\Omega).
$$
Here $W_{0}^{1}L_M (\Omega)$ and $W^{1}L_M (\Omega)$ denote the
Orlicz-Sobolev Spaces associated to the $N$-function $M$ (see section 2).
\end{Def}

Recently, Fuchs and Gongbao proved in \cite[Theorem 1.1]{fg2} that
if $u$ is a weak solution of (1.1) with the second member $f$ lies
in $L^{\infty}(\Omega)$, and
$\sup_{\partial\Omega}\theta (x)< \infty$, then $u$ is bounded from
above; i.e.,
$$ \sup_{\Omega}u(x)\leq \mathop{\rm const}
(\sup_{ \partial\Omega}\theta (x),\|u\|_{L^1 (\Omega)},n,
|\Omega|,M,\|f\|_{L^{\infty} \Omega)},\lambda_1,\lambda_2)< \infty.
$$
For this, the authors have supposed  additionally to (H1)-(H2)
that the $N$-function $M$ satisfies the $\Delta_2$-condition near
infinity.

The aim of this paper is to prove  (Theorem 3.2) the previous statement
for the general operators,
$$
Au = -\mathop{\rm div} (a(x,u,\nabla u)) \eqno{(1.2)}
$$
without assuming the $\Delta_2$-condition. To do this, we replace the
hypothesis (H1) by the more general growth condition,
 $$
 |a(x,s,\xi)|\leq c(x) + k_1 {\overline {P}}^{-1}M(k_2 |s|)
 + k_3 {\overline {M}}^{-1}M(k_4 |\xi|) \eqno{(1.3)}
 $$
and the hypothesis (H2) by
$$
a(x,s,\xi)\xi \geq \alpha M(\frac{|\xi|}{\beta}). \eqno{(1.4)}
$$
(see section 3). To generalize theorem of \cite{fg2}, in our case,
we need to prove the following approximating result (see theorem 3.2)
$$W_{0}^{1, 1}(\Omega)\cap W^1 L_M (\Omega) = W_{0}^{1}L_M (\Omega).
$$
which guaranties that $\varphi = \max (u-k, 0)$ can be taken as a test
function (for details see theorem 3.2).

When $M(t) = |t|^p$ ($p>1$) (i.e., $a$ satisfies the polynomial growth
condition), the regularity result of the solution of (1.1) are
investigated in \cite{l} and \cite{gt}. Non-standard examples of
 $M(t)$ which occur in the mechanics of solids and fluids are
 $M(t) = t\ln(1+t)$,
$M(t) = \int_{0}^{t}s^{1-\alpha}(\mathop{\rm arsinh} s)^{\alpha}ds$
 ($0\leq \alpha\leq 1$) and $M(t) = t\ln(1 + \ln(1 + t))$
 (see \cite{fs1,fs2,fs3,fg2}) for more details).
When  $Au = -\Delta u$ (corresponding to the Poisson equation),
the reader is referred to  \cite{abt}, where the regularity of $u$
is studied in Orlicz Spaces with respect to the second member $f$
(in particular where $f$ is a measure).

Finally. note that some problems of the calculus of variations
(see  \cite[remark 1.2]{fg2}) can be lead to the equation (3.1) as in
section 3, contributions in this sense include the works
\cite{fg1, ab}.

\section{Preliminaries}

{\bf 2-1} Let $M : \mathbb{R}^{+}\to \mathbb{R}^{+}$ be  an
$N$-function, i.e. $M$ is continuous, convex, with $M(t)>0 $ for
$t>0$, $M(t)/t\to 0$ as $t\to 0$ and $M(t)/t\to \infty$ as $t\to
\infty$.

The $N$-function conjugate to $M$ is defined  as $\overline {M}(t)
= \sup\{st - M(t), s\geq 0\}$. We will extend these $N$-functions
into even functions on all $\mathbb{R}$.

