\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
2004-Fez conference on Differential Equations and Mechanics \newline
{\em Electronic Journal of Differential Equations},
Conference 11, 2004, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or
http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2004 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE/Conf/11 \hfil Existence and regularity of entropy
solutions]
{Existence and regularity of entropy solutions for some nonlinear
elliptic equations}
\author[L. Aharouch, E. Azroul\hfil EJDE/Conf/11 \hfilneg]
{Lahsen Aharouch, Elhoussine Azroul} % in alphabetical order
\address{Lahsen Aharouch\hfill\break
D\'epartement de Math\'ematiques et Informatique\\
Facult\'e des Sciences Dhar-Mahraz\\
B.P. 1796 Atlas F\`es, Maroc}
\email{lahrouche@caramail.com}
\address{Elhoussine Azroul\hfill\break
D\'epartement de Math\'ematiques et Informatique\\
Facult\'e des Sciences Dhar-Mahraz\\
B.P. 1796 Atlas F\`es, Maroc}
\email{elazroul@caramail.com}
\date{}
\thanks{Published October 15, 2004.}
\subjclass[2000]{35J60}
\keywords{Orlicz Sobolev spaces; boundary value problems; Entropy solution}
\begin{abstract}
This paper concerns the existence and regularity of entropy solutions
to the Dirichlet problem
\begin{gather*}
Au = -\mathop{\rm div} (a(x, u, \nabla u))
= f - \mathop{\rm div}\phi(u) \quad \mbox{in }\Omega \\
u = 0 \quad \mbox{on } \partial \Omega.
\end{gather*}
In particular, we show the $L^{{\bar q}}$-regularity
of the solution to this boundary-value problem.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{thm}{Theorem}[section]
\newtheorem{lemma}[thm]{Lemma}
\newtheorem{remark}[thm]{Remark}
\allowdisplaybreaks
\section{Introduction}
Let $\Omega$ be a bounded open subset of $\mathbb{R}^N$ $(N\geq 2)$,
and let
$p$ be a real number such that $2-\frac{1}{N}
0.\label{e1.3}
\end{gather}
Here $\alpha >0$, $\beta\geq 0$ and $c(x)\in L^{p'}(\Omega)$.
In the present paper, we study the boundary-value problem
\begin{equation}
\begin{gathered}
Au := -\mathop{\rm div} a(x, u, \nabla u) = f - \mathop{\rm div}\phi(u)
\quad
\mbox{in } \Omega \\
u = 0 \quad \mbox{on } \partial \Omega ,
\end{gathered} \label{e1.4}
\end{equation}
where the right hand side is assumed to satisfy
\begin{gather}
f\in L^1 (\Omega),\label{e1.5} \\
\phi\in C^0 (\mathbb{R} , \mathbb{R}^N).\label{e1.6}
\end{gather}
Recall that, since no growth hypothesis is assumed on the function
$\phi$, the
term $\mathop{\rm div} \phi(u)$ may be meaningless, even as a
distribution for
a function $v\in W_{0}^{1,r}(\Omega)$, $r>1$ (see \cite{b} and
\cite{bgdm}).
\noindent \textbf{Definition}\;
A function $u$ is called an entropy solution of the Dirichlet problem
\eqref{e1.4} if,
\begin{gather*}
u\in W_{0}^{1,q}(\Omega),\quad 10,\\
{\int_{\Omega}a(x, u, \nabla u)\nabla T_k (u -\varphi )\,dx
\leq \int_{\Omega}f T_k (u -\varphi )\,dx
+ \int_{\Omega}\phi(u) \nabla T_k (u -\varphi )\,dx},\\
\forall \varphi\in W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega),
\end{gather*} %\label{e1.7}
where $T_k (s)$ is the truncation operator at height $k>0$ defined on
$\mathbb{R}$.
