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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 39, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/39\hfil Singular elliptic systems]
{Singular elliptic systems involving  concave terms and critical
Caffarelli-Kohn-Nirenberg exponents}

\author[M. E. O. El Mokhtar \hfil EJDE-2012/39\hfilneg]
{Mohammed E. O. El Mokhtar} 

\address{Mohammed El Mokhtar Ould El Mokhtar \newline
University of Tlemcen, 
Dynamics Systems Laboratoire and Applications,
BO 119, 13 000, Tlemcen, Algeria}
\email{med.mokhtar66@yahoo.fr}


\thanks{Submitted July 21, 2011. Published March 14, 2012.}
\subjclass[2000]{35J66, 35J55, 35B40}
\keywords{Singular elliptic system; concave term; mountain pass theorem; 
\hfill\break\indent critical Caffarelli-Kohn-Nirenberg exponent;
 Nehari manifold; sign-changing weight function}

\begin{abstract}
 In this article, we establish the existence of at least four
 solutions to a singular system with a concave term, a critical
 Caffarelli-Kohn-Nirenberg exponent, and sign-changing weight functions.
 Our main tools are the Nehari manifold and the mountain pass theorem.
\end{abstract}

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

\section{Introduction}

In this article, we consider the existence of multiple  nontrivial
nonnegative solutions of the
\begin{equation}\label{eSllm}
\begin{gathered}
-L_{\mu ,a}u=  (\alpha +1)|x|^{-2_{\ast}b}h|u|^{\alpha -1}u|v|^{\beta+1}
+\lambda _1|x|^{-c}f_1|u|
^{q-2}u\quad \text{in }\Omega \backslash \{0\}\\
-L_{\mu ,a}v= (\beta +1)|x|^{-2_{\ast}b}h|u|^{\alpha +1}|v|^{\beta
-1}v+\lambda _2|x|^{-c}f_2|v|
^{q-2}v\quad\text{in }\Omega \backslash \{0\}\\
u=v=0\quad\text{on }\partial \Omega ,
\end{gathered}
\end{equation}
where $L_{\mu ,a}w:=\operatorname{div}(|x|^{-2a}\nabla w)-\mu |x|^{-2(a+1)}w$,
 $\Omega $ is a bounded regular domain in $\mathbb{R}^N$ $(N\geq 3)$ containing
$0$ in its interior, $-\infty <a<(N-2)/2$, $a\leq b<a+1$, $1<q<2$,
$2_{\ast }=2N/(N-2+2(b-a))$ is the critical Caffarelli-Kohn-Nirenberg
exponent, $0<c=q(a+1)+N(1-q/2)$, $-\infty <\mu < \bar{\mu}_{a}:=((N-2(a+1))/2)^2$,
 $\alpha $, $\beta $ are positive reals such that $\alpha +\beta =2_{\ast }-2$,
$\lambda _1$, $\lambda _2$ are real parameters, $f_1$, $f_2$ and $h $ are functions
defined on $\bar{\Omega}$.

Elliptic systems have been widely studied in recent years, we refer the
readers to \cite{a1,d1} for regular systems which
derive from potential. However, only a few results for singular systems, we can cite
 \cite{b2,d1}. As noticed, when $a=b=c=0 $, $h\equiv 1$, $q=2$ and
 $f_1\equiv f_2\equiv 1$, Liu and Han \cite{l1} studied
\eqref{eSllm}. By applying the mountain pass theorem,
they proved that, if $0<\mu \leq \bar{\mu}_0-1$ then, system
\eqref{eSllm}  admits one positive solution for
all $\lambda _1,\lambda _2\in (0,\eta _1(\mu ))$. Here, $\eta _1(\mu )$ denote
 the first eigenvalue of the positive operator $-\Delta -\mu |x|^{-2}$ with
Dirichlet boundary condition. Wu \cite{w1} proved that the
system \eqref{eSllm} with $\mu=0$, has at least two
nontrivial nonnegative solutions when $a=b=c=0$, the pair of the parameters
$(\lambda_1 ,\lambda_2 )$ belong to a certain subset of $\mathbb{R}^2$ and
under some conditions on the weight functions $f_1$, $f_2$ and
$h$.
For $c=0$, $q=1$ and $h\equiv 1$, system \eqref{eSllm}  has been studied by
 Bouchekif and El Mokhtar \cite{b1}. By using the Nehari manifold, they proved that
there exists a positive constant $\Lambda _0$ such that
\eqref{eSllm} admits two nontrivial solutions when $\lambda _1$, $\lambda _2$
satisfy $0<|\lambda_1|\|f_1\|_{\mathcal{H}_{\mu }'}+|\lambda _2|\|f_2\|_{
\mathcal{H}_{\mu }'}<(1/2)\Lambda _0$.

