\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 142, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/142\hfil Existence of solutions]
{Existence of solutions for quasilinear elliptic degenerate systems with
$L^1$ data and nonlinearity in the gradient}

\author[A. Mouida, N. Alaa, S. Mesbahi,  W. Bouarifi \hfil EJDE-2013/142\hfilneg]
{Abdelhaq Mouida, Noureddine Alaa, Salim Mesbahi, Walid Bouarifi}  

\address{Abdelhaq Mouida \newline
Laboratory LAMAI \\
Faculty of Science and Technology of Marrakech \\
University Cadi Ayyad \\
B.P. 549, Street Abdelkarim Elkhattabi, Marrakech - 40000, Morocco}
\email{abdhaq.mouida@gmail.com}

\address{Noureddine Alaa \newline
Laboratory LAMAI \\
Faculty of Science and Technology of Marrakech \\
University Cadi Ayyad \\
B.P. 549, Street Abdelkarim Elkhattabi, Marrakech - 40000, Morocco}
\email{n.alaa@uca.ma}

\address{Salim Mesbahi \newline
Department of Mathematics\\
Faculty of Science\\
University Ferhat Abbas, Setif I \\
Pole 2 - Elbez \\
Setif - 19000, Algeria}
\email{salimbra@gmail.com}

\address{Walid Bouarifi \newline
Department of Computer Science \\
National School of Applied Sciences of Safi\\
Cadi Ayyad University\\
Sidi Bouzid, BP 63, Safi - 46000, Morocco}
\email{w.bouarifi@uca.ma}

\thanks{Submitted December 7, 2012. Published June 22, 2013.}
\subjclass[2000]{35J55, 35J60, 35J70}
\keywords{Degenerate elliptic systems; auasilinear; chemotaxis; 
\hfill\break\indent angiogenesis;  weak solutions}

\begin{abstract}
 In this article we show the existence of weak solutions for some
 quasilinear degenerate elliptic systems arising in modeling chemotaxis
 and angiogenesis.
 The nonlinearity we consider has critical growth with respect to the
 gradient and the data are in $L^1$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

Reaction-diffusion systems are important for a wide range of applied areas
such as cell processes, drug release, ecology, spread of diseases,
industrial catalytic processes, transport of contaminants in the
environment, chemistry in interstellar media, to mention a few. Some of
these applications, especially in chemistry and biology, are explained in
books by Murray \cite{26,27} and Baker \cite{10}. While a general
theory of reaction-diffusion systems is detailed in the books of Rothe
\cite{34} and Grzybowski \cite{21}. Various forms of this problems have been
proposed in the literature. Most discussions in the current literature are
for linear or nonlinear systems and different methods for the existence
problem have been used, see Alaa et al \cite{1}--\cite{9},
 Baras \cite{11,12}, Boccardo et al \cite{15}, Boudiba \cite{16} and 
Pierre et al \cite{29}-\cite{32}.
This is a relatively recent subject of mathematical and applied research.
Most of the work that has been done so far is concerned with the exploration
of particular aspects of very specific systems and equations. This is
because these systems are usually very complex and display a wide range of
phenomena remain poorly understood. Consequently, there is no established
program for solving a large class of systems. For example a system of
Chemotaxis, which is a biological phenomenon describing the change of motion
of a population densities or of single particles (such as amoebae, bacteria,
endothelial cells, any cell, animals, etc.) in response (taxis) to an
external chemical stimulus spread in the environment where they reside see
for example \cite{28}. The simple mathematical model which describes such a
phenomenon reads as follows
\begin{equation}
\begin{gathered}
\frac{\partial u}{\partial t}-D_{u}\Delta u+\nabla(\kappa(u) \nabla u
+\chi(v) \nabla v) =0\quad \text{in }\Omega \times (0,T) \\
\frac{\partial v}{\partial t}-D_{v}\Delta v
+\nabla  ( \zeta (u) \nabla u+\eta(v) \nabla v) =0 \quad
\text{in }\Omega \times (0,T) \\
u(0) =u_0,\quad v_0(0) =v_0 
\end{gathered}
\end{equation}
here $u$ and $v$ are the population densities. 
For a simple expansion, we include
\begin{gather*}
a(x)=\frac{\partial \kappa(u) }{\partial u},\quad
b(x)=\frac{\partial \chi(v) }{\partial v},\quad
c(x) =\frac{\partial \zeta(u) }{\partial u},\quad
d(x) =\frac{\partial \eta(v) }{\partial v} \\
f =-(\kappa(u) \Delta u+\chi(v) \Delta) v,\quad
g=-(\zeta(u) \Delta u+\eta(v)\Delta v)\,.
\end{gather*}
Then the system can be written as
\begin{equation}
\begin{gathered}
\frac{\partial u}{\partial t}-D_{u}\Delta u+a(x)|
\nabla u| ^2+b(x) | \nabla v|^2=f \quad\text{in }\Omega \times (0,T) \\
\frac{\partial v}{\partial t}-D_{v}\Delta v+c(x) |
\nabla u| ^2+d(x) | \nabla v| ^2=g\quad\text{in }\Omega \times (0,T) \\
u(0) =u_0,\quad v_0(0) =v_0 \,.
\end{gathered}
\end{equation}

In this work we are interested in the quasilinear elliplic
degenerate problem
\begin{equation}
\begin{gathered}
u-D_1\Delta u+a( x)| \nabla u|^2+b( x) | \nabla v| ^{\alpha }
=f(x) \quad\text{in }\Omega  \\
v-D_2\Delta v+c( x) | \nabla u| ^{\beta}+d( x) | \nabla v| ^2
=g( x) \quad\text{in }\Omega  \\
u=v=0\quad\text{on }\partial \Omega 
\end{gathered}  \label{eq1}
\end{equation}
where $\Omega $ is an open bounded set of $\mathbb{R}^{N}$, $N\geq 1$, 
with smooth boundary $\partial \Omega$, the diffusion
coefficients $D_1$ and $D_2$ are positive constants,
 $a, b, c, d, f, g:\Omega \to [ 0,+\infty ) $ are a non-negative
integrable functions and $1\leq \alpha,\beta \leq 2$.

