\documentclass[twoside]{article}
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\markboth{Strongly nonlinear elliptic problem without growth condition }
{ Aomar Anane \& Omar Chakrone }

\begin{document}
\setcounter{page}{41}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
2002-Fez conference on Partial Differential Equations,\newline
Electronic Journal of Differential Equations,
Conference 09, 2002, pp 41--47. \newline
http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu (login: ftp)}
 \vspace{\bigskipamount} \\
%
  Strongly nonlinear elliptic problem without growth condition
%
\thanks{ {\em Mathematics Subject Classifications:} 49R50, 74G65, 35D05.
\hfil\break\indent
{\em Key words:} $p$-Laplacian, growth condition.
\hfil\break\indent
\copyright 2002 Southwest Texas State University. \hfil\break\indent
Published December 28, 2002. } }

\date{}
\author{Aomar Anane \& Omar Chakrone}
\maketitle

\begin{abstract}
    We study a boundary-value problem for the
    $p$-Laplacian with a nonlinear term. We assume only coercivity
    conditions on the potential  and  do not assume growth condition
    on the nonlinearity. The coercivity is obtained by using similar
    non-resonance conditions as those in \cite{An-Go}.
\end{abstract}

\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{prop}[theorem]{Proposition}
\newtheorem{coro}[theorem]{Corollary}


\section{Introduction}

Consider the boundary-value problem
\begin{equation} \label{P}
\begin{gathered}
-\Delta_{p} u =  f(x,u) +h \quad \text{in }\Omega,\\
u=0 \quad \text{on }\partial\Omega,
\end{gathered}
\end{equation}
where $\Omega$ is a bounded domain of $\mathbb{R}^N$,
$-\Delta_{p}\colon W^{1,p}_{0}(\Omega)\to W^{-1,p'}(\Omega)$ is
the $p$-Laplacian operator defined by
$$
 \Delta_{p}u \equiv \mathop{\rm div}
(|\nabla u|^{p-2}\nabla u), \quad 1<p<\infty.
$$
The $p$-Laplacian is a degenerated quasilinear elliptic operator that
reduces to the classical Laplacian when $p=2$.
The notation $\langle .,.\rangle$ stands hereafter for the duality pairing
between $ W^{-1,p'}(\Omega)$ and $ W^{1,p}_{0}(\Omega)$.
While $f\colon \Omega\times \mathbb{R} \to \mathbb{R}$ is a Carath\'eodory
function and $h\in W^{-1,p'}(\Omega)$.

Consider the energy functional $\Phi\colon W^{1,p}_0(\Omega) \to
\overline{\mathbb{R}}$ associated with the problem
$$
\Phi(u)=\frac{1}{p}\int_\Omega |\nabla u |^p\,dx-\int_\Omega
F(x,u)\,dx -\langle h,u\rangle,
$$
where $F(x,s)=\int_0^s f(x,t)\,dt$. We are interested in
conditions to be imposed on the nonlinearity $f$ in order that
problem \eqref{P} admits at least one solution $u(x)$ for
any given $h$. Such conditions are usually called
\emph{non-resonance conditions}.

When the nonlinearity satisfies a growth condition of the type
\begin{equation}\label{(f)}
|f(x,s)|\leq a |s|^{q-1} +b(x) \quad \text{for all }s \in \mathbb{R},
\text{ and a.e. in }\Omega,
\end{equation}
with $q<p^*$ where the Sobolev exponent $p^{*}=\frac{Np}{N-p}$ when
$p<N$ and $p^{*}=+\infty$ when $p\geq N$ and $b(x)\in L^{(p^{*})'}(\Omega)$,
the functional $\Phi$ is well defined and is of class $\mathcal{C}^1$,
l.s.c.  and its critical points are weak solutions of \eqref{P}
in the usual sense.

However, when this growth condition is not satisfied, $\Phi$ is not
necessarily of class $\mathcal{C}^1$ on $W^{1,p}_0(\Omega)$ and may
take infinite values.
The first eigenvalue of the $p$-Laplacian  characterized by the
variational formulation
$$
\lambda_{1}=\lambda_{1}(-\Delta_{p})=\min \Big\{
\frac{\int_{\Omega}|\nabla u|^{p}}{\int_{\Omega}|u|^{p}}\,dx;\;u\in
W^{1,p}_{0}(\Omega)\setminus \{ 0\}\Big\}
$$
is known to be associated to a simple
eigenfunction that does not change sign \cite{An}.