The starting point of the variational approach to our problem is the
following Caffarelli-Kohn-Nirenberg inequality \cite{c1}, which
ensures the existence of a positive constant $C_{a,b}$ such that
\begin{equation}
(\int_{\mathbb{R}^N}|x|^{-2_{\ast }b}|v|^{2_{\ast}}dx)^{2/2_{\ast }}
\leq C_{a,b}\int_{\mathbb{R}^N}|x|^{-2a}|\nabla v|^2dx, \quad
\text{for all }v\in \mathcal{C}_0^{\infty }(\mathbb{R}^N).  \label{r1}
\end{equation}
In this equation, if $b=a+1$, then $2_{\ast }=2$ and we have the
 weighted Hardy inequality \cite{c2}:
\begin{equation}
\int_{\mathbb{R}^N}|x|^{-2(a+1)}v^2dx
\leq (1/\bar{\mu}_{a})\int_{\mathbb{R}^N}|x|^{-2a}|\nabla v|^2dx,
\quad \text{for all }v\in \mathcal{C}_0^{\infty }(\mathbb{R}^N). \label{r2}
\end{equation}
We introduce a weighted Sobolev spaces
$\mathcal{D}_{a}^{1,2}(\Omega)$ and $\mathcal{H}_{\mu }:=\mathcal{H}_{\mu }(\Omega )$
which are the completion of the space $\mathcal{C}_0^{\infty }(\mathbb{R}^N)$ with
respect to the norms
\begin{gather*}
\|u\|_{0,a}=\Big(\int_{\Omega }|x|
^{-2a}|\nabla u|^2dx\Big)^{1/2}, \\
\|u\|_{\mu ,a}=\Big(\int_{\Omega }(|x|^{-2a}|\nabla u|^2-\mu |y|
^{-2(a+1)}|u|^2)dx\Big)^{1/2}
\end{gather*}
for $-\infty <\mu <\bar{\mu}_{a}$, respectively.

It is known that by weighted Hardy inequality, the norm $\|u\|_{\mu ,a}$ is
equivalent to $\|u\|_{0,a}$.
More explicitly, we have
\begin{equation*}
(1-(1/\bar{\mu}_{a})\max (\mu ,0))^{1/2}\|u\|_{0,a}
\leq \|u\|_{\mu,a}\leq (1-(1/\bar{\mu}_{a})\min (\mu ,0) )^{1/2}\|u\|_{0,a},
\end{equation*}
for all $u\in \mathcal{H}_{\mu }$.