We are interested in the case where the data are non-regular and where the
growth of the nonlinear terms is arbitrary with respect to the gradient.
To help understanding the situation, let us mention some previous works
concerning the problem when $a,b,c,d\in L^{\infty }(\Omega )$.

$\bullet $ if $f, g$ are regular enough $( f, g\in W^{1,\infty}( \Omega ) ) $ 
and for all $\alpha ,\beta \geq 1$, the
method of sub- and super-solution can be used to prove the existence 
of solutions to \eqref{eq1}. 
For instance $(0,0)$ is a subsolution and a solution,  $w=(w_1,w_2) $, of the
linear problem
\begin{equation}
\begin{gathered}
w_1-D_1\Delta w_2=f( x)  \quad \text{in }\Omega  \\
w_1-D_1\Delta w_2=g( x)  \quad \text{in }\Omega  \\
w_1=w_2=0 \quad \text{on }\partial \Omega,
\end{gathered}
\end{equation}
is a supersolution. Then \eqref{eq1} has a solution 
$(u,v) \in W_0^{1,\infty }( \Omega ) \cap W^{2,p}(\Omega ) $;
 see Lions \cite{23}.

$\bullet $ If $f, g\in L^2( \Omega ) $ and $1\leq \alpha,\beta \leq 2$, 
then $| \nabla u| ^{\alpha }, |\nabla v| ^{\beta }\in L^1( \Omega )$.
 Many authors have studied this problem and showed that \eqref{eq1} 
has a solution $(u,v)\in H_0^1( \Omega ) \times H_0^1( \Omega) $, 
see Bensoussan et al \cite{14}, Boccardo et al \cite{15} and the
references there in.

$\bullet $ If $f, g\in L^1( \Omega ) $ and $1\leq \alpha,\beta <2$, 
Alaa and Mesbahi \cite{1} proved that \eqref{eq1} has a
non negative solution $(u,v)\in W_0^{1,1}( \Omega ) \times W_0^{1,1}( \Omega ) $.

$\bullet $ The case where $f, g\in M_{B}^{+}( \Omega ) $ 
($f, g$ are a finite non negative measures on $\Omega $) 
has treated by Alaa and Pierre \cite{9}. They proved that 
if $1\leq \alpha ,\beta \leq 2$ and the supersolution 
$w=(w_1,w_2)\in H_0^1( \Omega ) \times H_0^1( \Omega ) $, 
then the problem \eqref{eq1} has a non negative solution 
$(u,v)\in $ $H_0^1( \Omega ) \times H_0^1( \Omega ) $.

We are particularly interested in the case of a 
system \eqref{eq1} when $a, b, c, d, f, g$ are not
regular, more precisely, $a, b, c, d, f, g$ are in $L^1(\Omega ) $.

Let us make some specifications on the model problem
\begin{equation}
\begin{gathered}
u-D_1\Delta u+b( r) | \nabla v| ^{\alpha
}=f \quad\text{in }B \\
v-D_2\Delta v+c( r) | \nabla u| ^{\beta }=g\quad\text{in }B \\
u=v=0\quad\text{on }\partial B
\end{gathered}
\end{equation}
where $B$ is the unit ball in $\mathbb{R}^{N}$,
$r=\| x\| $ and 
\begin{equation}
b(r)=c(r)=\begin{cases}
-\ln r & \text{if }N=2 \\
r^{2-N} & \text{if  }N\geq 3\,.
\end{cases} \label{eqq}
\end{equation}

In this case, $b(r), c(r)$ are in $L_{\rm loc}^1( B) $ but not in 
$L^{\infty }( B) $. As a consequence the techniques usually used
to prove existence and based on a priori $L^{\infty }$-estimates on $u$ and 
$\nabla u$ fail. To overcome this difficulty, we will develop a new method
which differ completely of the previous approach.

We have organized this article as follows. In section \ref{section2} 
we give the precise setting of the problem and state the main result. 
In section \ref{section3} we present an approximate problem and we give 
suitable estimates to prove that \eqref{eq1} has a solution in the case
 where the growth of the nonlinearity
with respect to the gradient is arbitrary.

\section{Assumptions and statement of main results}\label{section2}

Let $f, g, a, b, c, d$ are functions that satisfies the
following assumptions
\begin{gather}
f, g\in L^1( \Omega ), \quad f, g\geq 0 \label{eq2}\\
a, b, c, d\in L_{\rm loc}^1( \Omega ), \quad a, b, c, d\geq 0 \label{eq3}
\end{gather}
First, we have to clarify in which sense we want to solved problem \eqref{eq1}.

\begin{definition} \label{def1.2} \rm 
 We say that $( u,v) $ is a weak solution of \eqref{eq1} if
\begin{equation}
\begin{gathered}
u, v\in W_0^{1,1}( \Omega ) \\
a( x) | \nabla u| ^2,\quad b( x)| \nabla v| ^{\alpha },\quad 
c( x) | \nabla u| ^{\beta },\quad 
d( x) | \nabla v| ^2\in L_{\rm loc}^1( \Omega ) \\
u-D_1\Delta u+a( x) | \nabla u| ^2+b(
x) | \nabla v| ^{\alpha }=f( x) \quad \text{in } D'( \Omega ) \\
v-D_2\Delta v+c( x) | \nabla u| ^{\beta}+d( x) | \nabla v| ^2=g( x) \quad
\text{in } D'( \Omega )
\end{gathered}
\end{equation}
\end{definition}