A procedure used to treat \eqref{P} when the nonlinearity lies
asymptotically on the left of $\lambda_{1}$ consists in supposing a
``coercivity'' condition on $F$ of the type
\begin{equation}\label{(F)}
\limsup_{s\to\pm\infty}\frac{pF(x,s)}{|s|^p}<\lambda_1\quad
\text{for almost every }x\in\Omega
\end{equation}
and minimizing $\Phi$ on $W^{1,p}_{0}(\Omega)$.  The minimum being a
weak solution of \eqref{P} in an appropriate sense \cite{An-Go,Degio-Zani1,Ch}.
Another way is to obtain \emph{a priori} estimates on the solutions
of some equations approximating \eqref{P} and to show that their
weak limit is indeed a weak solution.


Note that with the help of the conditions \eqref{(f)} and \eqref{(F)}, we know
since the work of Hammerstein (1930) that \eqref{P} admits a
weak solution that minimizes the functional $\Phi$ on
$W^{1,p}_{0}(\Omega)$.
The condition \eqref{(F)} does not imply a growth condition on $f$ unless
$f(x,u)$ is convex in $u$ (see for example
\cite{jabri-moussaoui2}).

In~\cite{An-Go}, Anane and Gossez supposed only a one-sided growth
condition with respect to the Sobolev (conjugate) exponent that do not
suffice to guarantee the differentiability of $\Phi$, which may even
take infinite values.
Nevertheless, they showed that any minimum of $\Phi$ solves
\eqref{P} in a suitable sense.

Here, we assume $1<p<\infty$ and
only that $f$ maps  $L^\infty(\Omega)$ into $L^1(\Omega)$; i.e.,
\begin{equation}\label{(f_0)}
\sup_{|s|\leq R}|f(.,s)|\in L^1_{\rm loc}(\Omega),\quad \forall R>0
\end{equation}
and a coercivity condition of the type \eqref{(F)}.  We prove that
any minimum $u$ of $\Phi$, which is not of class $\mathcal{C}^1$ on
$W^{1,p}_{0}(\Omega)$ and may take infinite values too, is a weak
solution of \eqref{P} in the sense
$$
\int_\Omega |\nabla u|^{p-2}\nabla u \nabla v\,dx =\int_\Omega
f(x,u)v\,dx +\langle h,v\rangle,
$$
for $v$ in a dense subspace of $W^{1,p}_0(\Omega)$. This result is proved
by Degiovanni-Zani~\cite{Degio-Zani1} in the case $p=2$.

In the autonomous case $f(x,s)=f(s)$, De~Figueiredo and
Gossez~\cite{Defi-Go} have proved the existence of solutions for any $h\in
L^{\infty}(\Omega)$ by a topological method. They supposed only a
coercivity condition and established that
$$
\int_{\Omega}|\nabla u|^{p-2}\nabla u\nabla v\,dx=\int_{\Omega}f(x,u)
v\,dx +\langle h,v\rangle
$$
for all $v\in W^{1,p}_{0}(\Omega)\cap L^{\infty}(\Omega) \cup \{u\}$
but the solution obtained may not minimize $\Phi$.  Indeed, an example
is given in~\cite{Defi-Go} in the case $p=2$ and an other one is given
in~\cite{Ch} where $p$ may be different from~2.

Note that in our case, the condition \eqref{(f_0)} implies no growth
condition on $f$ as it may be seen in the following example.

