\documentclass[twoside]{article} \usepackage{amsfonts, amsmath} % used for R in Real numbers \pagestyle{myheadings} \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 $$ \Delta_{p}u \equiv \mathop{\rm div} (|\nabla u|^{p-2}\nabla u), \quad 1
0
\end{equation}
and a coercivity condition of the type \eqref{(F)}. We prove that
any minimum $u$ of $\Phi$, which is not of class $\mathcal{C}^1$ on
$W^{1,p}_{0}(\Omega)$ and may take infinite values too, is a weak
solution of \eqref{P} in the sense
$$
\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 $00\}$,
\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$\square$
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\end{thebibliography}
\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\\
\end{document}