We are interested to proving the existence of weak positive solutions of
the problem \eqref{eq1}.
For this, we define the truncation function $T_k\in C^2$, such that 
\begin{equation}
\begin{gathered}
T_k( r) =r  \quad\text{if } 0\leq r\leq k \\
T_k( r) \leq k+1 \quad\text{if } r\geq k \\
0\leq T_k'( r) \leq 1 \quad\text{if }r\geq 0 \\
T_k'( r) =0 \quad\text{if } r\geq k+1 \\
0\leq -T_k''( r) \leq C( k)\,. 
\end{gathered}
\end{equation}
For example, the function $T_k$ can be defined as
\begin{equation}
\begin{gathered}
T_k( r) =r\quad\text{in }[ 0,k]  \\
T_k( r) =\frac{1}{2}( r-k) ^{4}-( r-k)
^{3}+r \quad\text{in }[ k,k+1]  \\
T_k( r) =\frac{1}{2}( k+1)  \quad\text{for } r>k+1\,.
\end{gathered}
\end{equation}

Then we define the the space 
\begin{equation*}
\tau ^{1,2}( \Omega ) =\big\{ w:\Omega \to \mathbb{R}
\text{ measurable, such that $T_k( w) \in H^1( \Omega)$  for all }k>0\big\}
\end{equation*}
This enables us to state the main result of this paper.

\begin{theorem}\label{thm2.1}
Assume that  \eqref{eq2} and \eqref{eq3} hold,  and
$1\leq \alpha ,\beta <2$.
If there exists a function $\theta \in \tau ^{1,2}( \Omega ) $
and a sequence $\theta _n\in L^{\infty }( \Omega ) $ such that
\begin{equation}
\begin{gathered}
0\leq a, b, c, d\leq \theta \quad \text{in }\Omega \\
\theta _n\to \theta \quad \text{a.e. }\Omega \\
\nabla T_k( \theta _n) \to \nabla T_k( \theta) \quad
 \text{strongly in }L^2( \Omega ) \\
\lim_{k\to \infty }\sup_n \Big( \frac{1}{k}
\int_{\Omega }| \nabla T_k(\theta _n)| ^2\Big) =0
\end{gathered} \label{eq4}
\end{equation}
Then the problem \eqref{eq1} has a non negative weak solution.
\end{theorem}

\begin{remark} \rm
(i) If $a,b,c,d\in L^{\infty }( \Omega ) $,
then  \eqref{eq4} is satisfied. Indeed, $\theta $ can
take the value of any non negative constant $C$, such that
\begin{equation}
C\geq \max \big\{ \| a\| _{L^{\infty }},\|b\| _{L^{\infty }},
\| c\| _{L^{\infty}},\| d\| _{L^{\infty }}\big\}
\end{equation}

(ii)  Hypothesis \eqref{eq4}  holds for the
functions $\xi =b$ or $c$ given in \eqref{eqq}.
 Indeed $-\Delta \xi =\lambda $ is in this case the measure of Dirac
 which is a finite non negative measure on $\Omega $. 
By consequent, we take $\theta =\xi $ and $\theta _n$ solution of
\begin{equation}
\begin{gathered}
-\Delta \theta _n=\lambda _n\quad\text{in }\Omega \\
\theta _n=0\quad\text{on }\partial \Omega ,
\end{gathered}
\end{equation}
where $\lambda _n\in C_0^{\infty }( \Omega )$,
$ \lambda _n\to \lambda $ in $L^1( \Omega ) $ and
$\lambda_n\leq \lambda $. Then, we can applied Theorem \ref{thm2.1}
 and conclude the existence of the non negative weak solution for our 
model problem \eqref{eq2}.
\end{remark}

\section{Proof of theorem \ref{thm2.1}}\label{section3}

\subsection{An approximation scheme}

In this paragraph, we define an approximated system of \eqref{eq1}.
For this, we truncate the functions $a,b,c,d,f,g$ by
introducing the sequence $a_n, b_n, c_n, d_n,f_n, g_n$
defined as follows
\begin{equation*}
a_n=\min \{ a,\theta _n\} ,\quad
b_n=\min \{ b,\theta_n\} ,\quad
c_n=\min \{ c,\theta _n\} ,\quad
d_n=\min \{ d,\theta _n\}
\end{equation*}
and
\begin{equation}
\begin{gathered}
\text{$f_n\in C_0^{\infty }( \Omega )$, $f_n\to f$  in $L^1( \Omega )$,
  $f_n\leq f$} \\
\text{$g_n\in C_0^{\infty }( \Omega )$, $g_n\to g$ in $L^1( \Omega ) $,
$g_n\leq g$}
\end{gathered}\label{eq7}
\end{equation}
Then the approximate problem is
\begin{equation}
\begin{gathered}
u_n, v_n\in W_0^{1,\infty }( \Omega )  \\
u_n-D_1\Delta u_n+a_n( x) | \nabla u_n| ^2+b_n( x) | \nabla
v_n| ^{\alpha }=f_n( x) \quad\text{in }D'(\Omega )  \\
v_n-D_2\Delta v_n+c_n( x) | \nabla u_n| ^{\beta }+d_n( x) | \nabla
v_n| ^2=g_n( x) \quad\text{in }D'( \Omega)\,.
\end{gathered}\label{eq5}
\end{equation}
One can see that $a_n, b_n, c_n, d_n$  are in 
$L^{\infty }( \Omega ) $. On the other hand, 
$( 0,0) $ is a subsolution of \eqref{eq5} and $( U_n,V_n) $ a
solution of the linear problem
\begin{equation}
\begin{gathered}
U_n-D_1\Delta U_n=f_n\quad\text{in }\Omega  \\
V_n-D_2\Delta V_n=g_n\quad\text{in }\Omega  \\
U_n, V_n\in W_0^{1,\infty }( \Omega ) 
\end{gathered}
\end{equation}
is a supersolution, then by the classical results in Amann and
Grandall \cite{13} and Lions \cite{23, 24}, there exists 
$(u_n,v_n) $ solution of \eqref{eq5} such that
\begin{gather*}
0\leq u_n\leq U_n \quad\text{for all }n \\
0\leq v_n\leq V_n \quad\text{for all }n
\end{gather*}