The $N$-function $M$ is said to satisfy the $\Delta_2$-condition if,
for some $k$
$$ M(2t)\leq kM(t)\quad  \forall t\geq 0. \eqno{(2.1)}
$$
When this inequality  holds only for $t\geq \mbox{some}\ t_0 >0$,
 $M$ is said to satisfy the $\Delta_2$-condition near infinity.
Moreover, we have the following Young's inequality
$$
\forall s,t \geq 0,\quad  st \leq M(s) + \overline {M}(t) \eqno{(2.2)}
$$
Let $P$ and $M$ be two $N$-functions. $P<<M$ means that $P$ grows
 essentially less rapidly than $M$, i.e. for each $\varepsilon >0$,
$\frac{P(t)}{M(\varepsilon t)}\to 0$ as $t\to \infty$
This is the case if and only if
$$ \lim_{t\to \infty}\frac{M^{-1}(t)}{P^{-1}(t)} = 0
$$
{\bf 2-2} Let $\Omega$ be an open subset of $\mathbb{R}^n$. The
Orlicz class $K_{M}(\Omega)$ [resp. The Orlicz space $L_M
(\Omega)$] is defined as the set of (equivalence classes of)
real-valued measurable functions $u$ on $\Omega$ such that,
$$
\int_{\Omega}M(u(x))dx<+\infty \quad (\mbox{resp. }
 \int_{\Omega}M(\frac{u(x)}{\lambda})dx \ \mbox{for some } \lambda
 >0).
$$
$L_M (\Omega)$ is a Banach space under the norm
$$
\|u\|_M = \inf\{\lambda >0 : \int_{\Omega}M(\frac{u(x)}{\lambda})dx
\leq 1\}
$$
and $K_M(\Omega)$ is a convex subset of $L_M (\Omega)$.
The closure in $L_M (\Omega)$ of the set of bounded measurable functions
with compact support in $\overline {\Omega}$ is denoted by
$E_M (\Omega)$.

\noindent{\bf 2-3} We now turn to the Orlicz-Sobolev spaces,
$W^1 L_M (\Omega)$ [resp. $W^1 E_M (\Omega)$] is the space of
functions $u$ such that $u$ and its distributional derivatives up to
order 1 lie in $L_M (\Omega)$ [resp. $E_M (\Omega)$].
It is a Banach space under the norm
$$
\|u\|_{1.M} = \sum_{|\alpha|\leq 1}\|D^{\alpha}u\|_M
$$
Thus, $W^1 L_M (\Omega)$ and $W^1 E_M (\Omega)$ can be identified with
subspaces of the product of $N + 1$ copies of $L_M (\Omega)$.
Denoting this product by $\Pi L_M$, we will use the weak topologies
 $\sigma (\Pi L_M, \Pi E_{\overline {M}})$ and
 $\sigma (\Pi L_M, \Pi L_{\overline {M}})$.

The space  $W_{0}^{1} E_M (\Omega)$ is defined as the (norm) closure
of the Schwartz space ${\cal D}(\Omega)$ in $W^1 E_M (\Omega)$ and
the space $W_{0}^{1} L_M (\Omega)$  as the
$\sigma (\Pi L_M, \Pi E_{\overline {M}})$ closure of
${\cal D}(\Omega)$ in $W^1 L_M (\Omega)$.
Now, we recall the following concept.

\begin{Def} \rm
A domain $\Omega$ has the segment property if for every
$x\in \partial \Omega$ there exists an open set $G_x$ and a nonzero
 vector $y_x$  such that $x\in G_x$ and if
 $z\in \overline {\Omega}\cap G_x$, then $z + ty_x \in \Omega$
 for all $0<t<1$.
\end{Def}

Recall that if $\Omega$ is a bounded Lipshitz domain it satisfies the segment property (see \cite{a}).

\section{Main results}
Let $\Omega$ be a bounded Lipshitz domain in $\mathbb{R}^n$. Our
first aim of this section is to prove the following result which
play an important role in the proof of the regularity result
(Theorem 3.2). Note that some ideas of the proof of Theorem 3.1
are inspired from the analogous of Theorem 1.3 of \cite{g1}.

\begin{thm}
Let $M$ be an $N$-function,  $\Omega$ be  a bounded open domain of
$\mathbb{R}^N$ satisfying the segment property. Then
$$ W_{0}^{1, 1}(\Omega)\cap W^1 L_M (\Omega) = W_{0}^{1}L_M (\Omega).
$$
\end{thm}
{\bf Proof.}  We use the  notation
$$
\tilde{u}=\begin{cases}
u &\mbox{in } \Omega\\
0 &\mbox{in } \mathbb{R}^N\backslash\Omega
\end{cases}
$$
{\bf Step 1} We show that, $W_{0}^{1, 1}(\Omega)\cap W^1 L_M (\Omega)
\subset W_{0}^{1}L_M (\Omega)$.\\
 Let $u\in W_{0}^{1, 1}(\Omega)\cap W^1 L_M (\Omega) $, set
 $K =: {\overline {\{x\in \Omega , u(x)\neq 0 \}}}^{\mathbb{R}^N}$
 (closure in $\mathbb{R}^N$). Then $K$ is a compact in $\mathbb{R}^N$,
 and $K\subset \overline{\Omega}$.