\smallskip
When $\phi = 0$ and $f$ is a bounded Radon measure, it is known
that \eqref{e1.4} admits a weak solution $u$ in $W_{0}^{1,q}(\Omega)$
with
$ 1 0 \label{e1.9} \\
a(x,s,\zeta)\zeta \geq \alpha M(\frac{|\zeta|}{\delta}),
\label{e1.10}
\end{gather}
for a.e. $x\in \Omega$, for all $s\in \mathbb{R}$, and all $\xi\in
\mathbb{R}^N$
\section{Preliminaries}
Let $M: \mathbb{R}^{+}\to \mathbb{R}^{+}$ be an $N$-function, i.e.
$M$ is continuous, convex, with $M(t)>0$ for $t>0$,
$\frac{M(t)}{t}\to 0$ as $t\to 0$, and $\frac{M(t)}{t}\to \infty$ as
$t\to \infty$.
Equivalently, $M$ admits the representation:
$$
M(t)={\int_{0}^{t}a(s)\,ds}
$$
where $a:\mathbb{R}^{+}\to \mathbb{R}^{+}$ is nondecreasing,
right continuous, with $a(0)=0$, $a(t)>0$ for $t>0$ and $a(t)$ tends
to
$\infty $ as $t\to \infty$.
The conjugate of $M$ is also an $N$-function and it is defined
by
$ \overline {M}={\int_{0}^{t}\bar{a}(s)\,ds}$, where
$\bar a:\mathbb{R}^{+}\to \mathbb{R}^{+}$ is the function
$\bar{a}(t)=\sup\{s: a(s)\leq t \}$.
An $N$-function $M$ is said to satisfy the $\Delta_2$-condition if, for
some
$k$,
\begin{equation}
M(2t)\leq kM(t) \quad \forall t\geq 0. \label{e2.1}
\end{equation}
When \eqref{e2.1} holds only for $t\geq t_0 >0$ then $M$ is said to
satisfy
the $\Delta_2$ condition near infinity.
We will extend these $N$-functions into even functions on all
$\mathbb{R}$.
Moreover, we have the following Young's inequality
$$
st \leq M(t) + \overline{ M}(s), \quad \forall s,t \geq 0.
$$
Given two $N$-functions, we write $P<0$,
$\frac{P(t)}{Q(\epsilon t)}\to 0$ as $t\to \infty$.
This is the case if and only if
$$ \lim_{t\to\infty}\frac{Q^{-1}(t)}{P^{-1}(t)}=0.
$$
%\par {\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
$$
\int_\Omega M(u(x))\,dx < +\infty \quad
(\mbox{resp. } \int_\Omega M(\frac{u(x)}{\lambda})\,dx < +\infty
\mbox{ for some } \lambda>0 ).
$$
The set $L_M(\Omega)$ is Banach space under the norm
$$
\|u\|_{M,\Omega}=\inf\big\{\lambda>0: \int_\Omega
M(\frac{u(x)}{\lambda})\,dx\leq 1\big\}
$$
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
$E_M(\Omega)$.
The dual of $E_M(\Omega)$ can be identified with
$L_{\overline{M}}(\Omega)$
by means of the pairing $\int_\Omega u v \,dx $, and the dual norm of
$L_{\overline{M}}(\Omega)$ is equivalent to
$\|.\|_{\overline{M},\Omega}$.
%{\bf 2-3}
We now turn to the Orlicz-Sobolev space, $W^{1}L_M(\Omega)$
[resp. $W^{1}E_M(\Omega)$] is the space of all 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 product of $N+1$ copies of $L_M(\Omega)$. Denoting this
product
by $\prod L_M$, we will use the weak topologies
$\sigma(\prod L_M, \prod E_{\overline{M}})$ and
$\sigma(\prod L_M, \prod L_{\overline{M}})$.
The space $W_{0}^{1}E_M(\Omega)$ is defined as the (norm) closure of
the
Schwartz space $D(\Omega)$ in $W^{1}E_M(\Omega)$ and the space
$W_{0}^{1}L_M(\Omega)$ as the $\sigma(\prod L_M, \prod
E_{\overline{M}})$
closure of $D(\Omega)$ in $W^{1}L_M(\Omega)$.