\subsection{A priori estimates}

To prove theorem \ref{thm2.1}, we propose to send $n$ to infinity 
in \eqref{eq5}. For this we will need some estimates passing to the
limit.

\begin{lemma}\label{lemma3.1}
Let $u_n, v_n, a_n, b_n, c_n, d_n$ be
sequences defined as above. Then
(i)
\begin{gather*}
\int_{\Omega }| \nabla T_k( u_n) | ^2
\leq k\| f\| _{L^1( \Omega ) } \\
\int_{\Omega }| \nabla T_k( v_n) | ^2
\leq k\| g\| _{L^1( \Omega ) }
\end{gather*}
and  (ii)
\begin{gather*}
\int_{\Omega }b_n.| \nabla T_k( v_n) |
^{\alpha } \leq k\| f\| _{L^1( \Omega ) } \\
\int_{\Omega }c_n.| \nabla T_k( u_n) |
^{\beta } \leq k\| g\| _{L^1( \Omega ) }
\end{gather*}
\end{lemma}

\begin{proof} (i)
By multiplying the first equation of \eqref{eq5} by $T_k( u_n) $ 
and the second equation by $T_k( v_n) $ and integrating over $\Omega $, we have
\begin{align*}
&\int_{\Omega }| T_k( u_n) | ^2
+D_1\int_{\Omega }| \nabla T_k( u_n)| ^2\\
&+\int_{\Omega }a_nT_k( u_n) |\nabla T_k( u_n) | ^2
+\int_{\Omega }b_nT_k( u_n) | \nabla T_k(
v_n) | ^{\alpha } \leq \int_{\Omega }f_nT_k(u_n)
\end{align*}
and
\begin{align*}
&\int_{\Omega }| T_k( v_n) |^2+D_2\int_{\Omega }| \nabla T_k( v_n)| ^2\\
&+\int_{\Omega }c_nT_k( v_n) |\nabla T_k( u_n) | ^{\beta }
+\int_{\Omega }d_nT_k( v_n) | \nabla T_k(v_n) | ^2 
\leq \int_{\Omega }g_nT_k( v_n)
\end{align*}
Thanks to the positivity of $a_n,b_n, c_n, d_n$, the
assumptions on $f_n\ $and $g_n$, the definition of the function $T_k$,
we deduce the result.

(ii) Integrating the first equation of  \eqref{eq5}
over $\Omega $, we obtain
\begin{equation}
\int_{\Omega }u_n-D_1\int_{\Omega }\Delta u_n+\int_{\Omega
}a_n( x) | \nabla u_n| ^2+\int_{\Omega
}b_n( x) | \nabla v_n| ^{\alpha
}=\int_{\Omega }f_n( x)\label{eq6}
\end{equation}
On the other hand, it is well know that for every function $y$ in 
$W_0^{1,1}( \Omega ) $ such that
\begin{gather*}
-\Delta y=H,\ H\in L^1( \Omega ) \\
y\geq 0
\end{gather*}
there exists a sequence $y_n$ in $C^2( \Omega ) \cap
C_0( \overline{\Omega }) $ which satisfies
\begin{gather*}
y_n \to y\quad \text{strongly in }W_0^{1,1}( \Omega ) \\
\Delta y_n \to \Delta y\quad \text{strongly in }L^1( \Omega)
\end{gather*}
The regularity of $y_n$ allows us to write
\begin{equation*}
\int_{\Omega }\Delta y_n=\int_{\partial \Omega }\frac{\partial y_n}{\partial
\upsilon }d\sigma,
\end{equation*}
but $y_n\geq 0$ on $\Omega $ and $y_n=0$ in $\partial \Omega $.
Then $\frac{\partial y_n}{\partial \upsilon }\leq 0$. 
We deduce by passing to the limit that $\int_{\Omega }\Delta y\leq 0$. 
Therefore
\begin{equation*}
\int_{\Omega }\Delta u_n\leq 0
\end{equation*}
The relation \eqref{eq6} yields
\begin{equation*}
\int_{\Omega }u_n+\int_{\Omega }a_n( x) | \nabla
u_n| ^2+\int_{\Omega }b_n( x) | \nabla
v_n| ^{\alpha }\leq \int_{\Omega }f_n( x)\,.
\end{equation*}
By \eqref{eq7}; we conclude that
\begin{equation*}
\int_{\Omega }u_n+\int_{\Omega }a_n( x) | \nabla
u_n| ^2+\int_{\Omega }b_n( x) | \nabla
v_n| ^{\alpha }\leq \| f\| _{L^1( \Omega
) }\,.
\end{equation*}
In the same way, if we integrate the second equation of \eqref{eq5}
over $\Omega $, we obtain
\begin{equation*}
\int_{\Omega }v_n+\int_{\Omega }c_n( x) | \nabla
u_n| ^{\beta }+\int_{\Omega }d_n( x) |
\nabla v_n| ^2 \leq \| g\| _{L^1( \Omega
) },
\end{equation*}
hence the result follows.
\end{proof}

\begin{remark} \rm
(1) Using the assertion (ii) of lemma \ref{lemma3.1}, and the compactness of the
operator 
\begin{gather*}
L^1( \Omega ) \to W_0^{1,q}( \Omega ) \\
G \mapsto \vartheta
\end{gather*}
where $1\leq q<\frac{N}{N-1}$, and 
$\vartheta $ is the solution of the problem
\begin{gather*}
\vartheta \in W_0^{1,q}( \Omega ) \\
\alpha \vartheta -\Delta \vartheta =G\quad \text{in }D'(\Omega)
\end{gather*}
we conclude the existence of $u$, up to a subsequence, still denoted by 
$u_n $ for simplicity, such that
\begin{gather*}
u_n\to u\quad \text{strongly in }W_0^{1,q}(\Omega) ,\quad
1\leq q<\frac{N}{N-1},\\
( u_n,\nabla u_n) \to ( u,\nabla u) \quad\text{a.e. in }\Omega
\end{gather*}
see Brezis \cite{17}