If $K\subset \Omega$. Let $j_{\varepsilon}$ a mollifier function,
the convolution $j_{\varepsilon}*u$ belongs in $C_{0}^{\infty}(\Omega)$,
for all $0<\varepsilon <dist(K, \partial \Omega)$.
By \cite[Lemma 6]{g1} we get $j_{\varepsilon}*u \to u$ in
$W^1 L_M (\Omega)$ for $\sigma (\Pi L_M, \Pi L_{\overline M})$, as
 $\varepsilon \to 0^+$, this proves that $u\in W_{0}^{1}L_M (\Omega)$.

If $K\cap \partial \Omega \neq \emptyset$. For all $x\in \partial \Omega$,
 let $G_x$ and  $y_x$ be respectively the open set and the nonzero
vector given by Definition 2.1.
Set $F = K\cap(\overline{\Omega}\backslash \cup_{x\in \partial \Omega}
G_x )$, then $F$ is a subset compact of  $\Omega$. Hence there exist an
open $G_0$ such that $F\subset G_0\subset\subset \Omega$.
Since $K$ is compact, we can found finitely   sets $G_x$;
(let us rename them  $G_1, \dots ,G_k$) such that
$K\subset G_0\cup \dots \cup G_k$. Moreover, as in the proof of
\cite[Theorem 3.18]{a}, we can construct some open sets
$\tilde G_0, \tilde G_1,\dots ,\tilde G_k $ which cover $K$, such that
$\overline{\tilde G_j}\subset G_j$ for every $j$.
 Now, let $\Theta = \{\theta_j, 0\leq j\leq k\}$ be a partition of unity
subordinate to $\{\tilde G_j, 0\leq j\leq k\}$ and put
$u_j = \theta_j u$, for every $j = 0,\dots ,k$.
We have $u = \sum_{j=0}^{k}u_j$ and
$\mathop{\rm supp}u_j\subset \tilde G_j$, for every $j = 0,\dots ,k$.
 So, it suffices to show that   any $u_j$ belongs in
$W_{0}^{1}L_M (\Omega\cap G_j)$.
Since $\overline{\tilde G_0}\subset \Omega$, as in the case
$K\subset \Omega$ above we prove that
$u_0\in W_{0}^{1}L_M (\Omega\cap G_j)$.

