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\begin{document}

\title[Positive solutions of singular elliptic systems]{Positive solutions of singular elliptic systems with multiple
parameters and Caffarelli-Kohn-Nirenberg exponents}

\centerline{}
%--------------------------------------------------------------------------------------------------------------
\author[G.A. Afrouzi]{G.A. Afrouzi}
\address{G.A. Afrouzi,
Department of Mathematics, Faculty of Mathematical Sciences, University of
Mazandaran, Babolsar, Iran}
\email{\href{mailto: G.A. Afrouzi <afrouzi@umz.ac.ir>}{afrouzi@umz.ac.ir}}
%--------------------------------------------------------------------------------------------------------------
%--------------------------------------------------------------------------------------------------------------
\author[V.D. R\u adulescu]{Vicen\c tiu D. R\u adulescu $^\ast$}
\address{Vicen\c tiu D. R\u adulescu,
Institute of Mathematics ``Simion Stoilow" of the Romanian
Academy, P.O. Box 1-764, 014700 Bucharest, Romania \& Department
of Mathematics, University of Craiova,  Street A.I. Cuza No. 13,
200585 Craiova, Romania} \email{\href{mailto: V.D. R\u adulescu
<vicentiu.radulescu@imar.ro>}{vicentiu.radulescu@imar.ro}}
%--------------------------------------------------------------------
\author[S. Shakeri]{S. Shakeri}
\address{S. Shakeri,
Department of Mathematics, Faculty of Mathematical Sciences,
University of Mazandaran, Babolsar, Iran} \email{\href{mailto: S.
Shakeri <s.shakeri@umz.ac.ir>}{s.shakeri@umz.ac.ir}}
%-----------------------------------------------------------------------------------------------------------
\keywords{Caffarelli-Kohn-Nirenberg exponents,  elliptic system, multiple parameters;
sub-supersolution method. \\
 {\it 2000 Mathematics Subject Classifications}:
35J55; 35J65.\\$^\ast$ Corresponding author.}

%\date{\bf Aug. 7, 2013}

\begin{abstract}
This paper is concerned with the existence of positive solutions for a class of quasilinear singular elliptic systems
with Dirichlet boundary condition. By studying the competition between the Caffarelli-Kohn-Nirenberg exponents, the
sign-changing potentials and the nonlinear terms, we establish
an interval on the range of multiple parameters over which solutions exist in an appropriate weighted Sobolev space. The
arguments rely on the method of weak sub- and super-solutions.
\end{abstract}

\maketitle

\numberwithin{equation}{section}
%--------------------------------------------------------------------------------------------------------------
\section{Introduction}
The study of positive solutions of singular partial differential equations or systems has been an extremely active
research topic
during the past few years.  Such singular nonlinear problems arise naturally and they  occupy a central role in the
interdisciplinary
research between analysis, geometry, biology, elasticity, mathematical physics, etc.

In this  paper, we are concerned with the existence of positive solutions to the
boundary value problem
\begin{equation}\label{e1.1}
\left\{\begin{array}{lll}
&-\mbox{div}\,(|x|^{-ap} |\nabla u|^{p-2}\nabla u) = |x|^{-(a+1)p+c_{1}}(\lambda_{1}A(x)f(v)
+ \mu_{1}C(x)h(u)) &\ \text{ in } \Omega,\\
&-\mbox{div}\,(|x|^{-bq} |\nabla v|^{q-2}\nabla v) = |x|^{-(b+1)q+c_{2}}(\lambda_{2} B(x)g(u)
+ \mu_{2}D(x)\tau(v)) &\ \text{ in } \Omega,\\
&u = v = 0 &\ \text{ on } \partial\Omega\,,
\end{array}\right.
\end{equation}
where $ \Omega $ is a  bounded domain in $ \R^{N} $ with $ 0 \in
\Omega $ , $ 1 < p,\,q < N$, $0 \leq a < \frac{N-p}{p}$, $0 \leq b<
\frac{N-q}{q}$, $c_{1},\,c_{2},\,\lambda_{1},\,\lambda_{2},\,\mu_{1},\,\mu_{2} $
are positive parameters.  We assume that the nonlinear terms $  f ,\, g ,\, h ,\, \tau:[0,\infty )
\longrightarrow [0,\infty )$  are nondecreasing continuous
functions. We also suppose that the potentials $ A$, $B$, $C$, and $D $ are $ C^{1} $ sign-changing
functions, that may be negative only near the boundary.