(2) Assertion (i) implies that
\begin{equation*}
( T_k( u_n) ,T_k( v_n) )
\to ( T_k( u) ,T_k( v) )\quad \text{ weakly in }
H_0^1( \Omega ) \times H_0^1( \Omega)
\end{equation*}
\end{remark}

\begin{lemma}\label{lemma3.3}
Let $( u_n,v_n) $ be a solution of \eqref{eq5}, then
\begin{equation*}
\lim_{h\to +\infty }  \sup_{n} \Big( \frac{1}{h}\int_{\Omega }|
\nabla T_{h}( u_n) | ^2dx\Big)
=\lim_{h\to +\infty }  \sup_{n} \Big( \frac{1}{h}\int_{\Omega }| 
\nabla T_{h}(v_n) | ^2dx\Big)=0
\end{equation*}
\end{lemma}

\begin{proof} 
We first remark that $u_n$ satisfies
\begin{equation*}
-\Delta u_n\leq f_n\quad \text{in }D'( \Omega )
\end{equation*}
If we multiply this inequality by $T_{h}( u_n) $ and
integrate on $\Omega $, we obtain for every $0<M<h$,
\begin{align*}
\int_{\Omega }| \nabla T_{h}( u_n) | ^2
&\leq \int_{\Omega \cap \{ u_n\leq M\} }fT_{h}(
u_n) +\int_{\Omega \cap \{ u_n>M\} }fT_{h}(
u_n) \\
&\leq M\int_{\Omega }f+h\int_{\Omega }f\chi _{\{ u_n>M\} }
\end{align*}
hence
\begin{gather*}
\frac{1}{h}\int_{\Omega }| \nabla T_{h}( u_n)| ^2
\leq \frac{M}{h}\int_{\Omega }f+\int_{\Omega }f\chi_{\{ u_n>M\} }
\\
| \{ u_n>M\} | =\int_{\{
u_n>M\} }dx\leq \frac{1}{M}\| u_n\| _{L^1}\leq
\frac{C}{M}
\end{gather*}
Then $\lim_{M\to +\infty } \big( \sup_{n}| \{ u_n>M\} | \big) =0$

On other hand, since $f\in L^1(\Omega )$, we have
for each $\varepsilon >0$  there exists $\delta$ such that for
for all $E\subset \Omega$,
\[
| E| <\delta \int_{E}| f| \leq \frac{\varepsilon }{2}.
\]
Taking into account the above limit, we obtain
that for each $\varepsilon >0$, there exists $M_{\varepsilon}$
such that for all $M\geq M_{\varepsilon }$,
\[
\sup_{n}\Big( \int_{\Omega }f\chi _{_{
[ u_n>M] }}\Big) \leq \frac{\varepsilon }{2}
\]
Taking $M=M_{\varepsilon }$ and letting $h$ tend to infinity, we obtain
\[
\lim_{h\to \infty } \sup_{n} \Big( \frac{1}{h}
\int_{\Omega }| \nabla T_{h}(u_n)| ^2\Big) =0\,.
\]
\end{proof}

\begin{lemma}\label{lemma3.4}
Let $\eta _n$ be sequence such that $\eta _n\to \eta$,
a.e. in $\Omega $ and $\int_{\Omega }| \eta _n|^2\leq C$ then 
$\eta _n\to \eta $  in $L^{\alpha }(\Omega ) $ for all $1\leq \alpha <2$.
\end{lemma}

\begin{proof} 
We show that $\eta _n$ is equi-integrable in $L^{\alpha}( \Omega )$.
Let $E$ be a measurable subset of $\Omega $; we
have
\begin{equation*}
\int_{E}| \eta _n| ^{\alpha }
\leq |E| ^{(2-\alpha)/2} \Big( \int_{E}| \eta_n| ^2\Big) ^{\alpha/2}
\leq C | E| ^{(2-\alpha)/2}
\end{equation*}

Since $1\leq \alpha <2$ then $0<2-\alpha \leq 1$.
We choose $|E| =( \frac{\varepsilon }{C}) ^{2/(2-\alpha)}$,
we obtain $\int_{E}| \eta _n| ^{\alpha }\leq \varepsilon $.
\end{proof}