 Since $u_j\in W^{1}L_M (\Omega\cap G_j)$ for all $j\geq 1$ we claim
that $\tilde u_j\in W^{1}L_M (\mathbb{R}^N)$ (indeed:  the
function $\tilde u_j$ lies in $W_{0}^{1, 1}(\mathbb{R}^N)$ due to
$u_j\in W_{0}^{1, 1}(\Omega\cap G_j)$.  Moreover, $ \nabla\tilde
u_j = \tilde \nabla u_j  $ in the distributional sense
 and a.e.  on $\mathbb{R}^N$. On the other hand, $\tilde u_j\in L_M(\mathbb{R}^N)$ and
$\nabla\tilde u_j\in(L_M(\mathbb{R}^N))^N$, because $u_j\in W^1
L_M(\Omega\cap G_j)$).
 Let $K_j = \mathop{\rm supp} u_j$ and $u_{j, t} = \tilde u_j(x - ty_j)$,
with $0<t<\min\{1, |y_j|^{-1}\mathop{\rm dist}(\tilde G_j, G_{j}^c)\}$,
where $y_j$ be the nonzero  vector associated to the set  $G_j$
(see Definition 2.1). We claim that
$\mathop{\rm supp} u_{j, t}\subset \Omega\cap G_j,$ for all $t$
satisfying $0<t<\min\{1, |y_j|^{-1}\mathop{\rm dist}(\tilde G_j,
G_{j}^c)\}$.
In fact, by the segment property, we have
$$\mathop{\rm supp} u_{j, t} = K_j + ty_j \subset
(G_j\cap\overline{\Omega}) + ty_j\subset \Omega.
$$
On the other hand, let $x\in \mathop{\rm supp} u_{j, t}$. Then
$\mathop{\rm dist}(x, \tilde G_j)\leq \mathop{\rm dist}(x, x-ty_j) + \mathop{\rm dist}(x-ty_j, K_j) + \mathop{\rm dist}( K_j, \tilde G_j) = \mathop{\rm dist}(x, x-ty_j)$,\\             which implies that,
$$\mathop{\rm dist}(x, \tilde G_j)\leq \mathop{\rm dist}(x, x-ty_j) = |ty_j|.$$
Then, $\mathop{\rm dist}(x, \tilde G_j)<\mathop{\rm dist}(\tilde G_j, G_{j}^c)$, hence $x\in G_j$.\\
Since $u_{j, t}\in W_{0}^1 L_M(\mathbb{R}^N)$ and \  $\mathop{\rm
supp} u_{j, t}\subset \Omega\cap G_j$, in virtue of lemma $1.5$ of
\cite{g1}, we see that $u_{j, t}\to 0$ in
$W^1 L_M(\Omega\cap G_j)$ for $\sigma(\Pi L_M, \Pi L_{\overline{M}})$ as $t\to 0$. Moreover, by using lemma 1.6 of \cite{g1} we can approximate $u_{j, t}$ by a sequences of elements of  ${\cal D}(\Omega\cap G_j)$),  $W^1 L_M(\Omega\cap G_j)$ for $\sigma(\Pi L_M, \Pi L_{\overline{M}})$, hence gives the result.\\ \ \\
{\bf Step 2 }We shall prove that, $W_{0}^{1}L_M (\Omega)\subset W_{0}^{1, 1}(\Omega)\cap W^1 L_M (\Omega)  .$\\
Let $u\in W_{0}^{1}L_M (\Omega)$, by theorem $1.4$ of \cite{g2} there exists $u_n\in {\cal D}(\Omega)$ and $\lambda > 0$ such that
$$\int_{\Omega}M(\frac{D^{\alpha}u_n -D^{\alpha}u}{\lambda})\; dx\to \ 0 \ \ \mbox{as}\ \ n\to \ \infty \ \ \forall \ \ |\alpha|\leq 1.$$
By using Jensen's inequality, we have
$$M(\frac{1}{\mathop{\rm meas}(\Omega)}\int_{\Omega}(\frac{D^{\alpha}u_n -D^{\alpha}u}{\lambda})\; dx)\leq \frac{1}{\mathop{\rm meas}(\Omega)}\int_{\Omega}M(\frac{D^{\alpha}u_n -D^{\alpha}u}{\lambda})\; dx$$
for $n$ large enough. Then, $$M(\frac{1}{\mathop{\rm
meas}(\Omega)}\int_{\Omega}(\frac{D^{\alpha}u_n
-D^{\alpha}u}{\lambda})\; dx)\to 0, \ \ \mbox{as}\ \ n\to \ \infty
\ \ \forall \ \ |\alpha|\leq 1,$$ which gives, since $M^{-1}$ is
right continuous in $\mathbb{R}^+$,
$$\int_{\Omega}|D^{\alpha}u_n -D^{\alpha}u|\; dx \to  0 \quad
 \mbox{as } n\to \infty \; \forall \, |\alpha|\leq 1.
$$
This completes the proof. \hfil$\square$\smallskip

 Let $M$ and $P$ be two $N$-functions such that $P<<M$. Consider
 the Leray Lions operator $A$ defined from
$D(A)\subset W_{0}^{1} L_M (\Omega)\to W^{-1} L_{\overline {M}}(\Omega)$
 by
$$Au = -\mathop{\rm div} (a(x,u, \nabla u)),
$$
where $a :\Omega \times \mathbb{R}\times\mathbb{R}^n \to
\mathbb{R}^n$ is a Carath\'eodory function satisfying for a.e.
$x\in \Omega$, all $s\in \mathbb{R}$ and all $\xi \neq \xi^{*}
\in\mathbb{R}^n$:
\begin{itemize}
\item[(H1')]
$|a(x,s,\xi)|\leq c(x) + k_1 {\overline {P}}^{-1}M(k_2|s|)
+ k_3 {\overline {M}}^{-1}M(k_4|\xi|)$
\item[(H2')]
$a(x,s,\xi)\xi \geq \alpha M(\frac{|\xi|}{\beta})$
for some positive constants $k_1,\dots ,k_4, \alpha$ and $\beta$,
where $c(x)$ belongs to $E_{\overline {M}}(\Omega)$.
\end{itemize}
Consider the nonlinear boundary-value problem
$$
\begin{gathered}
-\mathop{\rm div}(a(x,u,\nabla u)) = f\quad \mbox{in } \Omega \\
u \equiv \theta \quad \mbox{in } \partial\Omega.
\end{gathered}
\eqno{(3.1)}
$$
Our objective is the following.