Nonlinear problems like \eqref{e1.1} are introduced in relationship with models for physical phenomena
related to the equilibrium of anisotropic media that possibly are somewhere ``perfect" insulators or ``perfect"
conductors, see Dautray and Lions \cite[p. 79]{dautray}. The qualitative analysis of these problems has been much
developed after the pioneering paper by Murthy and Stampacchia \cite{murthy}.
A crucial milestone in the understanding of the
elliptic problems involving the singular quasilinear elliptic operator $-\mbox{div}\,(|x|^{-ap} |\nabla u|^{p-2}\nabla u)$
is the paper by Caffarelli, Kohn and Nirenberg
\cite{CafKohNir} (see also \cite{catrina}).

The study of this type of problems is
motivated by its various applications, for example, in fluid
mechanics, population genetics, Newtonian fluids,  flow through porous media, and
glaciology, see \cite{Akh,AtkKal,Dan,fpr}. On the other hand,
quasilinear elliptic systems have an extensive practical background.
They are used to describe the multiplicate chemical reaction
catalyzed by the catalyst grains under constant or variant
temperature, in the theory of quasi-regular and quasi-conformal mappings in Riemannian manifolds with boundary, or in the
description of several physical phenomena such as the propagation of pulses in birefringent optical fibers and Kerr-like
photorefractive media, see
\cite{Esc,Tol}.  We refer to \cite{AfrRas},
\cite{BueFerZum,FanLiu,graef} for additional results on elliptic problems.
For the regular case, that is, when  $ a = b = 0$, $c_{1} = p $ and $
c_{2} = q $ the quasilinear elliptic equation has been studied by
several authors (see, e.g., \cite{AliShi}). We refer to
\cite{AliShiRam,Dal},  where the authors discussed the system
(\ref{e1.1}) when $ p = q = 2$, $\alpha_{1} = \alpha_{2}$, $\beta_{1} =
\beta_{2} = 0 $, $f$, $g$ are increasing, and $ f, g \geq 0 $. In
\cite{HaiShi1}, the authors extended the study of \cite{Dal} to the
case when no sign conditions on $ f(0) $ or $ g(0)$ were required
and in \cite{HaiShi2} they extend this study to the case when $  p =
q > 1 $.

In the present paper  we focus on further extending the study in
\cite{AliShi} for the quasilinear elliptic systems involving
singularity. Due to this singularity in the weights, the extensions
are challenging and nontrivial. Our approach is based on the method
of sub-super solutions, see \cite{CanDraGam,DraHer}. Several methods
have been used to treat quasilinear equations and systems. In the
scalar case, weak solutions can be obtained through variational
methods which provide critical points of the corresponding energy
functional, an approach which is also fruitful in the case of
potential systems, that is, the nonlinearities on the right hand side
are the gradient of a $ C^{1}$-functional \cite{AlvFig}. However,
due to the loss of the variational structure, the treatment of
nonvariational systems like (\ref{e1.1}) is more complicated and is
based mostly on topological methods \cite{AmbAzoPer}. Here we focus
on further extending the study in \cite{Ras} to the system
(\ref{e1.1}), which features multiple parameters, weight functions
and stronger coupling. More precisely, under suitable conditions on
$ f, g ,h , \tau $  we  show that the singular problem (\ref{e1.1}) has a
positive solution for $ \lambda_{i},\,\mu_{i}$, $(i=1,2)$ sufficiently
large.

\section{Auxiliary results and technical assumptions}
Let $\Omega$ be a bounded domain in ${\mathbb R}^N$ with smooth boundary and $0\in\Omega$.
Let $ W_{0}^{1,p}(\Omega,|x|^{-ap})$ denote the
completion of  $C_{0}^{\infty}(\Omega)$ with respect to the norm $
\|u\|=(\int_{\Omega}|x|^{-ap}|\nabla u|^{p}dx)^{1/p}$.

Consider the nonlinear eigenvalue
problem
\begin{equation}
\left\{\begin{array}{ll}\label{e2.1}
-\mbox{div}\,(|x|^{-sr} |\nabla \phi|^{r-2}\nabla \phi) =\lambda |x|^{-(s+1)r+t}|\phi|^{r-2}\phi &\quad \text{ in }
\Omega,\\
\phi = 0 &\quad \text{ on } \partial\Omega,
\end{array}\right.
\end{equation}
For $ r=p ,s=a $ and $ t=c_{1}$, let $ \phi_{1,p} $ be the
eigenfunction corresponding to the first eigenvalue $ \lambda_{1,p}
$ of problem  (2.1)     such that $ \phi_{1,p}> 0 $ in $ \Omega $, and
$\|\phi_{1,p}\|_{\infty}=1 $. For $ r=q$, $s=b $ and $ t=c_{2} $,
let $ \phi_{1,q} $ be the eigenfunction corresponding to the first
eigenvalue $ \lambda_{1,q} $ of problem \eqref{e2.1} such that $\phi_{1,q}> 0 $
in $ \Omega $  and $\|\phi_{1,q}\|_{\infty}=1 $, see
\cite{MiyRod,Xua1}.