%{\bf 2-4}
Let $W^{-1}L_{\overline{M}}(\Omega)$ [resp.
$W^{-1}E_{\overline{M}}(\Omega)$]
denote the space of distributions on $\Omega$ which can be written as
sums
of derivatives of order $\leq 1$ of functions in
$L_{\overline{M}}(\Omega)$
[resp. $E_{\overline{M}}(\Omega)$]. It is a Banach space under the
usual
quotient norm.(for more details see \cite{a}).
We recall some lemmas introduced in \cite{be} which will be used
later.
\begin{lemma} \label{lem2.1.0}
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 $00\\
\int_{\Omega}a(x, u, \nabla u)\nabla T_k (u -\varphi )\,dx\leq
\int_{\Omega}f T_k (u -\varphi )\,dx
+ \int_{\Omega}\phi(u) \nabla T_k (u -\varphi )\,dx,\\
\forall\; \varphi\in W_{0}^{1}L_{M}(\Omega)\cap L^{\infty}(\Omega)
\end{gathered} \label{e3.1}
\end{equation}
admits at least one solution $u\in W_{0}^{1,\overline{q}}(\Omega)$.
\end{thm}
When $p = N$ we assume, in addition, that
There exists an $N$-function $H$
such that $H(t^N)$ is equivalent to $M(t)$.
\begin{thm} \label{thm3.2}
Assume that for $p = N$ the above hypothesis hold,
\eqref{e1.8}--\eqref{e1.10} hold,
$f\in L^1 (\Omega)$, $\phi \in C^0 (\mathbb{R}, \mathbb{R}^N)$,
$\int_{.}^{\infty}\frac{t^{N-1}}{M(t)}\,dt <\infty$ and
$\frac{t^N}{{\overline{H}}^{-1}(e^{t^{N'}})}$ remains bounded near
infinity.
Then \eqref{e3.1} admits at least one solution in
$W_{0}^{1,N}(\Omega)$.
\end{thm}
\begin{proof}[Proof of Theorems \ref{thm3.1} and \ref{thm3.2}] \quad\\
{\bf Step 1 The approximate problem and a priori estimate.}
Let $f_n$ be a sequence in $W^{-1}E_{\overline{M}}(\Omega)\cap L^1
(\Omega)$
such that $f_n \to f$ in $L^1 (\Omega)$, and $\|f_n\|_1 \leq \|f\|_1$.
Consider the approximate problem
\begin{equation}
\begin{gathered}
Au_n = f_n - \mathop{\rm div} \phi_n(u_n) \\
u_n \in W_{0}^{1}L_M (\Omega)
\end{gathered}\label{e3.3}
\end{equation}
where $\phi_n (x) = \phi(T_n (x))$. From the work \cite{gm},
there exists at least one solution $u_n$ of the approximate problem
\eqref{e3.3}.
Moreover, as in \cite{bk}, there exists a constant $C = C(p, \alpha,
\|f\|_1)$
such that
$$\|\nabla u_n\|_{L_{\overline{q}}(\Omega)}\leq C,$$
which implies that $u_n$ is bounded in
$W_{0}^{1,\overline{q}}(\Omega)$.
Then there exists $u\in W_{0}^{1,\overline{q}}(\Omega)$ and a
subsequence still
denoted by $u_n$ such that
\begin{equation}
\begin{gathered}
u_n\rightharpoonup u \quad \mbox{weakly in }
W_{0}^{1,{\overline{q}}}(\Omega)\\
u_n\to u \quad \mbox{strongly in $L^{{\overline{q}}}(\Omega)$ and a.e.
in }\Omega .