\begin{thm}
Let $u\in W^1 L_M (\Omega)$ denotes a weak solution of (3.1) and assume
that $f$ is in $L^{\infty}(\Omega)$.
Then $ \sup_{\partial \Omega}\theta <\infty$ implies that
$u$ is bounded from above i.e.
$$ \sup_{\Omega}u \leq \mathop{\rm const}
(\sup_{\partial \Omega}\theta ,\|u\|_{L^1 (\Omega)},n,
|\Omega|,M,\|f\|_{L^{\infty} (\Omega)},\alpha ,\beta) <\infty .
$$
\end{thm}

\begin{rem}\rm
Note that the statement of Theorem 3.2 holds for any $N$-function M.
In particular for the following critical cases:
$$
M(t) = t\log(1 + t) \quad \mbox{and}\quad M(t) = e^{t} - t - 1.
$$
\end{rem}

We state the following lemmas which are needed below.

\begin{lem}[\cite{fg1}]
Let $k_0 >0, \gamma >0, \varepsilon >0$ and $\alpha\in[0, 1+\varepsilon]$
denote constants and suppose that $u\in L^1 (\Omega)$ satisfies the
estimate
$$\int_{A_k}(u-k)dx \leq \gamma k^{\alpha}|A_k|^{1+\varepsilon},
$$
for all $k\geq k_0$, where $A_k$ denotes the set of points $x\in\Omega$
for which $u(x)>k$. Then $ \sup_{\Omega}u$ is bounded by a finite
constant depending on $\gamma ,\varepsilon ,\alpha ,k_0$ and
$\|u\|_{L^1 (A_{k_0})}$.
\end{lem}

\noindent{\bf Proof of Theorem 3.2 }
Let $k_0 = \sup_{\partial \Omega}\theta <\infty$ and let $u$ be a
weak solution of (3.1).
Let us remark that by the lemma 3.1, it suffices to show that
$$
\int_{A_k}(u-k)dx \leq \mathop{\rm const} (n,|\Omega|,M,\|f\|_{L^{\infty}
 (\Omega)},\alpha ,\beta)|A_k|^{1+\frac{1}{n}}\quad  \forall k\geq k_0.
\eqno{(3.2)}
$$
Observe that
$$
\begin{aligned}
\int_{A_k}f(u-k)dx  \leq &
|A_k|^{1/n}\Big(\int_{A_k}|u-k|^{\frac{n}{n-1}}dx\Big)^{\frac{n-1}{n}} \\
\leq & c(n)|A_k|^{1/n}\int_{A_k}|\nabla u|dx.
\end{aligned} \eqno{(3.3)}
$$
On the other hand, we have
$$
\begin{aligned}
{\int_{A_k}|\nabla u|dx }\leq &
{\int_{A_k}\beta\frac{|\nabla u|}{\beta}dx
\leq \int_{A_k}(\overline {M}(\beta) + M(\frac{|\nabla u|}{\beta}))dx}\\
\leq & {\overline {M}(\beta)|A_k|+\int_{A_k} M(\frac{|\nabla u|}{\beta})dx}.
\end{aligned} \eqno{(3.4)}
$$
Moreover by theorem 3.1, the function $\varphi = \max(u-k,0)$ lies in
the space $ W_{0}^{1}L_M (\Omega)$.  Hence $\varphi$ is admissible
in (3.1) and we obtain,
$$
\int_{A_k}M(\frac{|\nabla u|}{\beta})dx \leq \frac{1}{\alpha}
\int_{A_k}f(u-k)dx. \eqno{(3.5)}
$$
The right hand side of this inequality can be estimated with
H\"older's  inequality and the imbedding
$W_{0}^{1.1}(\Omega)\hookrightarrow L^{1^{*}}(\Omega)$ as follows:
$$\begin{aligned}
\frac{1}{\alpha} \int_{A_k}f(u-k)dx
\leq &\frac{1}{\alpha}\|f\|_{\infty}|A_k|^{1/n}(\int_{A_k}
|u-k|^\frac{n}{n-1}dx)^{\frac{n-1}{n}}\\
\leq &\frac{2\beta}{\alpha}c(n)\|f\|_{\infty}|A_k|^{1/n}
\int_{A_k}\frac{|\nabla u|}{2\beta}dx.
\end{aligned}
$$
Using  Young's inequality we can write
$$
\begin{aligned}
{\frac{1}{\alpha} \int_{A_k}f(u-k)dx}
\leq &{\overline  {M}(\frac{2\beta}{\alpha}c(n)\|f\|_{\infty}|A_k|^{1/n})|
A_k| + \int_{A_k}M(\frac{|\nabla u|}{2\beta})dx}\\
\leq &{\overline  {M}(\frac{2\beta}{\alpha}c(n)\|f\|_{\infty}
|A_k|^{1/n})|A_k| + \frac{1}{2}\int_{A_k}M(\frac{|\nabla u|}{\beta})dx.}
\end{aligned} \eqno{(3.6)}
$$
Then the conclusion follow immediately from  (3.3)--(3.6).