\paragraph{Example}
    Consider the function
    $$
    f(x,s)= \begin{cases}
    d(x)\Big( \sin (\frac{\pi s}{2})-\frac{\mathop{\rm sign}(s)}{2}\Big) \exp \Big(
    \frac{2\cos\left( \frac{\pi s}{2}\right)}{\pi} +\frac{|s|-1}{2}
    \Big) & \text{if } |s|\geq 1\\
    d(x)\frac{s}{2}(10 s^{2}-9) &\text{if } |s|\leq 1\,,
    \end{cases}
    $$
    where $d(x)\in L^{1}_{\rm loc}(\Omega)$ and $d(x)\geq 0$ almost
    everywhere in $\Omega$, so that
    $$
    F(x,s)= \begin{cases}
    -d(x)\exp\Big(\frac{2\cos \left(
    \frac{\pi s}{2}\right)}{\pi} \Big) \exp \Big(
    \frac{|s|-1}{2}  \Big) & \text{if } |s|\geq 1\\
    -d(x)\frac{s^{2}}{4}(-5 s^{2}+9) &\text{if } |s|\leq 1\,.
    \end{cases}
    $$
    Then $F(x,s)\leq 0$ for all $s\in\mathbb{R}$ almost everywhere in
    $\Omega$. So, $\Phi$ is coercive. Nevertheless, as we can check
    easily, $f$ satisfies no growth condition.

\section{Theoretical approach}

We will show that when \eqref{(f_0)} is fulfilled, any minimum $u$
of $\phi$ is a weak
solution of \eqref{P} in an acceptable sense.

\paragraph{Definition}
The space $L^{\infty}_0(\Omega)$ is defined by
$$
L^{\infty}_0(\Omega)=\big\{v\in L^{\infty}(\Omega);\;v(x)=0\text{ a.e.  outside a compact
subset of }\Omega \big\}.
$$
For $u\in W^{1,p}_0(\Omega)$, we set
$$
V_{u}=\big\{v\in W^{1,p}_0(\Omega)\cap L^{\infty}_0(\Omega);\;u\in L^\infty(\{x\in
\Omega;\;v(x)\not=0\}) \big\}.
$$

\begin{prop}[Brezis-Browder {\cite{B-B}}]\label{bre-bro-prop}
If $u\in W^{1,p}_0(\Omega)$, there exists a sequence $(u_n)_n\subset
W^{1,p}_0(\Omega)$ such that:
\begin{itemize}
\item[(i)] $(u_n)_n\subset W^{1,p}_{0}(\Omega)\cap L^{\infty}_{0}(\Omega)$.
\item[(ii)] $|u_n(x)|\leq |u(x)|$ and $u_n(x).u(x) \geq 0$ a.e. in $\Omega$.
\item[(iii)] $u_n\to u$ in $W^{1,p}_0(\Omega)$, as $n\to\infty$.
\end{itemize}
\end{prop}

The linear space $V_{u}$ enjoys some nice properties.

\begin{prop} \label{prop2}
The space $V_u$ is dense in $W^{1,p}_0(\Omega)$.  And if we assume
that \eqref{(f_0)} holds, then
$$
A_u=\big\{\varphi\in W^{1,p}_0(\Omega);\;f(x,u)\varphi\in L^1(\Omega) \big\}
$$
is a dense subspace of $W^{1,p}_0(\Omega)$ as $V_u\subset A_u$.  More
precisely, Brezis-Browder's result holds true if we replace
$W^{1,p}_{0}(\Omega)\cap L^{\infty}_{0}(\Omega)$ by $V_{u}$.
\end{prop}

\paragraph{Proof}
    It suffices to show that $V_{u}$ is dense in $W^{1,p}_{0}(\Omega)$
 and that $V_{u}\subset A_{u}$ when \eqref{(f_0)} holds.\\
\textbf{The density of $V_{u}$ in $W^{1,p}_{0}(\Omega)$:}
     We have to show that for any $\varphi\in W^{1,p}_{0}(\Omega)$, there
     exists a sequence $(\varphi_{n})_{n}\subset V_{u}$ satisfying
    (ii) and (iii). This is done in two steps. First, we show it is true
    for all $\varphi\in W^{1,p}_{0}(\Omega)\cap L_{0}^{\infty}(\Omega)$.
    Then, using Proposition~\ref{bre-bro-prop}, we show it is true in
    $W^{1,p}_{0}(\Omega)$.