Define the space $\mathcal{H}:=\mathcal{H}_{\mu }\times \mathcal{H}_{\mu }$
which is endowed with the norm
\begin{equation*}
\|(u,v)\|_{\mu ,a}=\big(\|u\|_{\mu ,a}^2+\|v\|_{\mu ,a}^2\big)^{1/2}.
\end{equation*}
From the boundlessness of $\Omega $ and the standard approximation
arguments, it is easy to see that \eqref{r1} hold for any $
u\in \mathcal{H}_{\mu }$ in the sense
\begin{equation}
(\int_{\Omega }|x|^{-c}|u|
^{p}dx)^{1/p}\leq C_{a,p}(\int_{\Omega }|x|
^{-2a}|\nabla v|^2dx)^{1/2},  \label{r3}
\end{equation}
where $C_{a,p}$ positive constant, $1\leq p\leq 2N/(N-2)$, 
$c\leq p(a+1)+N(1-p/2)$, and in \cite{w1}, if $p<2N/(N-2)$ the embedding $\mathcal{H}_{\mu}\hookrightarrow L_p(\Omega ,|x|^{-c})$
is compact, where $L_p(\Omega ,|x|^{-c})$
is the weighted $L_p$ space with norm
\begin{equation*}
|u|_{p,c}=(\int_{\Omega }|x|^{-c}|u|^{p}dx)^{1/p}.
\end{equation*}
Since our approach is variational, we define the functional
 $J:=J_{\lambda_1,\lambda _2,\mu }$ on $\mathcal{H}$ by
\begin{equation*}
J(u,v):=(1/2)\|(u,v)
\|_{\mu ,a}^2-P(u,v)-Q(u,v),
\end{equation*}
with
\begin{gather*}
P(u,v):=\int_{\Omega }|x|^{-2_{\ast}b}h|u|^{\alpha +1}|v|^{\beta +1}dx,\\
Q(u,v):=(1/q)\int_{\Omega }|x|^{-c}(\lambda _1f_1|u|^q+\lambda _2f_2|v|^q)dx.
\end{gather*}
A couple $(u,v)\in \mathcal{H}$ is a weak solution of the
system \eqref{eSllm} if it satisfies
\begin{equation*}
\langle J'(u,v),(\varphi ,\psi )\rangle :=R(u,v)(\varphi ,\psi )-S(
u,v)(\varphi ,\psi )-T(u,v)(\varphi,\psi )=0
\end{equation*}
 for all $(\varphi ,\psi )\in \mathcal{H}$ with
\begin{gather*}
R(u,v)(\varphi ,\psi ):=\int_{\Omega }\big(
|x|^{-2a}(\nabla u\nabla \varphi +\nabla v\nabla
\psi )-\mu |x|^{-2(a+1)}(u\varphi+v\psi)  \big)
\\
S(u,v)(\varphi ,\psi ):=\int_{\Omega }|
x|^{-2_{\ast }b}h((\alpha +1)|
u|^{\alpha }|v|^{\beta +1}\varphi +(\beta +1)|u|^{\alpha +1}|v| ^{\beta }\psi )\\
T(u,v)(\varphi ,\psi ):=\int_{\Omega }|
x|^{-c}(\lambda _1f_1|u|^{q-1}\varphi +\lambda _2f_2|v|^{q-1}\psi ).
\end{gather*}
Here $\langle .,.\rangle $ denotes the product in the duality
$\mathcal{H}'$, $\mathcal{H}$, where $\mathcal{H}'$ is the dual of $\mathcal{H})$.
Let
\begin{gather*}
S_{\mu }:=\inf_{u\in \mathcal{H}_{\mu }\backslash \{0\}}
\frac{\|u\|_{\mu ,a}^2}{(\int_{\Omega }|
x|^{-2_{\ast }b}|u|^{2_{\ast }}dx) ^{2/2_{\ast }}},
\\
\tilde{S}_{\mu }:=\inf_{(u,v)\in (\mathcal{H}
\backslash \{(0,0)\})^2}\frac{\|
(u,v)\|_{\mu ,a}^2}{(\int_{\Omega }|
x|^{-2_{\ast }b}|u|^{\alpha +1}|
v|^{\beta +1}dx)^{2/2_{\ast }}}.
\end{gather*}
From \cite{k1}, it is known that $S_{\mu }$ is achieved.