\subsection{Convergence}

The aim of this paragraph is to prove that $(u,v)$ (obtained in
the previous section) is in fact a solution of problem \eqref{eq1}.
According to definition \ref{def1.2}, we have  to show only that
\begin{gather*}
u-D_1\Delta u+a( x) | \nabla u|^2+b( x) | \nabla v| ^{\alpha }
=f(x) \quad\text{in }D'( \Omega ) \\
v-D_2\Delta v+c( x) | \nabla u| ^{\beta}+d( x) | \nabla v| ^2=g( x) 
\quad\text{in }D'( \Omega ) 
\end{gather*}
By lemma \ref{lemma3.1}, we know that 
$a_n( x) |\nabla u_n| ^2$, 
$d_n( x) |\nabla v_n| ^2$, 
$b_n( x) |\nabla v_n| ^{\alpha }$,
$c_n( x) | \nabla u_n| ^{\beta }$ are uniformly
bounded in $L^1( \Omega) $. Moreover
\begin{equation*}
a_n( x) | \nabla u_n| ^2\geq 0,\quad
d_n( x) | \nabla u_n| ^2\geq 0,\quad
b_n( x) | \nabla u_n| ^{\alpha }\geq 0,\quad
c_n( x) | \nabla u_n| ^{\beta }\geq 0
\end{equation*}
and for almost every $x$ in $\Omega $, we have
\begin{gather*}
a_n( x) | \nabla u_n(x)| ^2 \to a( x) | \nabla u(x)| ^2 \\
d_n( x) | \nabla v_n(x)| ^2 \to d( x) | \nabla v(x)| ^2 \\
b_n( x) | \nabla v_n(x)|^{\alpha } \to b( x) | \nabla v(x)| ^{\alpha } \\
c_n( x) | \nabla u_n(x)| ^{\beta } \to c( x) | \nabla u(x)| ^{\beta }
\end{gather*}
Then there exists $\mu _1, \mu _2$ non negative measures, see
Schwartz \cite{35}, such that
\begin{gather*}
\begin{aligned}
&\lim_{n\to +\infty}( u_n-D_1\Delta u_n+a_n( x) | \nabla u_n|
^2+b_n( x) | \nabla v_n| ^{\alpha}) \\
&= u-D_1\Delta u+a( x) |\nabla u| ^2+b( x) | \nabla v|
^{\alpha }+\mu _1\quad\text{in }D'( \Omega ) 
\end{aligned} \\
\begin{aligned}
&\lim_{n\to +\infty}( v_n-D_2\Delta v_n+c_n( x) | \nabla u_n|
^{\beta }+d_n( x) | \nabla v_n| ^2) \\
&= v-D_2\Delta v+c( x) | \nabla u| ^{\beta }+d( x) | \nabla
v| ^2+\mu _2\quad\text{in }D'( \Omega ) 
\end{aligned}
\end{gather*}
Consequently,
\begin{gather*}
u-D_1\Delta u+a( x) | \nabla u|
^2+b( x) | \nabla v| ^{\alpha }\leq f\quad\text{in }
D'( \Omega )  \\
v-D_2\Delta v+c( x) | \nabla u| ^{\beta
}+d( x) | \nabla v| ^2\leq g\quad\text{in } D'( \Omega ) 
\end{gather*}
Therefore, to conclude the proof of Theorem \ref{thm2.1}, we must establish the
opposite inequality. For this, Let $H$ be a function in 
$C^1(\mathbb{R})$, such that
\begin{gather*}
0\leq H(s)\leq 1 \\
H(s)=\begin{cases}
0 &\text{if }|s| \geq 1\\
1&\text{if }|s| \leq \frac{1}{2}
\end{cases}
\end{gather*}
To this end, we introduce the  test functions
\begin{gather*}
\Phi _1 =\psi _1\exp [ -\frac{\theta_n}{D_1}u_n] H(
\frac{\theta_n}{k}) H( \frac{u_n}{k}), \\
\Phi _2 =\psi _2\exp [ -\frac{\theta_n}{D_2}v_n] H(
\frac{\theta_n}{k}) H( \frac{v_n}{k})
\end{gather*}
where $H$ denotes the function defined above and 
$\psi _1$, $\psi _2\leq 0$, $\psi _1$, 
$\psi _2\in H_0^1( \Omega ) \cap L^{\infty}( \Omega ) $. 
We multiply the first equation in \eqref{eq5} by $\Phi _1$ and
 we integrate on $\Omega $, we obtain
\begin{equation*}
\int_{\Omega }f_n\Phi _1=\underset{1\leq j\leq 7}{\sum }I_{j},
\end{equation*}
where
\begin{gather*}
I_1=\int_{\Omega }u_n\Phi _1, 
\\
I_2=D_1\int_{\Omega }\nabla u_n\nabla \psi _1
\exp [ -\frac{ \theta_n}{D_1}u_n] H( \frac{\theta_n}{k}) H(\frac{u_n}{k})
\\
I_{3}=-\int_{\Omega }u_n\nabla u_n\nabla \theta_n\Phi _1
\\
I_4=\frac{D_1}{k}\int_{\Omega }\nabla u_n\nabla \theta_n\psi _1
\exp [ -\frac{\theta_n}{D_1}u_n] H'( \frac{\theta_n}{k}) H( \frac{u_n}{k})
\\
I_5=\int_{\Omega }(a_n-\theta_n)| \nabla u_n|^2 \Phi _1
\\
I_6=\frac{D_1}{k}\int_{\Omega }| \nabla u_n|
^2 \psi _1 \exp [ -\frac{\theta_n}{D_1}u_n]  H( \frac{
\theta_n}{k})  H'( \frac{u_n}{k})
\\
I_7=\int_{\Omega }b_n.| \nabla v_n| ^{\alpha } \Phi_1
\end{gather*}

Next we study each term. For the first term, we have
\begin{align*}
\lim_{n\to +\infty}I_1 
&=\lim_{n\to +\infty}\int_{\Omega }T_k( u_n)  \psi _1 \exp [ -\frac{
\theta_n}{D_1}u_n]  H( \frac{\theta_n}{k})  H(\frac{u_n}{k})  \\
&=\int_{\Omega }u \psi _1 \exp [ -\frac{\theta}{D_1}u]  H(
\frac{\theta}{k})  H( \frac{u}{k})
\end{align*}
since
\begin{equation*}
\psi _1 \exp [ -\frac{\theta_n}{D_1}u_n]  H( \frac{
\theta_n}{k})  H( \frac{u_n}{k})
\end{equation*}
converges strongly in $L^2( \Omega ) $ to
\begin{equation*}
\psi _1\exp [ -\frac{\theta}{D_1}u] H( \frac{\theta}{k})
H( \frac{u}{k}) \quad \text{in }L^2( \Omega )
\end{equation*}
and $\nabla T_k( u_n) $ converges weakly to $\nabla
T_k( u) $ in $L^2( \Omega ) $, (see 
\cite[lemma 1.3, p 12]{24}).