\end{gathered}
\label{e3.4}
\end{equation}
Moreover, the use of $T_k (u_n)$ as test function in \eqref{e3.3}
implies that the
sequence $T_k (u_n)$ is bounded in $W_{0}^{1}L_{M}(\Omega)$, then there
exists
a subsequence of $T_k (u_n)$ still denoted by $T_k (u_n)$ such that
\begin{equation}
\begin{gathered}
T_k (u_n)\rightharpoonup T_k(u) \quad \mbox{weakly in }
W_{0}^{1}L_M(\Omega)
\mbox{ for } \sigma(\prod L_M, \prod E_{\overline{M}})\\
T_k(u_n)\to T_k(u) \quad \mbox{strongly in } E_M(\Omega)\mbox{ and a.e.
in } \Omega .
\end{gathered} \label{e3.5}
\end{equation}
\noindent {\bf Step 2 Convergence of the gradient.}
Let $\Omega_r = \{x\in \Omega : |\nabla T_k (u(x))|\leq r\}$ and
denote by $\chi_r$ the characteristic function of $\Omega_r$. Clearly,
$\Omega_r\subset\Omega_{r+1}$ and $\rm {meas}(\Omega\backslash \Omega_r)\to
0$ as $r \to \infty$.\\
Fix $r$ and let $s\geq r$. We have,
\begin{align*}
0&\leq\int_{\Omega_r}[a(x, T_k(u_n) , \nabla T_k(u_n)) - a(x, T_k(u_n)
,
\nabla T_k(u))][\nabla T_k(u_n) -\nabla T_k(u)]\,dx\\
&\leq \int_{\Omega_s}[a(x, T_k(u_n) , \nabla T_k(u_n)) - a(x, T_k(u_n)
,
\nabla T_k(u))][\nabla T_k(u_n) -\nabla T_k(u)]\,dx\\
&=\int_{\Omega_s}[a(x, T_k(u_n) , \nabla T_k(u_n)) - a(x, T_k(u_n) ,
\nabla T_k(u)\chi_s)][\nabla T_k(u_n) -\nabla T_k(u)\chi_s]\,dx\\
&\leq \int_{\Omega}[a(x, T_k(u_n) , \nabla T_k(u_n)) - a(x, T_k(u_n) ,
\nabla T_k(u)\chi_s)][\nabla T_k(u_n) -\nabla T_k(u)\chi_s]\,dx.
\end{align*}
On the other hand, let $h>k$ and $M = 4k + h$.
If one takes $w_n = T_{2k}(u_n -T_{h}(u_n)+T_{k}(u_n)-T_{k}(u))$ as
test
function in \eqref{e3.3}, it is easy to see that $\nabla w_n = 0$ when
$|u_n|>M$.
We can write
$$
\int_{\Omega}a(x, T_M(u_n) , \nabla T_M(u_n))\nabla w_n \,dx
= \int_{\Omega}f_n w_n \,dx + \int_{\Omega}\phi_n (u_n)\nabla w_n \,dx.
$$
We have
\begin{align*}
&\int_{\Omega}a(x, T_M(u_n) , \nabla T_M(u_n))\nabla T_{2k}(u_n
-T_{h}(u_n)
+T_{k}(u_n)-T_{k}(u))\,dx \\
&\geq \int_{\Omega}a(x, T_k(u_n) , \nabla T_k(u_n))(\nabla T_{k}(u_n)
-\nabla T_{k}(u))\,dx\\
&\quad - \int_{|u_n|>k}|a(x, T_M(u_n) , \nabla T_M(u_n))||\nabla
T_{k}(u)|\,dx\\
& = \int_{\Omega}a(x, T_k(u_n) , \nabla T_k(u_n))(\nabla T_{k}(u_n)
-\nabla T_{k}(u)\chi_s)\,dx\\
&\quad - \int_{\Omega}a(x, T_k(u_n) , \nabla T_k(u_n))(\nabla T_{k}(u)
-\nabla T_{k}(u)\chi_s)\,dx\\
&\quad - \int_{|u_n|>k}|a(x, T_M(u_n) , \nabla T_M(u_n))||\nabla
T_{k}(u)|\chi_s
\,dx\\
&\quad - \int_{|u_n|>k}|a(x, T_M(u_n) , \nabla T_M(u_n))|(|\nabla
T_{k}(u)|
- |\nabla T_{k}(u)|\chi_s) \,dx\,.