\begin{lemma}\label{lem0}
Let $\Omega $ be a domain (not necessarily bounded),
$-\infty<\mu <\bar{\mu}_{a}$ and $\alpha +\beta \leq 2_{\ast }-2$. Then we have
\begin{equation*}
\tilde{S}_{\mu }:=\big[(\frac{\alpha +1}{\beta +1})^{(
\beta +1)/2_{\ast }}+(\frac{\alpha +1}{\beta +1})
^{-(\alpha +1)/2_{\ast }}\big]S_{\mu }.
\end{equation*}
With $[(\frac{\alpha +1}{\beta +1})^{(\beta
+1)/2_{\ast }}+(\frac{\alpha +1}{\beta +1})^{-(
\alpha +1)/2_{\ast }}]$ simply written as $K(\alpha,\beta )$.
\end{lemma}

\begin{proof}
The proof is essentially the same as in \cite{a1}, with minor
modifications.
\end{proof}

We put assumptions on $h$ which is somewhere positive but which may change
sign in $\bar{\Omega}$
\begin{itemize}
\item[(H1)] $h\in C(\bar{\Omega})\text{ and }h^{+}=\max \{
h,0\}\not\equiv 0$  in $\Omega$

\item[(H2)]  There exists $\varrho _0$  positive such that
$|h^{+}|_{\infty }=h(0)=\max_{x\in \bar{\Omega}}h(x)>\varrho _0$.
\end{itemize}

In our work, we research for critical points as the minimizers of the energy
functional associated with \eqref{eSllm} with the constraint defined by the Nehari
manifold, which are solutions of our system.

Let $\Lambda _0$ be positive number and $f_1$, $f_2$ be continuous
functions such that
\begin{equation*}
\Lambda _0:=(C_{a,q})^{-q}(|h^{+}|
_{\infty })^{-1/(2_{\ast }-2)}[(S_{\mu
})K(\alpha ,\beta )]^{2_{\ast }/2(2_{\ast}-2)}L(q)
\end{equation*}
and $|f_{i}(x)|_{\mathcal{\infty }}=\sup_{x\in \bar{\Omega}}|f_{i}(x)|$
for $i=1,2$, where
$$
L(q):=(\frac{2_{\ast }-2}{2_{\ast}-q})^{1/(2-q)}[(\frac{2-q}{2_{\ast }(
2_{\ast }-q)})]^{1/(2_{\ast }-2)}.
$$
Now we  state our main results as follows.

\begin{theorem} \label{thm1}
Let $f_1$, $f_2\in L^{\infty }(\Omega )$.
Assume that $-\infty <a<(N-2)/2$, $0<c=q(a+1)+N(1-q/2)$, $\alpha +\beta +2=2_{\ast }$,
$-\infty <\mu <\bar{\mu}_{a}$,
{\rm (H1)} satisfied and $\lambda _1$, $\lambda _2$ satisfying
$0<(|\lambda _1||f_1|_{\mathcal{\infty }})^{1/(2-q)}+(|\lambda
_2||f_2|_{\mathcal{\infty }}) ^{1/(2-q)}<\Lambda _0$, then
\eqref{eSllm} has at least one positive solution.
\end{theorem}

\begin{theorem}\label{thm2}
In addition to the assumptions of the Theorem \ref{thm1}, if
{\rm (H2)} holds and $\lambda _1$, $\lambda _2$ satisfy
$$
0<(|\lambda _1||f_1|_{\mathcal{\infty }})^{1/(2-q)}+(|\lambda
_2||f_2|_{\mathcal{\infty }})^{1/(2-q)}<(1/2)\Lambda _0,
$$
then \eqref{eSllm} has at least two positive solutions.
\end{theorem}

\begin{theorem}\label{thm3}
In addition to the assumptions of the Theorem \ref{thm2},
assuming $N\geq \max (3, 6(a-b+1))$, there
exists a positive real $\Lambda _1$ such that, if $\lambda _1$, $\lambda
_2$ satisfy
$$
0<(|\lambda _1|| f_1|_{\mathcal{\infty }})^{1/(2-q)}+(|\lambda _2||f_2|_{\mathcal{
\infty }})^{1/(2-q)}<\min ((1/2)\Lambda _0,\Lambda _1),
$$
then \eqref{eSllm} has at least two positive solution and two opposite solutions.
\end{theorem}

This article is organized as follows. In Section 2, we give some
preliminaries. Section 3 and 4 are devoted to the proofs of Theorems
\ref{thm1} and \ref{thm2}. In the last Section, we prove the Theorem \ref{thm3}.