The maximum principle implies that  $ \frac{\partial {\phi_{1,r}}}{\partial n} < 0
$ on $ \partial\Omega $ for  $ r\in\{ p,q\} $, where $n$ is the outward
normal. Thus, there are positive constants $ m_0 ,\,\delta $ and $ \sigma_{p},\,\sigma_{q}
\in (0,1)$ such that
\begin{equation}
\left\{\begin{array}{ll}\label{e3.1}
\lambda_{1,r} |x|^{-(s+1)r+t}\phi_{1,r}^{r}-|x|^{-sr} |\nabla \phi_{1,r}|^{r}\leq -m_{0} &\quad \text{ in }
\Omega_{\delta}\\
\phi_{1,r}\geq \sigma_{r}&\quad \text{ in } \Omega \setminus \overline{\Omega_{\delta}},\\
\end{array}\right.
\end{equation}
with $ r\in\{p,q\}$, $s\in\{a,b\}$, $t\in\{c_{1},c_{2}\}$, and
$\overline{\Omega_{\delta}}=\{x \in \Omega: d(x,\partial \Omega)\leq
\delta\}$, see \cite{MiyRod}.

We also consider the unique
solution $ (\zeta_{p}(x),\zeta_{q}(x))\in
W_{0}^{1,p}(\Omega,|x|^{-ap})\times
W_{0}^{1,q}(\Omega,|x|^{-bq}) $ of the quasilinear singular system
\begin{equation}\label{e4.1}
\left\{\begin{array}{ll}
-\mbox{div}\,(|x|^{-ap} |\nabla\zeta_{p}|^{p-2}\nabla\zeta_{p}) = |x|^{-(a+1)p+c_{1}} &\quad \text{ in } \Omega\\
-\mbox{div}\,(|x|^{-bq} |\nabla \zeta_{q}|^{q-2}\nabla \zeta_{q}) = |x|^{-(b+1)q+c_{2}} &\quad \text{ in } \Omega\\
u = v = 0 & \quad \text{ on } \partial\Omega\,.
\end{array}\right.
\end{equation}
Then, by \cite{MiyRod}, we have $ \zeta_{r}> 0
$ in $ \Omega $ and  $\frac{\partial {\zeta_{r}}}{\partial n} < 0 $
on $
\partial\Omega $ for  $ r\in\{p,q\} $.

Throughout this paper, we assume that the weight functions $ A$, $B$, $C$, $D $ take
negative values in $\overline\Omega_\delta  $ but require $ A$, $B$, $C$, $D $  to be strictly positive in $ \Omega\backslash \Omega_\delta
$. To be precise we assume that there exist positive constants $
a_{0},b_{0},c_{0},d_{0} $  and $ a_{1},b_{1},c_{1},d_{1}$ such that
\begin{equation}\label{hyp1}
 A(x)\geq - a_{0},\quad B(x)\geq - b_{0},\quad C(x)\geq - c_{0},\quad D(x)\geq -
d_{0}\ \  \mbox{on}\  \overline\Omega_\delta   \end{equation} and
\begin{equation}\label{hyp2} A(x)\geq a_{1},\quad B(x)\geq
b_{1},\quad C(x)\geq c_{1},\quad D(x)\geq  d_{1} \ \  \mbox{on}\ \Omega\backslash
\overline\Omega_\delta  .\end{equation}
Let $ s_{0}\geq 0 $ be such that \begin{equation}\label{hyp3}
\lambda_{1} a_{1} f(s)+ \mu_{1} c_{1} h(s) > 0 \ \mbox{and}\ \lambda_{2}
b_{1} g(s)+ \mu_{2} c_{2} \tau(s) > 0  \ \mbox{ for every} \ s > s_{0}.\end{equation}
%--------------------------------------------------------------------------------------------------------------
\section{Main result}
In this section, we  establish our abstract existence result via the
method of sub- and super-solutions. The concepts of sub- and super-solution were introduced
by Nagumo \cite{nagumo} in 1937 who proved, using also the shooting method, the existence
of at least one solution for a class of nonlinear Sturm-Liouville problems. In fact, the premises of the sub- and
super-solution method can be traced back to Picard. He applied, in the early 1880s, the method of successive
approximations to
argue the existence of solutions for nonlinear elliptic equations that are suitable
perturbations of uniquely solvable linear problems. This is the starting point of
the use of sub- and super-solutions in connection with monotone methods. Picard's techniques were applied later by
Poincar\'e \cite{p166} in connection with problems arising in astrophysics. We refer to \cite{radbook}