\end{align*}
Then
\begin{align*}
&\int_{\Omega}a(x, T_M(u_n) , \nabla T_M(u_n))\nabla T_{2k}(u_n
-T_{h}(u_n)+T_{k}(u_n)-T_{k}(u))\,dx\\
&\geq \int_{\Omega}a(x, T_k(u_n) , \nabla T_k(u_n))(\nabla T_{k}(u_n)
-\nabla T_{k}(u)\chi_s)\,dx\\
&\quad - \int_{\Omega \backslash \Omega_s}a(x, T_k(u_n),
\nabla T_k(u_n))\nabla T_{k}(u)\,dx \\
&\quad - \int_{|u_n|>k}|a(x, T_M(u_n) , \nabla T_M(u_n))||\nabla
T_{k}(u)|\chi_s
\,dx \\
&\quad - \int_{\Omega \backslash \Omega_s}|a(x, T_M(u_n) ,
\nabla T_M(u_n))||\nabla T_{k}(u)| \,dx\,.
\end{align*}
From this inequality, it follows
\begin{equation}
\begin{aligned}
&\int_{\Omega}[a(x, T_k(u_n) , \nabla T_k(u_n)) - a(x, T_k(u_n) ,
\nabla T_k(u)\chi_s)][\nabla T_k(u_n) -\nabla T_k(u)\chi_s]\,dx\\
& \leq \int_{|u_n|>k}|a(x, T_M(u_n) , \nabla T_M(u_n))||\nabla
T_{k}(u)|\chi_s \,dx\\
&\quad + \int_{\Omega \backslash \Omega_s}a(x, T_k(u_n) ,
\nabla T_k(u_n))\nabla T_{k}(u)\,dx \\
&\quad + \int_{\Omega \backslash \Omega_s}|a(x, T_M(u_n) ,
\nabla T_M(u_n))||\nabla T_{k}(u)| \,dx \\
&\quad + \int_{\Omega}f_n T_{2k}(u_n
-T_{h}(u_n)+T_{k}(u_n)-T_{k}(u))\,dx \\
&\quad + \int_{\Omega}\phi_n(u_n))\nabla T_{2k}(u_n
-T_{h}(u_n)+T_{k}(u_n)
-T_{k}(u))\,dx \\
&\quad - \int_{\Omega}a(x, T_k(u_n) , \nabla T_k(u)\chi_s)][\nabla
T_k(u_n)
-\nabla T_k(u)\chi_s]\,dx
\end{aligned} \label{e3.6}
\end{equation}
Now, we study each term of the right hand side of the above inequality.
We denote by $\varepsilon_i(t)$ ($i=1,2,3,\dots$) various sequences of
real
numbers which tends to $0$ when $t$ tends to infinity. Remark that
$a(x, T_{\mu}(u_n), \nabla T_{\mu}(u_n))$ is bounded in
$L_{ {\overline{M}}}(\Omega)$ for all $\mu >0$.