\noindent  \textbf{First Step:}
Suppose $\varphi\in W^{1,p}_{0}(\Omega)\cap
    L_{0}^{\infty}(\Omega)$ and consider a sequence
    $(\Theta_{n})_{n}\subset \mathcal{C}^{\infty}_{0}(\mathbb{R})$ such that:\\
    (1) $\mathop{\rm supp} \Theta_{n}\subset [-n,n]$,\\
 (2) $\Theta_n\equiv 1$ on $[-n+1,n-1]$,\\
 (3) $0\leq \Theta_{n}\leq 1$ on $\mathbb{R}$ and\\
    (4) $|\Theta_{n}'(s)|\leq 2$.

    The sequence we are looking for is obtained by setting
    $$
 \varphi_{n}(x)=(\Theta_n\circ u)(x)\varphi(x) \quad\text{for a.e. }x \text{
    in }\Omega.
    $$
    Indeed, let's check the following three statements\\
    (a) $\varphi_{n}\in V_{u}$,\\
    (b) $|\varphi_{n}(x)|\leq |\varphi(x)|$ and
 $\varphi_{n}(x)\varphi(x)\geq 0$
    a.e. in $\Omega$ and\\
    (c) $\varphi_{n}\to \varphi$ in $W^{1,p}_{0}(\Omega)$.\\
    For (a), since $\varphi\in L^{\infty}_{0}(\Omega)$, we have that
    $\varphi_{n}\in L^{\infty}_{0}(\Omega)$ and it's clear by~(4) that
    $\varphi_{n}\in W^{1,p}_{0}(\Omega)$. Finally, by (1),
    $u(x)\in[-n,n]$ for a.e. $x$ in $\{ x\in \Omega;\;    \varphi_{n}(x)\not= 0\}$.
    The assumption (b) is a consequence of (3).
    For (c), by (2), $\varphi_{n}(x)\to\varphi(x)$ a.e. in $\Omega$ and
    $$
    \frac{\partial \varphi_{n}}{\partial x_{i}}(x)=\Theta'_{n}(u(x))\frac{\partial
    u}{x_{i}}\varphi(x)+\Theta_{n}(u(x))\frac{\partial \varphi}{\partial
    x_{i}}\ \to\ \frac{\partial \varphi}{\partial x_{i}}\text{ in }\Omega.
    $$
    And by (4),
 $$
 \Big|\frac{\partial \varphi_{n}}{\partial x_{i}}(x)\Big|
 \leq 2\Big|\frac{\partial u}{\partial x_{i}}(x)\Big|  |\varphi(x)|
 +\Big|\frac{\partial \varphi}{\partial x_{i}}(x) \Big|\in L^{p}(\Omega).
 $$
 Finally, by the dominated convergence theorem we get (c).

\noindent \textbf{Second Step:}
Suppose that $\varphi\in W^{1,p}_{0}(\Omega)$. By
    Proposition~\ref{bre-bro-prop}, there is a sequence
    $(\psi_{n})_{n}\subset W^{1,p}_{0}(\Omega)$ satisfying (i), (ii)
    and (iii).

    For $k=1,2,\ldots$, there is $n_{k}\in \mathbb{N}$ such that
    $||\psi_{n_{k}}-\varphi||_{1,p}\leq {1}/{k}$. Since
    $\psi_{n_{k}}\in W^{1,p}_{0}(\Omega)\cap
    L^{\infty}_{0}(\Omega)$, by the first step, there is $\varphi_{k}\in
    V_{u}$ such that $|\varphi_{k}(x) |\leq |\psi_{n_{k}}(x)|$ and
    $\varphi_{k}(x)\psi_{n_{k}}(x)\geq 0$ almost everywhere in $\Omega$
    and $||\varphi_{k}-\psi_{n_{k}}||_{1,p}\leq {1}/{k}$, so that
    $(\varphi_{k})_{k}$ is the sequence we are seeking. Indeed,
    $|\varphi_{k}(x) |\leq |\psi_{n_{k}}(x)|\leq |\varphi(x) |$,
    $\varphi_{k}(x)\varphi(x)\geq 0$ a.e. in $\Omega$ and
    $||\varphi_{k}-\varphi(x)||_{1,p}\leq ||\varphi_{k}-\psi_{n_{k}}||_{1,p}
    + ||\psi_{n_{k}}-\varphi(x)||_{1,p}\leq {2}/{k}$.