A pair of nonnegative functions $(\psi_{1},\psi_{2}) ,\,
(z_{1},z_{2}) $ are called a sub-solution and super-solution of
problem (\ref{e1.1}) if they satisfy
$(\psi_{1},\psi_{2})=(z_{1},z_{2})=(0,0 )$ on $
\partial\Omega  $ and
\begin{align*}
\begin{split}
\int_\Omega |x|^{-ap}|\nabla \psi_{1}|^{p-2}
\nabla \psi_{1}\cdot \nabla \omega\,dx& \leq \int_\Omega |x|^{-(a+1)p+c_{1}}[\lambda_{1}A(x)f(\psi_{2})
+ \mu_{1}C(x)h(\psi_{1})]\omega\,dx \\
\int_\Omega |x|^{-bq}|\nabla \psi_{2}|^{q-2} \nabla \psi_{2}\cdot
\nabla \omega\,dx & \leq \int_\Omega
|x|^{-(b+1)q+c_{2}}[\lambda_{2}B(x)g(\psi_{1})+
\mu_{2}D(x)\tau(\psi_{2})]\omega\,dx
\end{split}
\end{align*}
and
\begin{align*}
\begin{split}
\int_\Omega |x|^{-ap} |\nabla z_{1}|^{p-2}
\nabla z_{1}\cdot \nabla \omega\,dx& \geq \int_\Omega |x|^{-(a+1)p+c_{1}}[\lambda_{1}A(x)f(z_{2})
+ \mu_{1}C(x)h(z_{1})]\omega dx \\
\int_\Omega |x|^{-bq}|\nabla z_{2}|^{q-2} \nabla z_{2}\cdot \nabla
\omega\,dx & \geq \int_\Omega
|x|^{-(b+1)q+c_{2}}[\lambda_{2}B(x)g(z_{1})+
\mu_{2}D(x)\tau(z_{2})]\omega dx
\end{split}
\end{align*}
for all $\omega\in W = \{ \omega \in C_0^\infty(\Omega): \omega \geq
0 \text{ in }\Omega\}$.

In what follows, if $f_1,\, f_2,\, g_1,\, g_2$ are real-valued functions, we write
$$(f_1,f_2)\leq (g_1,g_2)\ \mbox{if and only if}\ f_1\leq g_1\ \mbox{and}\ g_1\leq g_2.$$
In such a case, we write
$$(u,v)\in[(f_1,f_2),(g_1,g_2)]\ \mbox{if and only if}\ f_1\leq u\leq g_1\ \mbox{and}\ g_1\leq v\leq g_2.$$

A key role in our arguments will be played by the following auxiliary result, which is due to Miyagaki and Rodrigues,
see \cite{MiyRod}.

\begin{lemma}\label{lem1.2}
Suppose there exist sub- and super-solutions $ (\psi_{1},\psi_{2}) $
and $ (z_{1},z_{2}) $, respectively  of problem (\ref{e1.1}), such
that $(\psi_{1},\psi_{2})\leq (z_{1},z_{2}  ) $.
 Then  problem
(\ref{e1.1}) has at least one positive solution $ (u,v) $ such that
$(u,v) \in [(\psi_{1},\psi_{2}),(z_{1},z_{2})]$.
\end{lemma}

In order to introduce our main result, we introduce the following hypotheses:
\begin{description}
\item[(H1)] $f,g: [0,+\infty)\to \R $ are $ C^1 $ increasing functions such that
$$
\lim_{t\to +\infty}f(t) = \lim_{t\to +\infty}g(t)=+\infty;
$$
\item[(H2)] We have
$$
\lim_{t\to
+\infty}\frac{f(Kg(t)^\frac{1}{q-1})}{t^{p-1}} = 0,
$$
for every constant $ K > 0 $;
\item[(H3)]  $ h , \tau \in C^1[0 , \infty) $ are nonnegative and
nondecreasing functions such that
\[
\lim_{u \to+\infty} \frac{h(u)}{u^{p - 1}} = 0,  \quad
 \lim_{u \to+\infty} \frac{\tau(u)}{u^{q - 1}} = 0,\lim_{u\to
+\infty}h(u)= \lim_{u\to +\infty}\tau(u)=+\infty\,;
\]