Let $\varepsilon >0$, we have
$$
M(\frac{|\nabla T_{k}(u)|\chi_s \chi_{\{|u_n|>k\}}}{\varepsilon})
\leq M(\frac{s}{\varepsilon})\in L^1 (\Omega)
$$
and
$$
|\nabla T_{k}(u)|\chi_s \chi_{\{|u_n|>k\}} \to 0 \quad \rm {a.e.}
$$
Then by the Lebesgue dominated convergence theorem we deduce that
$$
|\nabla T_{k}(u)|\chi_s \chi_{\{|u_n|>k\}} \to 0 \quad \mbox{in }
L_{M}(\Omega),
$$
which implies that the first term in the right hand side of
\eqref{e3.6} tends to
$0$ as $n$ tends to $\infty$. Concerning the second and third terms on
the
right hand side of \eqref{e3.6}, since $|a(x, T_M(u_n) , \nabla
T_M(u_n))|$
and $|a(x, T_k(u_n) , \nabla T_k(u_n))|$ are bounded in
$L_{\overline{M}}(\Omega)$
then there exist two functions $\varphi$ and $\psi$ in
$L_{\overline{M}}(\Omega)$
such that
\begin{equation}
\begin{gathered}
|a(x, T_M(u_n) , \nabla T_M(u_n))|\to \varphi \quad \mbox{for }
\sigma (L_{\overline{M}}, E_M)\\
|a(x, T_k(u_n) , \nabla T_k(u_n))|\to \psi \quad \mbox{for }
\sigma (L_{\overline{M}}, E_M)\,.
\end{gathered} \label{e3.7}
\end{equation}
This implies
\begin{equation}
\int_{\Omega \backslash \Omega_s}|a(x, T_M(u_n) ,
\nabla T_M(u_n))||\nabla T_{k}(u)|\,dx \to \int_{\Omega \backslash
\Omega_s}
\varphi |\nabla T_{k}(u)|\,dx \label{e3.8}
\end{equation}
and
\begin{equation}
\int_{\Omega \backslash \Omega_s}|a(x, T_k(u_n) , \nabla
T_k(u_n))||\nabla T_{k}(u)|
\,dx \to \int_{\Omega \backslash \Omega_s}\psi
|\nabla T_{k}(u)|\,dx\,.\label{e3.9}
\end{equation}
On the other hand,
$$
\lim_{n\to \infty}\int_{\Omega}f_n T_{2k}(u_n -T_h (u_n) + T_k (u_n) -
T_k (u))\,dx
= \int_{\Omega}f T_{2k}(u -T_h (u))\,dx =\varepsilon_3 (h)
$$
and, for $n$ large enough, one can write.
\begin{align*}
&\int_{\Omega}\phi_n (u_n) \nabla T_{2k}(u_n -T_h (u_n) + T_k (u_n) -
T_k (u))\,dx\\
&= \int_{\Omega}\phi (T_{4k+h}(u_n)) \nabla T_{2k}(u_n -T_h (u_n) + T_k
(u_n)
- T_k (u))\,dx,
\end{align*}
which yields,
\begin{align*}
&\lim_{n\to \infty}\int_{\Omega}\phi_n (u_n) \nabla T_{2k}(u_n -T_h
(u_n)
+ T_k (u_n) - T_k (u))\,dx \\
&= \int_{\Omega}\phi (u) \nabla T_{2k}(u -T_h (u)\,dx = 0.
\end{align*}
The right-most term in \eqref{e3.6} tends to $0$:
Since $a(x, T_k(u_n) , \nabla T_k(u)\chi_s)$ converges strongly to
$a(x, T_k(u) , \nabla T_k(u)\chi_s)$ in $(E_{\overline{M}}(\Omega))^N$,
using Lemma \ref{lem2.3} while $ \nabla T_k(u_n)$ tends weakly to
$\nabla T_k(u)$ by
\eqref{e3.4}.
We conclude then that
\begin{align*}
0&\leq \limsup_{n\to \infty} \int_{\Omega_r}\big[a(x, T_k(u_n) ,
\nabla T_k(u_n))\\
&\quad - a(x, T_k(u_n) , \nabla T_k(u)][\nabla T_k(u_n) -\nabla
T_k(u)\big]\,dx \\
&\leq \int_{\Omega \backslash \Omega_s}\varphi |\nabla T_{k}(u)|\,dx
+ \int_{\Omega \backslash \Omega_s}\psi |\nabla T_{k}(u)|\,dx
+ \int_{\Omega}f T_{2k}(u -T_h (u))\,dx\,.