\noindent\textbf{The inclusion $V_{u}\subset A_{u}$:}
    Indeed, for $\varphi\in V_{u}$, set $E=\big\{ x\in \Omega;\;    \varphi(x)\neq 0\big\}$ so that
    $$
    \begin{array}{rl}
    |f(x,u)\varphi | & = \left| f(x,u) \chi_{E}\varphi(x)\right|\\[2mm]
    & \leq \max \big\{ | f(x,s) \varphi(x)|; |s|\leq||u
    ||_{L^{\infty}(E)}\big\}
    \end{array}
    $$
    where $\chi_{E}$ is the characteristic function of the set $E$.
 By \eqref{(f_0)}, the last term lies to $L^{1}(\Omega)$, so that
    $\varphi\in A_{u}$.
\hfill$\square$



\begin{theorem}\label{thmfond1}
Assume \eqref{(f_0)}. If  $u\in W^{1,p}_{0}(\Omega)$
 is a minimum of $\Phi$
 such that  $F(x,u)\in L^1(\Omega)$, then
\begin{itemize}
\item[(i)]
$\int_\Omega |\nabla u|^{p-2}\nabla u \nabla \phi\,dx =
\int_\Omega f(x,u)\phi\,dx + \langle h,\phi\rangle $
for all $\phi\in A_u$.

\item[(ii)] $f(x,u)\in W^{-1,p'}(\Omega)$ in the sense that the mapping
$T: V_u\to{\mathbb R} : T(\phi)=\int_\Omega f(x,u)\phi\,dx$ is linear,
continuous and admits an unique extension $\tilde{T}$ to the whole
space $W_0^{1,p}(\Omega)$.

\item[(iii)] $\langle f(x,u),\phi\rangle=
\int_\Omega f(x,u)\phi\,dx \ \quad \forall \phi\in A_u$.

\item[(iv)] $-\Delta_p u=f(x,u)+h$  in $W^{-1,p'}(\Omega)$.
\end{itemize}
\end{theorem}

\paragraph{Remark} % rmk1
There are in In~\cite{An-Go} some conditions that guarantee the existence of a minimum
$u$ of $\Phi$ in $W_0^{1,p}(\Omega)$ and consequently $F(x,u)\in L^1(\Omega)$.


\paragraph{Proof of Theorem \ref{thmfond1}}
We will prove that the assertion (i) holds for all $\phi\in V_{u}$ as a first
step, then prove (iii), (iv) and (i).
Let $\phi\in V_{u}$ and $s\in{\mathbb R}$ such that $0<s<1$. There exists
$\beta=\beta(x,s,\phi,u)\in [-1,1]$ such that
 \begin{align*}
 \big|\frac{F(x,u+s\phi)-F(x,u)}{s}\big|
 &=|f(x,u+\beta\phi)\phi|\\
 &\leq \max\big\{|f(x,t)\phi(x)|;\; |t|\leq \|u\|_{L^{\infty}(E)}+
 \|\phi\|_{L^{\infty}(\Omega)}\big\},
 \end{align*}
 where $E=\{ x \in \Omega; \ \phi(x)\neq 0\ \mbox{ a.e. } \}$.
 Since $F(x,u)\in L^1(\Omega)$, by \eqref{(f_0)}, we have
 $F(x,u+s\phi)\in L^1(\Omega)$ for all $0<s<1$.
 On the other hand
    $$
 \lim_{s\to 0}
 \frac{F(x,u(x)+s\phi(x))-F(x,u(x))}{s}
 =f(x,u(x))\phi \quad \mbox{a.e. in }\Omega.
    $$
 It follows from Lebesgue's dominated convergence that
    $$
 \lim_{s\to 0}
 \frac{F(x,u+s\phi)-F(x,u)}{s}
 =f(x,u)\phi \quad \mbox{strongly in }L^1(\Omega).
    $$
 Since $u\in W_0^{1,p}(\Omega)$ is a minimum point of $\Phi$, we get
 $$
 \frac{\Phi(u+s\phi)-\Phi(u)}{s}\geq 0 \quad \mbox{for all }  0<s<1,
 $$
 then, we get  (i) for all $\phi\in V_u$.