\item[(H4)] If $   \alpha_{p}=\frac{p-1}{p}\sigma_{p}^\frac{p}{p-1}$ , $
\alpha_{q} =\frac{q-1}{q}\sigma_{q}^\frac{q}{q-1}$, and $ \overline\alpha =
\min\{\alpha_{p},\alpha_{q}\}$ then there exists $ \gamma > \frac{s_{0}}{\overline\alpha}$ such that
\begin{eqnarray*}
&\max\Bigg\{{\frac{\gamma \lambda_{1,q}}{b_{1}
g(\gamma^\frac{1}{p-1}\alpha_{p})+{d_{1}
\tau(\gamma^\frac{1}{q-1}\alpha_{q})}}},  \frac{\gamma
\lambda_{1,p}}{a_{1} f(\gamma^\frac{1}{q-1}\alpha_{q})+{c_{1}
h(\gamma^\frac{1}{p-1}\alpha_{p})}} \Bigg\} <&\\
&\min\Bigg\{{\frac{m_{0}\gamma   }{b_{0}
g(\gamma^\frac{1}{p-1})+d_{0} \tau(\gamma^\frac{1}{q-1})}},
\frac{m_{0}\gamma }{a_{0} f(\gamma^\frac{1}{q-1})+c_{0} h(\gamma
^\frac{1}{p-1})}\Bigg\}\,. &
\end{eqnarray*}
\end{description}
We recall that $m_0,\, \sigma_p,\,\sigma_q$ are introduced in relation \eqref{e3.1} while $s_0$ is defined in \eqref{hyp3}.

We now state  our main result for the problem
(\ref{e1.1}).

\begin{theorem}\label{the2.1}
Assume that the conditions $(H1)$--$(H4)$ are fulfilled. Then there exists a closed interval
$[\alpha,\beta]$ such that problem (\ref{e1.1}) has a  positive solution
$(u,v)$ for every $\lambda_{i},\mu_{i} \in [\alpha,\beta]$, ${i=1,2}$.
\end{theorem}


\begin{proof}[Proof]
Choose $ r > 0 $ such that $$ r \leq \min
\Bigg\{|x|^{-(a+1)p+c_{1}},|x|^{-(b+1)q+c_{2}}\Bigg\}\quad
 \mbox{in $ \overline\Omega_\delta $.}$$
Pick $\gamma > \frac{s_{0}}{\overline\alpha} $
as in hypothesis {\bf (H4)}. Define
$$ \alpha := \max\Bigg\{{\frac{\gamma \lambda_{1,q}}{b_{1}
g(\gamma^\frac{1}{p-1}\frac{p-1}{p}\alpha_{p}^\frac{p}{p-1})+{d_{1}
\tau(\gamma^\frac{1}{q-1}\frac{q-1}{q}\alpha_{q}^\frac{q}{q-1})}}},
$$
$$
\frac{\gamma \lambda_{1,p}}{a_{1}
f(\gamma^\frac{1}{q-1}\frac{q-1}{q}\alpha_{q}^\frac{q}{q-1})+{c_{1}
h(\gamma^\frac{1}{p-1}\frac{p-1}{p}\alpha_{p}^\frac{p}{p-1})}}
\Bigg\}
$$
and
$$ \beta:=\min\Bigg\{{\frac{m_{0}\gamma   }{b_{0}
g(\gamma^\frac{1}{p-1})+d_{0} \tau(\gamma^\frac{1}{q-1})}},
\frac{m_{0}\gamma }{a_{0} f(\gamma^\frac{1}{q-1})+c_{0} h(\gamma
^\frac{1}{p-1})}\Bigg\}.
$$

Set $$\psi_{1}=(\gamma
r)^\frac{1}{p-1}\frac{p-1}{p} \phi_{1,p}^\frac{p}{p-1}\qquad \mbox{and}\qquad
\psi_{2}=(\gamma r)^\frac{1}{q-1}\frac{q-1}{q}
\phi_{1,q}^\frac{q}{q-1} .$$