\end{align*} %\label{e3.10}
Letting $s$ and $h$ approach infinity we get,
$$
\int_{\Omega_r}[a(x, T_k(u_n) , \nabla T_k(u_n)) - a(x, T_k(u_n) ,
\nabla T_k(u)][\nabla T_k(u_n) -\nabla T_k(u)]\,dx \to 0
$$
as $n\to \infty$.
Passing to a subsequence if necessary, we can assume that
$$
[a(x, T_k(u_n) , \nabla T_k(u_n)) - a(x, T_k(u_n) , \nabla
T_k(u)][\nabla T_k(u_n)
-\nabla T_k(u)]\to 0
$$ a.e. in $\Omega_r$.
As in \cite{be}, we deduce that there exists a subsequence still
denoted by
$u_n$ such that
$ \nabla u_n \to \nabla u \quad \mbox{a.e. in } \Omega $.
%\label{e3.11}
\smallskip
\noindent{\bf Step 3 Passage to the limit.}
Let $\varphi \in W_{0}^{1}L_M (\Omega)\cap L^{\infty}(\Omega)$, and set
$M = k + \|\varphi\|_{\infty} $ with $k>0$.
We shall prove that
$$
\liminf_{n\to \infty}{ \int_{\Omega}a(x, u_n , \nabla u_n)\nabla
T_k(u_n
- \varphi)\,dx \geq \int_{\Omega}a(x, u , \nabla u)\nabla T_k(u -
\varphi)\,dx}.
$$
We have: If $|u_n|>M$ then $|u_n - \varphi|>k$ which implies
\begin{align*}
&a(x, u_n, \nabla u_n)\nabla T_k(u_n - \varphi)\\
&=a(x, T_M(u_n) , \nabla T_M (u_n))(\nabla u_n -
\nabla\varphi)\chi_{\{|u_n
- \varphi|\leq k\}}\\
&=a(x, T_M(u_n) , \nabla T_M (u_n))(\nabla T_M(u_n) -
\nabla\varphi)\chi_{\{|u_n
- \varphi|\leq k\}}.
\end{align*}
Let $\Omega_s = \{x\in \Omega: |\nabla \varphi|\leq s\}$ and denote by
$\chi_s$
the characteristic function of $\Omega_s$. Then
\begin{align*}
&\int_{\Omega}a(x, u_n , \nabla u_n)\nabla T_k(u_n - \varphi)\,dx\\
& = \int_{\Omega} a(x, T_M(u_n) , \nabla T_M (u_n))(\nabla T_M(u_n)
- \nabla\varphi)\chi_{\{|u_n - \varphi|\leq k\}}\,dx \\
& = \int_{\Omega} a(x, T_M(u_n) , \nabla T_M (u_n))(\nabla T_M(u_n)
- \nabla\varphi \chi_s)\chi_{\{|u_n - \varphi|\leq k\}}\,dx \\
&\quad - \int_{\Omega}a(x, T_M(u_n) , \nabla T_M (u_n))(\nabla \varphi
- \nabla\varphi \chi_s)\chi_{\{|u_n - \varphi|\leq k\}}\,dx,
\end{align*}
and
\begin{equation}
\begin{aligned}
&\int_\Omega a(x, u_n , \nabla u_n)\nabla T_k(u_n - \varphi)\,dx\\
& \geq -\int_{\Omega\backslash \Omega_s} |a(x, T_M(u_n) ,
\nabla T_M (u_n))||\nabla \varphi |\,dx\\
&\quad + \int_{\Omega} \big[a(x, T_M(u_n) , \nabla T_M (u_n)) - a(x,
T_M(u_n) ,
\nabla \varphi \chi_s)\big]\\
&\quad\times [\nabla T_M(u_n)
- \nabla\varphi \chi_s]\chi_{\{|u_n - \varphi|\leq k\}}\,dx \\
&\quad + \int_{\Omega}a(x, T_M(u_n) , \nabla \varphi \chi_s)[\nabla
T_M(u_n)
- \nabla\varphi \chi_s]\chi_{\{|u_n - \varphi|\leq k\}}\,dx.