Concerning the second term, we get
\begin{align*}
\lim_{n\to +\infty}I_2 
&=\lim_{n\to +\infty}D_1\int_{\Omega }\nabla T_k( u_n)  \nabla \psi
_1 \exp [ -\frac{\theta_n}{D_1}u_n]  H( \frac{\theta_n}{k}) 
  H( \frac{u_n}{k})  \\
&=D_1\int_{\Omega }\nabla u \nabla \psi _1 \exp [ -\frac{\theta}{D_1}u
]  H( \frac{\theta}{k})  H( \frac{u}{k})
\end{align*}
since
\begin{equation*}
\nabla \psi _1\exp [ -\frac{\theta_n}{D_1}u_n] H( \frac{
\theta_n}{k}) H( \frac{u_n}{k})
\end{equation*}
converges strongly in $L^2( \Omega ) $ to
\begin{equation*}
\nabla \psi _1\exp [ -\frac{\theta}{D_1}u] H( \frac{\theta}{k}
) H( \frac{u}{k})\,.
\end{equation*}

For $I_{3}$, we first remark that
\begin{align*}
\lim_{n\to +\infty}I_{3} 
&=-\lim_{n\to +\infty}\int_{\Omega }T_k(u_n) \nabla T_k(u_n) \nabla
T_k(\theta_n) \psi _1 \exp [ -\frac{\theta_n}{D_1}u_n]  H(
\frac{\theta_n}{k})  H( \frac{u_n}{k})  \\
&=-\int_{\Omega }T_k(u) \nabla
T_k(u) \nabla T_k(\theta) \psi _1 \exp [ -\frac{\theta}{D_1}u]
 H( \frac{\theta}{k})  H( \frac{u}{k})
\end{align*}
since
\begin{gather*}
T_k(u_n)\to T_k(u)\quad\text{weakly in } H_0^1( \Omega) \\
T_k(\theta_n)\to T_k(\theta) \quad\text{strongly in } H_0^1( \Omega) 
\end{gather*}

To study $I_4$ and $I_6$ we use Lemma \ref{lemma3.3}. 
For $I_4$, we have
\begin{align*}
I_4 &\leq D_1\Big[ \frac{1}{k}\int_{\Omega }| \nabla
T_k( u_n) | ^2 \psi _1 \exp [ -\frac{\theta_n
}{D_1}u_n]  H'( \frac{\theta_n}{k})
 H( \frac{u_n}{k}) \Big] ^{1/2}  \\
&\quad \times [ \frac{1}{k}\int_{\Omega }| \nabla T_k(
\theta_n) | ^2 \psi _1 \exp [ -\frac{\theta_n}{D_1}u_n
]  H'( \frac{\theta_n}{k})  H( \frac{u_n}{k}) ] ^{1/2}
\\
&\leq D_1[ \| \psi _1\| _{L^{\infty }( \Omega
) } \frac{1}{k}\int_{\Omega }| \nabla T_k(
u_n) | ^2] ^{1/2} [ \| \psi
_1\| _{L^{\infty }( \Omega ) } \frac{1}{k}\int_{\Omega
}| \nabla T_k( \theta_n) | ^2 ] ^{12}
\end{align*}
since $\exp [ -\frac{\theta_n}{D_1}u_n] \leq 1$, thus
\begin{equation*}
I_4\leq D_1[ \| \psi _1\| _{L^{\infty }}\delta
_k] ^{1/2}[ \| \psi _1\| _{L^{\infty
}}\rho _k] ^{1/2}
\end{equation*}
Where
\begin{equation*}
\delta _k=\underset{n}{\sup }( \frac{1}{k}\int_{\Omega }|
\nabla T_k(u_n)| ^2) \text{ \ and \ }\rho _k=
\underset{n}{\sup }( \frac{1}{k}\int_{\Omega }| \nabla
T_k(\theta_n)| ^2)
\end{equation*}
By Lemma \ref{lemma3.3}, we have
\begin{equation*}
\lim_{k\to \infty} \delta_k =0 , \quad \lim_{k\to \infty} \rho_k =0
\end{equation*}
Then
\begin{equation*}
\lim_{k\to \infty}\underset{n}{\sup }( I_4)=0
\end{equation*}
Similarly, for $I_6$, we have
\begin{equation*}
I_6\leq D_1\| \psi _1\| _{L^{\infty }}\delta _k
\end{equation*}
Then
\begin{equation*}
\lim_{k\to \infty}\underset{n}{\sup }( I_6)=0
\end{equation*}
Now we investigate the remaining term $I_5$. 
Since $a_n\leq \theta_n$ and $\psi_1\leq 0$, we have
\begin{equation*}
( a_n-\theta _n) | \nabla u_n| ^2\exp
[ -\frac{\theta _n}{D_1}u] H( \frac{\theta _n}{k}
) H( \frac{u_n}{k}) \geq 0\text{ \ \ \ \ in }\Omega
\end{equation*}
Therefore, by Fatou's lemma, we obtain
\begin{equation*}
\lim_{n\to +\infty} I_5\geq \int_{\Omega }( a-\theta ) |
\nabla u| ^2\exp [ -\frac{\theta }{D_1}u] H(
\frac{\theta }{k}) H( \frac{u}{k})
\end{equation*}