We shall verify that
$(\psi_{1},\psi_{2})$ is a sub-solution of problem (\ref{e1.1}) for
$ \lambda_{i},\mu_{i}\in [\alpha,\beta],{i=1,2}$
 Indeed, let $\omega \in
W $ with $\omega \geq 0$ in $ \Omega $. Then  a simple calculation
shows that
\begin{align}\label{e2.10}
\begin{split}
& \int_\Omega |x|^{-ap}|\nabla \psi_{1}|^{p-2}
\nabla \psi_{1}\cdot\nabla \omega\, dx \\
&=\gamma r\int_{
\Omega}|x|^{-ap}\phi_{1,p}|\nabla\phi_{1,p}|^{p-2}\nabla\phi_{1,p}\cdot \nabla \omega\,dx \\
&= \gamma r \left\{
\int_\Omega|x|^{-ap}|\nabla\phi_{1,p}|^{p-2}\nabla\phi_{1,p}\cdot[\nabla(\phi_{1,p}
\omega)\,- \nabla \phi_{1,p}
\omega ] dx\right\} \\
& = \gamma r  \int_\Omega[\lambda_{1,p}
|x|^{-(a+1)p+c_{1}}\phi_{1,p}^{p}-|x|^{-ap} |\nabla
\phi_{1,p}|^{p}]\omega\,dx.
\end{split}
\end{align}
Similarly we obtain
\begin{align}\label{e2.11}
\begin{split}
& \int_\Omega |x|^{-bq}|\nabla \psi_{2}|^{q-2}
\nabla \psi_{2}\cdot\nabla \omega\, dx \\
&= \gamma r\int_\Omega[\lambda_{1,q}
|x|^{-(b+1)q+c_{2}}\phi_{1,q}^{q}-|x|^{-bq} |\nabla
\phi_{1,q}|^{q}]\omega\,dx.
\end{split}
\end{align}

By relation \eqref{e3.1} we have  $ \lambda_{1,p}
|x|^{-(a+1)p+c_{1}}\phi_{1,p}^{p}-|x|^{-ap} |\nabla
\phi_{1,p}|^{p}\leq -m_{0} $ on $ \overline\Omega_\delta $. Using hypothesis \eqref{hyp1} in combination with the fact that $f$ is nondecreasing we have for all $x\in\overline\Omega_\delta$
\begin{equation}\label{camb1}A(x)f(\psi_2)\geq -a_0f(\psi_2)\geq f(\gamma^\frac{1}{q-1}).\end{equation}
In the last inequality we have also used
$$\gamma^\frac{1}{q-1}\geq (\gamma r)^\frac{1}{q-1}\frac{q-1}{q}
\phi_{1,q}^\frac{q}{q-1}=:\psi_2.$$
A similar argument shows that
\begin{equation}\label{camb2}C(x)h(\psi_1)\geq -c_0h(\gamma^\frac{1}{p-1})\quad\mbox{in $\overline\Omega_\delta$}.\end{equation}
Combining relations \eqref{camb1}, \eqref{camb2} and the definition of $\beta$ we find
\begin{align}\label{e2.2}
\begin{split}
\gamma(\lambda_{1,p} |x|^{-(a+1)p+c_{1}}\phi_{1,p}^{p}-|x|^{-ap}
|\nabla \phi_{1,p}|^{p})&\leq -\gamma m_{0}\\
& \leq-\lambda_1 a_{0}f(\gamma^\frac{1}{q-1})-\mu_{1}c_{0}
h(\gamma^\frac{1}{p-1})\\
& \leq \lambda_1 A(x)f(\psi_{2})+\mu_1 C(x)h(\psi_{1}).
\end{split}
\end{align}
Thus,  for all $\lambda_1,\mu_1\in [\alpha,\beta]$
\begin{align}\label{neweqq}
\begin{split}
 &\gamma r\int_{\overline\Omega_\delta}[\lambda_{1,p}
|x|^{-(a+1)p+c_{1}}\phi_{1,p}^{p}-|x|^{-ap} |\nabla
\phi_{1,p}|^{p}]\omega\,dx \leq\\ & \int_{\overline\Omega_\delta}
|x|^{-(a+1)p+c_{1}}[\lambda_{1}A(x)f(\psi_{2})+
\mu_{1}C(x)h(\psi_{1})]\omega dx.
\end{split}
\end{align}

We recall that $
\phi_{1,p} \geq \sigma_{p} ,\phi_{1,q} \geq \sigma_{q}$ on $ \Omega\backslash \overline\Omega_\delta $, for some $  \sigma_{p},\, \sigma_{q} \in (0,1) $. Thus, by the condition
$(H1)$ and the definition of $ \psi_{1} ,\psi_{2} $, it follows
that
\begin{align}\label{e2.3}
\begin{split}
&\gamma(\lambda_{1,p} |x|^{-(a+1)p+c_{1}}\phi_{1,p}^{p}-|x|^{-ap}
|\nabla \phi_{1,p}|^{p})\leq \gamma \lambda_{1,p}
\leq\\
& \lambda_1 a_{1}f\left(\gamma ^\frac{1}{q-1}\frac{q-1}{q}
\alpha_{q}^\frac{q}{q-1}\right)
 +\mu_{1}c_{1} h\left(\gamma ^\frac{1}{p-1}\frac{p-1}{p}
\alpha_{p}^\frac{p}{p-1}\right)
\\
& \leq \lambda_1 A(x)f(\psi_{2})+\mu_1 C(x)h(\psi_{1}),
\end{split}
\end{align}
for all $\lambda_{1},\mu_{1} \in [\alpha,\beta]$.