\end{aligned} \label{e3.12}
\end{equation}
Similarly to the proof of \eqref{e3.8}, the first term in the right
hand side
of \eqref{e3.12} is greater than a value $\varepsilon_6(s)$,
which implies
\begin{equation}
\begin{aligned}
&\liminf_{n\to \infty} \int_{\Omega}a(x, u_n , \nabla u_n)\nabla
T_k(u_n
- \varphi)\,dx\\
& \geq \lim_{n\to \infty} \int_{\Omega}a(x, T_M(u_n) , \nabla \varphi
\chi_s)
[\nabla T_M(u_n) - \nabla\varphi \chi_s]\chi_{\{|u_n - \varphi|\leq
k\}}\,dx
+ \varepsilon_6(s) \\
&\quad + \int_{\Omega} [a(x, T_M(u) , \nabla T_M (u)) - a(x, T_M(u) ,
\nabla \varphi \chi_s)]\\
&\quad \times [\nabla T_M(u) - \nabla\varphi \chi_s]\chi_{\{|u
- \varphi|\leq k\}}\,dx.
\end{aligned}\label{e3.13}
\end{equation}
From Lemma \ref{lem2.3}, the first term in the right hand side of
\eqref{e3.13} is equal to
$$
\int_{\Omega}a(x, T_M(u) , \nabla \varphi \chi_s)[\nabla T_M(u)
- \nabla\varphi \chi_s]\chi_{\{|u - \varphi|\leq k\}}\,dx +
\varepsilon_6(s),
$$
then
\begin{align*}
&\liminf_{n\to \infty}\int_{\Omega}a(x, u_n , \nabla u_n)\nabla T_k(u_n
- \varphi)\,dx\\
&\geq \int_{\Omega} a(x, T_M(u) , \nabla T_M (u)) [\nabla T_M(u)
- \nabla\varphi \chi_s]\chi_{\{|u - \varphi|\leq k\}}\,dx +
\varepsilon_6(s).
\end{align*}
By letting $s\to + \infty$, we obtain
\begin{align*}
&\liminf_{n\to \infty}\int_{\Omega}a(x, u_n ,\nabla u_n)\nabla T_k(u_n
- \varphi)\,dx\\
&\geq \int_{\Omega} a(x, T_M(u) , \nabla T_M (u)) [\nabla T_M(u)
- \nabla\varphi ]\chi_{\{|u - \varphi|\leq k\}}\,dx \\
& = \int_{\Omega} a(x, u , \nabla u)\nabla T_k(u - \varphi)\,dx.
\end{align*}
Now taking $T_k(u_n - \varphi )$ as test function in \eqref{e3.8} and
passing
to the limit we deduce the desired statement.
\end{proof}
\begin{remark} \label{rmk3.1} \rm
If $M$ and $\overline {M} $ satisfy the $\Delta_2$ condition,
instead of \eqref{e1.8} we can assume the condition:
\begin{equation}
|a(x, s, \xi)|\leq c(x) + k_1{\overline {M}}^{-1}M(k_2|s|)
+ k_3{\overline {M}}^{-1}M(k_4|\xi|)
\label{e1.8'}.
\end{equation}
Then we prove the same result as in Theorems \ref{thm3.1} and
\ref{thm3.2}.
\end{remark}
\begin{remark} \label{rmk3.2} \rm
If w$f$ belongs to $W^{-1}L_{\overline {M}}(\Omega)$ the statements
of
Theorems \ref{thm3.1} and \ref{thm3.2} still hold.
\end{remark}
\subsection*{Example}
Let $2 -\frac{1}{N}1$.
Then it is easy to verify that $M(t)$ satisfies the condition of
Theorems \ref{thm3.1} and \ref{thm3.2}.
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\end{document}