For $I_7$, we obtain
\begin{equation*}
\lim_{n\to +\infty}I_7=\lim_{n\to +\infty}\int_{\Omega }T_k( b_n)  |
\nabla T_k( v_n) | ^{\alpha } \psi _1
\exp [ -\frac{\theta_n}{D_1}u_n]  H(
\frac{\theta_n}{k})  H( \frac{u_n}{k})
\end{equation*}
By a direct application of Lemma \ref{lemma3.3}, we have  
$| \nabla T_k( v_n) | ^{\alpha }\to | \nabla
T_k( v) | ^{\alpha }$strongly in $L^1(\Omega )$,
then
\begin{equation*}
\lim_{n\to +\infty}I_7=\int_{\Omega }b| \nabla
v| ^{\alpha }\psi _1\exp [ -\frac{\theta}{D_1}u]
H( \frac{\theta}{k}) H( \frac{u}{k})
\end{equation*}
We have shown that
\begin{align*}
&\omega ( \frac{1}{k}) +\int_{\Omega }u \psi _1 \exp [ -
\frac{\theta}{D_1}u]  H( \frac{\theta}{k})  H( \frac{u}{k})
+D_1\int_{\Omega }\nabla u \nabla \psi _1 \exp [ -\frac{\theta}{D_1}u
]  H( \frac{\theta}{k})  H( \frac{u}{k})
\\
&-\int_{\Omega }u \nabla u \nabla \theta \psi _1 \exp [ -\frac{\theta}{D_1}u
]  H( \frac{\theta}{k})  H( \frac{u}{k})
+\int_{\Omega }( a-\theta ) | \nabla
u| ^2\exp [ -\frac{\theta }{D_1}u] H( \frac{
\theta }{k}) H( \frac{u}{k})
\\
&+\int_{\Omega }b | \nabla v| ^{\alpha } \psi _1 \exp
[ -\frac{\theta}{D_1}u]  H( \frac{\theta}{k})  H( \frac{u}{k})
\\
&\leq \int_{\Omega }f \psi _1 \exp [ -\frac{\theta}{D_1}u]  H(
\frac{\theta}{k})  H( \frac{u}{k})
\end{align*}
where $\omega (\varepsilon)$ denotes a quantity that tends to 
$0$ when $\varepsilon $ tends to $0 $. Now we choose
\[
\psi _1
=-\varphi _1 \exp [ \frac{\theta}{D_1}u]  H( \frac{
\theta}{k})  H( \frac{u}{k})
\]
where $\varphi _1\geq 0$, $\varphi _1\in D( \Omega ) $ and we
replace $\psi _1$ by this value in the previous inequality to obtain
\begin{align*}
&w( \frac{1}{k}) -\int_{\Omega }u\varphi
_1H^2( \frac{\theta }{k}) H^2( \frac{u}{k})
-D_1\int_{\Omega }\nabla u\nabla \varphi _1H^2( \frac{
\theta }{k}) H^2( \frac{u}{k})  \\
&-\int_{\Omega }\varphi _1u\nabla u\nabla \theta H^2(
\frac{\theta }{k}) H^2( \frac{u}{k})
-\int_{\Omega }\varphi _1| \nabla u| ^2\theta
H^2( \frac{\theta }{k}) H^2( \frac{u}{k})  \\
&-\frac{D_1}{k}\int_{\Omega }\varphi _1\nabla u\nabla \theta
H'( \frac{\theta }{k}) H( \frac{\theta }{k}
) H^2( \frac{u}{k})  
-\frac{D_1}{k}\int_{\Omega }\varphi _1| \nabla
u| ^2H^2( \frac{\theta }{k}) H( \frac{u}{k}
) H'( \frac{u}{k})  \\
&+\int_{\Omega }u\nabla u\nabla \theta \varphi _1H^2(
\frac{\theta }{k}) H^2( \frac{u}{k})
-\int_{\Omega }b| \nabla v| ^{\alpha }\varphi
_1H^2( \frac{\theta }{k}) H^2( \frac{u}{k})  \\
&-\int_{\Omega }a| \nabla u| ^2\varphi
_1H^2( \frac{\theta }{k}) H^2( \frac{u}{k})
+\int_{\Omega }\theta \varphi _1| \nabla u|
^2H^2( \frac{\theta }{k}) H^2( \frac{u}{k})  \\
&\leq - \int_{\Omega }f\varphi _1H^2( \frac{\theta }{k}) H^2( \frac{u}{k})
\end{align*}
By developing calculations and remarking that the sixth and seventh
terms are equivalent to $\omega ( \frac{1}{k}) $, we can write
\begin{align*}
&-\int_{\Omega }u \varphi _1 H^2( \frac{\theta}{k})  H^2(
\frac{u}{k}) -D_1\int_{\Omega }\nabla u \nabla \varphi
_1 H^2( \frac{\theta}{k})  H^2( \frac{u}{k})
\\
&-\int_{\Omega }[ a | \nabla u| ^2+b |
\nabla v| ^{\alpha }]  \varphi _1 H^2( \frac{\theta}{k}
)  H^2( \frac{u}{k}) +\omega ( \frac{1}{k})
\\
&\leq -\int_{\Omega }f \varphi _1 H^2( \frac{\theta}{k})
 H^2( \frac{u}{k})
\end{align*}
Finally passing to the limit as $k$ tends to infinity, we use the fact
that
\begin{equation*}
\lim_{k\to \infty}H( \frac{\theta}{k}) =1,\quad 
\lim_{k\to \infty}H( \frac{u}{k}) =1
\end{equation*}
to conclude that for every $\varphi _1\geq 0,\ \varphi _1\in D(
\Omega ) $,
\begin{equation*}
\int_{\Omega }[ u-D_1\Delta u+a( x) | \nabla
u| ^2+b( x) | \nabla v| ^{\alpha }
]  \varphi _1 \geq \int_{\Omega }f \varphi _1
\end{equation*}

In the same way, we multiply the second equation in \eqref{eq5} by 
$\Phi _2$ and we integrate on $\Omega $. By studying separately each term
as in the previous case still using Lemmas \ref{lemma3.1}, \ref{lemma3.3} 
and \ref{lemma3.4}, we choose
\begin{equation*}
\psi _2=-\varphi _2\exp [ \frac{\theta}{D_2}u] H( \frac{
\theta}{k}) H( \frac{v}{k})
\end{equation*}
where $\varphi _2\geq 0$, $\varphi _2\in D( \Omega ) $ and we
replace $\psi _2$ by this value in the inequality obtained to conclude
that for every $\varphi _2\geq 0$, $\varphi _2\in D( \Omega ) $
that
\begin{equation*}
\int_{\Omega }[ v-D_2\Delta v+c( x) | \nabla
u| ^{\beta }+d( x) | \nabla v| ^2
] \varphi _2 \geq \int_{\Omega }g\varphi _2
\end{equation*}
This completes the proof of theorem \ref{thm2.1}.

\subsection*{Acknowledgments}
We are grateful to the anonymous referee for the 
corrections and useful suggestions that have improved this
article.


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\end{thebibliography}

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