Relations (\ref{e2.2}) and (\ref{e2.3}) imply that
\begin{equation}\label{e2.4}
 \int_{\Omega} |x|^{-ap}
|\nabla \psi_{1}|^{p-2} \nabla \psi_{1}\cdot \nabla \omega\, dx \leq
\int_{\Omega} |x|^{-(a+1)p+c_{1}}[\lambda_1 A(x)f(\psi_{2})+\mu_1
C(x)h(\psi_{1})]\omega\,dx
\end{equation}
 A similar argument shows that
\begin{equation}\label{e2.5}
 \int_{\Omega} |x|^{-bq}
|\nabla \psi_{2}|^{q-2} \nabla \psi_{2}\cdot \nabla \omega dx \leq
\int_{\Omega} |x|^{-(b+1)q+c_{2}}[\lambda_{2}B(x)g(\psi_{1})+
\mu_{2}D(x)\tau(\psi_{2})]\omega dx
\end{equation}
 From (\ref{e2.4}) and (\ref{e2.5}), we deduce that $(\psi_{1},\psi_{2})$ is a
sub-solution of problem (\ref{e1.1}). Moreover, we have $ \psi_{1}> 0
$ and $ \psi_{2} > 0 $ in $ \Omega $.

Next, we  construct a super-solution of problem (\ref{e1.1}).
For this purpose, we  prove that there exists a large enough positive constant $ C  $  so that
$$
(z_{1}, z_{2}):=(C\zeta_{p},
\Big[(\lambda_{2}\|b\|_\infty+\mu_2\|d\|_\infty)g(C\|\zeta_{p}\|_{\infty})\Big]^\frac{1}{q-1}\zeta_{q}(x))
$$
 is a super-solution of problem
(\ref{e1.1}), where $(\zeta_p,\zeta_q)$ is a solution of problem \eqref{e4.1}.  We first observe that
\begin{equation}\label{e2.6}\begin{array}{ll}
\di & \di\int_\Omega |x|^{-ap}
|\nabla z_{1}|^{p-2} \nabla z_{1}\cdot \nabla \omega\, dx
=C^{p-1}\int_\Omega |x|^{-ap} |\nabla \zeta_{p}|^{p-2} \nabla
\zeta_{p}\cdot \nabla \omega\, dx =\\ & \di C^{p-1}\int_\Omega
|x|^{-(a+1)p+c_{1}}  \omega\, dx\end{array}
\end{equation}
Using the conditions $ (H2)--(H3) $, we can choose the number $ C >
0 $ large enough so that
\begin{align}\label{e2.7}
\begin{split}
C^{p-1} & \geq   \lambda_{1}\|a\|_\infty
f(\Big[(\lambda_{2}\|b\|_\infty+\mu_2\|d\|_\infty)g(C\|\zeta_{p}\|_{\infty})\Big]^\frac{1}{q-1}\zeta_{q})
+\mu_{1}\|c\|_\infty h(C\|\zeta_{p}\|_\infty)  \\  &\geq
\lambda_{1}\|a\|_\infty
f(\Big[(\lambda_{2}\|b\|_\infty+\mu_2\|d\|_\infty)g(C\|\zeta_{p}\|_{\infty})\Big]^\frac{1}{q-1}\zeta_{q})
+\mu_{1}\|c\|_\infty
h(C\zeta_{p} ) \\
& \geq  \lambda_{1}A(x)
f(z_{2})+\mu_{1}C(x) h(z_{1}).
\end{split}
\end{align}
Therefore
\begin{equation}\label{e2.8}
 \int_\Omega |x|^{-ap}
|\nabla z_{1}|^{p-2} \nabla z_{1}\cdot \nabla \omega\, dx \geq
\int_\Omega |x|^{-(a+1)p+c_{1}}(\lambda_{1}A(x) f(z_{2})+\mu_{1}C(x)
h(z_{1}))  \omega\, dx.
\end{equation}
Next, from the definition of $ z_{2} $, the condition $(H3)$ for $
C
> 0 $ large enough and the fact that $ g $ is increasing
$$ g\Big(C\|\zeta_{p}\|_{\infty}\Big)\geq
\tau(\Big[(\lambda_{2}\|b\|_\infty+\mu_2\|d\|_\infty)
g(C\|\zeta_{p}\|_{\infty})\Big]^\frac{1}{q-1}\parallel\zeta_{q}\|_{\infty}).$$
It follows that
\begin{align}\label{e2.9}
\begin{split}
&\di \int_\Omega |x|^{-bq}
|\nabla z_{2}|^{q-2} \nabla z_{2}\cdot \nabla \omega\, dx \geq \\
& \ (\lambda_{2}\|b\|_\infty+\mu_2\|d\|_\infty) g(C\|\zeta_{p}\|_{\infty})\int_\Omega |x|^{-(b+1)q+c_{2}} \omega dx \geq\\
&  \int_\Omega |x|^{-(b+1)q+c_{2}}(\lambda_{2}\|b\|_\infty+\mu_2\|d\|_\infty)g\Big(C\|\zeta_{p}\|_{\infty}\Big)
\omega dx =\\
& \int_\Omega |x|^{-(b+1)q+c_{2}}[\lambda_{2}\|b\|_\infty g\Big(C\|\zeta_{p}\|_{\infty}\Big)
+\mu_2\|d\|_\infty g\Big(C\|\zeta_{p}\|_{\infty}\Big)]\omega dx \geq\\
 &  \int_\Omega |x|^{-(b+1)q+c_{2}}[ \lambda_{2}\|b\|_\infty g(z_{1})+\mu_2\|d\|_\infty
\tau(z_{2})]\omega dx\geq\\
 &  \int_\Omega |x|^{-(b+1)q+c_{2}}[
 \lambda_{2}\|b\|_\infty g(z_{1})+\mu_2\|d\|_\infty
\tau(z_{2})]\omega\ dx\geq\\
&  \int |x|^{-(b+1)q+c_{2}}[\lambda_2 B(x)g(z_{1})+\mu_2
D(x)\tau(z_{2})]\omega\,dx.
\end{split}
\end{align}