 The linear mapping defined by $T(\phi)=\int_{\Omega}f(x,u)\phi$
 is continuous, because for all $\phi\in V_u$,
\[
 |T(\phi)|=\big|\int_{\Omega}|\nabla u|^{p-2}\nabla u \nabla \phi -
 \langle h,\phi\rangle\big|
\leq \big(\|u\|^{p/p'}_{1,p}+\|h\|_{W^{-1,p'}(\Omega)}\big)\|\phi\|_{1,p}.
\]
 By Proposition \ref{prop2}, $T$ admits an unique extension $\tilde{T}$ to the
 whole space $W_0^{1,p}(\Omega)$. Henceforth, we will make the
 identification $f(x,u)=\tilde{T}$. Since
 $$
 \langle -\Delta_p u,\phi\rangle= \langle f(x,u),\phi\rangle-
 \langle h,\phi\rangle \ \ \forall \phi\in V_u,
 $$
 we conclude {\bf (iv)}.
 Let $\phi\in W_0^{1,p}(\Omega)$ such that $f(x,u)\phi\in L^1(\Omega)$,
 i.e. $\phi\in A_u$. By Proposition \ref{prop2} there exists
 $(\phi_n)\subset V_u$. We can suppose that $\phi_n\to\phi$ almost
 everywhere, $|f(x,u)\phi_n|\leq |f(x,u)\phi|$ and
 $f(x,u)\phi_n\to f(x,u)\phi$ a.e..
 By the dominated convergence theorem,
 $$
 f(x,u)\phi_n\to f(x,u)\phi \quad \mbox{in } L^1(\Omega).
 $$
 Since  $\langle f(x,u),\phi_n\rangle=\int_{\Omega} f(x,u)\phi_n$
 for all $n\in {\mathbb N}$ and $f(x,u)\in W^{-1,p'}(\Omega)$
 we get (iii). Finally, (i) is an immediate consequence of
 (iii) and (iv). \hfill$\square$

\section{Description of the space $A_u$}

Now, we will see some condition that guarantee some properties of $A_u$.

\begin{prop} \label{prop4}
Assume \eqref{(f_0)}. Let $u$ be a minimum of $\Phi$ in $W_0^{1,p}(\Omega)$
with $F(x,u)\in L^1(\Omega)$. And let $\phi\in W_0^{1,p}(\Omega)$,
$v\in L^1(\Omega)$ such that $f(x,u(x))\phi(x)\geq v(x)$ or
$f(x,u(x))\phi(x)\leq v(x)$ a.e. in $\Omega$, then $\phi\in A_u$.
\end{prop}

\paragraph{Proof}
Suppose $f(x,u(x))\phi(x)\geq v(x)$ a.e. in $\Omega$ (the same argument
works if $f(x,u(x))\phi(x)\leq v(x)$ a.e. in $\Omega$). By Proposition
\ref{prop2},
there exists $(\phi_n)\subset V_u$ such that $\phi_n\to\phi$ in
$W_0^{1,p}(\Omega)$, $|\phi_n|\leq |\phi|$ and $\phi_n(x)\phi(x)\geq 0$ a.e.
in $\Omega$. We have
\begin{align*}
f(x,u(x))\phi_n(x)&= f^+(x,u(x))\phi_n(x)-f^-(x,u(x))\phi_n(x)\\
&\geq  -f^+(x,u(x))\phi^-(x)-f^-(x,u(x))\phi^+(x)\\
&\geq -v^-(x).
\end{align*}
By Fatou lemma, we have
\begin{align*}
-\infty<\int_{\Omega} f(x,u(x))\phi(x)
&\leq  \liminf_n \int_{\Omega} f(x,u(x))\phi_n(x)\\
&= \liminf_n \langle f(x,u),\phi_n\rangle
<+\infty,
\end{align*}
which implies $f(x,u)\phi\in L^1(\Omega)$, i.e. $u\in A_u$.
\hfill$\square$

\begin{coro} \label{coro5}
If $\eta$, $\eta_1$ and $\eta_2$ in $L^1_{\rm loc}(\Omega)$, such that
one of the following conditions is satisfied:
\begin{itemize}
\item[(1)] $f(x,u(x))\geq\eta(x)$ a.e.  in $\Omega$