\begin{rem} \rm
The method used in the proof of the above theorem gives also
$ \inf_{\Omega}u > -\infty$ provided that $\theta$  is bounded from below. In particular, boundedness of $\theta$ implies $u\in L^{\infty}(\Omega)$ (compare with remark 1.1 \cite{fg2}).
\end{rem}

\paragraph{Example} Let $M(t) = e^{|t|} - |t| - 1$, we set
 $a(x, s, \xi )= (a_i (x, s, \xi ))_{1\leq i\leq n}$ such that
$$
a_i (x, s, \xi )  =
\begin{cases}
\frac{e^{|\xi_i|}-|\xi_i|-1}{|\xi_i|}\mathop{\rm sign}\xi_i
&\mbox{if }|\xi_i|\neq 0\\
 0 &\mbox{if }|\xi_i| = 0\,.
\end{cases}
$$
Then  $M(t)$ and $a(x, s, \xi)$ satisfy the conditions (1.3) and (1.4).
Note that the $N$-function $M(t)$ does not satisfy the
$\Delta_2$-condition.

\begin{rem} \rm
 Let $m(t)$ the right derivative of $M(t)$. Then $m(t)$ and
$a(x, s, \xi)$  don't satisfy the condition $(H_2)$. Indeed, take
$\xi = (0, \dots ,0, n, 0, \dots , 0), n\in \mathbb{N}$. Then we
have $a(x, s, \xi)\xi = e^n -n-1$, and $|\xi|m(|\xi|) = n(e^n
-1)$. But for all constant $C>0$ there exists $n$ large enough,
such that $\frac{e^n -n-1}{n(e^n -1)}< C$.
\end{rem}

\begin{thebibliography}{99} \frenchspacing

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\bibitem{ab} E. A{\sc zroul} and A. B{\sc enkirane},
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 space,}{ Math. Slovaca,  {\bf 51} (2001), No1, 93-105.}

\bibitem{abt} E. A{\sc zroul}, A. B{\sc enkirane} and M. T{\sc ienari},
 {\em On the regularity of solutions to the Poisson equation in
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\bibitem{e} A. E{\sc lkhalil},
{\em Autour de la premi\`ere courbe propre du $p-$Laplacien,}
{Th\`ese de Doctorat, Facult\'e des Sciences Dhar-Mahraz Fes 6 Mai 1999.}

\bibitem{fg1} M. F{\sc uchs} and L. G{\sc ongbao},
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\noindent\textsc{Lahsen Aharouch} (e-amil: lahrouche@caramail.com)\\
\textsc{Elhoussine Azroul} (e-mail: elazroul@caramail.com)\\
\textsc{Abdelmoujib Benkirane} (e-mail: abenkirane@fsdmfes.ac.ma)\\[2pt]
 D\'epartement de Math\'ematiques et Informatique\\
 Facult\'e des Sciences Dhar-Mahraz\\
 B.P. 1796 Atlas F\`es, Maroc
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