Relations (\ref{e2.8}) and (\ref{e2.9}) yield that $ (z_{1}, z_{2}) $ is a
super-solution of problem (\ref{e1.1}) with $ \psi_{1} \leq z_{1} $
and $ \psi_{2} \leq z_{2} $ for $ C > 0 $ large.

Thus, by Lemma \ref{lem1.2} there exists a positive solution $(u,v)$ of the
system (\ref{e1.1}) such that    $(\psi_{1},\psi_{2}) \leq (u,v)
 \leq (z_{1},z_{2}) $. This
completes the proof of Theorem \ref{the2.1}.
\end{proof}

\medskip
\textbf{Acknowledgements.} The authors are grateful to the anonymous referee for the careful reading of the paper and numerous useful suggestions.
V.~R\u adulescu acknowledges the support through Grant CNCS PCE-47/2011.

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\end{document}
 This study concerns the existence of positive solution for the
following nonlinear system
\begin{equation}
\left\{\begin{array}{ll}
-\mbox{div}\,(|x|^{-ap} |\nabla u|^{p-2}\nabla u) = |x|^{-(a+1)p+c_{1}}(\lambda_{1}A(x)f(v)
+ \mu_{1}C(x)h(u)) \quad \text{ in } \Omega,\\
-\mbox{div}\,(|x|^{-bq} |\nabla v|^{q-2}\nabla v) = |x|^{-(b+1)q+c_{2}}(\lambda_{2} B(x)g(u)
+ \mu_{2}D(x)\tau(v)) \quad \text{ in } \Omega,\\
u = v = 0, \quad \text{ on } \partial\Omega,
\end{array}\right.
\end{equation}
where $ \Omega $ is a bounded domain  of $ \R^{N} $ with $ 0 \in
\Omega $ , $ 0 < p,q < N ,0 \leq a < \frac{N-p}{p},0 \leq b<
\frac{N-q}{q} ,c_{1},c_{2},\lambda_{1},\lambda_{2},\mu_{1},\mu_{2} $
are positive parameters.  Here $ f , g , h , \tau:[0,\infty )
\longrightarrow [0,\infty )$ are nondecreasing continuous  functions
and
$$
\lim_{t\to
+\infty}\frac{f\Big(K\big(g(t))^\frac{1}{q-1}\Big)}{t^{p-1}} = 0
$$
for every constant $ K > 0 $.
The potentials
 $ a , b, c, d $ are $ C^{1} $ sign-changing the weight functions,
that maybe negative near the boundary. We discuss the existence of
positive solution when $  f, g,h $ and $ \tau $ satisfy certain
additional conditions. We use the method of sub-super solutions to
establish our results.