\item[(2)] $f(x,u(x))\leq\eta(x)$ a.e.  in $\Omega$

\item[(3)] $f(x,u(x))\leq\eta_1(x)$ a.e.  in $\{x\in\Omega;\;u(x)<0\}$
and $f(x,u(x))\geq\eta_2(x)$ a.e.  in  $\{x\in\Omega;\; u(x)>0\}$,

\item[(4)] $f(x,u(x))\geq\eta_1(x)$ a.e.  in $\{x\in\Omega;\;u(x)<0\}$
and $f(x,u(x))\leq\eta_2(x)$ a.e.  in $\{x\in\Omega;\;u(x)>0\}$.
\end{itemize}
Then $f(x,u)\in L^1_{\rm loc}(\Omega)$ and consequently
$L^{\infty}_c(\Omega)\cap W_0^{1,p}(\Omega)\subset A_u$.
\end{coro}

\paragraph{Proof}
Assume (3) (the same argument works for (4)). Let
$\phi\in C_c^{\infty}(\Omega)$. We set
$\Omega_1=\{x\in \Omega;\;u(x)\leq -1 \mbox{ a.e.}\}$,
$\Omega_2=\{x\in \Omega;\;|u(x)|\leq 1 \mbox{ a.e.}\}$ and
$\Omega_3=\{x\in \Omega;\;u(x)\geq 1 \mbox{ a.e.}\}$.
It suffices to prove that $f(x,u)|\phi|\chi_{\Omega_i}\in L^1(\Omega)$ for
$i=1,2$,  $3$.
By \eqref{(f_0)} we have $f(x,u)\phi\chi_{\Omega_2}\in L^1(\Omega)$.
Let $\theta\in C^{\infty}({\mathbb R})$ :
$$
\theta(s)=\begin{cases}
1&\mbox{if } s\geq 1,\\
0\leq\theta(s)\leq 1 &\mbox{if } 0\leq s\leq 1,\\
0 &\mbox{if } s\leq 0.
\end{cases}
$$
It is clear that $(\theta\circ u)|\phi|\in W_0^{1,p}(\Omega)$ and that
$$
f(x,u(x))(\theta\circ u(x))|\phi(x)|
\geq (\theta\circ u(x))|\phi(x)|\eta_2(x)\in L^1(\Omega).
$$
By Proposition \ref{prop4}, we have   $f(x,u)(\theta\circ u)|\phi|\in L^1(\Omega) $,
then $f(x,u)\phi\chi_{\Omega_3}\in L^1(\Omega)$ (the same argument to prove
$f(x,u)\phi\chi_{\Omega_1}\in L^1(\Omega)$). We conclude that
$f(x,u)\phi\in L^1(\Omega)$ for all $\phi\in C_c^{\infty}(\Omega)$, which
implies $f(x,u)\in L^1_{\rm loc}(\Omega)$.

Now assume (1) (the same argument works for (2)).
For all $\phi\in C_c^{\infty}(\Omega)$ we have
$f(x,u)|\phi|\geq \eta(x)|\phi|\in L^1(\Omega)$, then
$f(x,u)|\phi|\in L^1(\Omega)$; therefore, $f(x,u)\phi\in L^1(\Omega)$.
Then we conclude that $f(x,u)\in L_{\rm loc}^1(\Omega)$.
\hfill$\square$



\begin{thebibliography}{00} \frenchspacing

\bibitem{An-Go}
A.  Anane and J.-P.  Gossez,
\newblock Strongly non-linear elliptic eigenvalue problems.
\newblock \textit{Comm. Partial Diff.  Eqns.}, \textbf{15}, 1141--1159,
(1990).

\bibitem{Degio-Zani1}
M. Degiovanni and S. Zani,
\newblock Euler equations involving nonlinearities without growth
conditions.
\newblock \emph{Potential Anal.}, \textbf{5},  505--512, (1996).


\bibitem{Ch}
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\noindent\textsc{Aomar Anane} (e-mail: anane@sciences.univ-oujda.ac.ma)\\
\textsc{Omar Chakrone } (e-mail: chakrone@sciences.univ-oujda.ac.ma)\\[2pt]
 University Mohamed I, Department of Mathematics,\\
 Faculty of Sciences, Box 524, 60000 Oujda, Morocco\\

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