\section{Preliminaries}

\begin{definition}\label{def1} \rm
Let $c\in\mathbb{R}$, $E$ a Banach space and
$I\in C^{1}(E,\mathbb{R})$.

(i) $(u_{n},v_{n})_{n}$ is a Palais-Smale
sequence at level $c$ ( in short $(PS)_{c}$) in $E$ for $I$ if
\begin{equation*}
I(u_{n},v_{n})=c+o_{n}(1)\text{ and }I'(u_{n},v_{n})=o_{n}(1),
\end{equation*}
where $o_{n}(1)$ tends to $0$ as $n$ approaches infinity.

(ii) We say that $I$ satisfies the $(PS)_{c}$
condition if any $(PS)_{c}$ sequence in $E$ for $I$ has a
convergent subsequence.
\end{definition}

\begin{lemma}\label{lem0'}
 Let $X$ Banach space, and $J\in C^{1}(X,\mathbb{R})$
satisfying the Palais -Smale condition. Suppose that $J(0,0)=0$
and that:
\begin{itemize}
\item[(i)] there exist $R>0$, $r>0$ such that if $\|(u,v)\|=R$,
then $J(u,v)\geq r$;

\item[(ii)] there exist $(u_0,v_0)\in X$ such that $\|(u_0,v_0)\|>R$ and
 $J(u_0,v_0) \leq 0$.
\end{itemize}
Let $c=\inf_{\gamma \in \Gamma }\max_{t\in [0,1]}(J(\gamma (t)))$ where
\begin{equation*}
\Gamma =\{\gamma \in C([0,1];X)\text{ such
that }\gamma (0)=(0,0)\text{ and }\gamma (1)=(u_0,v_0)\},
\end{equation*}
then $c$ is a critical value of $J$ such that $c\geq r$.
\end{lemma}

\subsection{Nehari manifold}

It is well known that $J$ is of class $C^{1}$ in $\mathcal{H}$ and the
solutions of \eqref{eSllm}
are the critical points of $J$ which is not bounded below on $\mathcal{H}$.
Consider the Nehari manifold
\begin{equation*}
\mathcal{N}=\{(u,v)\in \mathcal{H}\backslash \{0,0\}: \langle J'(u,v),(u,v)\rangle =0\},
\end{equation*}
Thus, $(u,v)\in \mathcal{N}$ if and only if
\begin{equation}
\|(u,v)\|_{\mu ,a}^2-2_{\ast }P(u,v)-Q(u,v)=0.  \label{e13}
\end{equation}
Note that $\mathcal{N}$ contains every nontrivial solution of
\eqref{eSllm}. Moreover, we
have the following results.

\begin{lemma}\label{lem4}
 $J$ is coercive and bounded from below on $\mathcal{N}$.
\end{lemma}

\begin{proof}
If $(u,v)\in \mathcal{N}$, then by \eqref{e13}
and the H\"{o}lder inequality, we deduce that
\begin{equation} \label{f14}
\begin{split}
J(u,v)&= ((2_{\ast }-2)/2_{\ast }2)\|(u,v)\|_{\mu ,a}^2-((2_{\ast}-q)/2_{\ast }q)Q(u,v) \\
&\geq ((2_{\ast }-2)/2_{\ast }2)\|(u,v)\|_{\mu ,a}^2
 -(\frac{(2_{\ast }-q)}{2_{\ast }q})\Big((|\lambda _1||f_1|_{\mathcal{
\infty }})^{1/(2-q)}\\
&\quad +(|\lambda_2||f_2|_{\infty })^{1/(2-q)}\Big)(C_{a,p})^q\|(u,v)\|_{\mu ,a}^q.
\end{split}
\end{equation}
Thus, $J$ is coercive and bounded from below on $\mathcal{N}$.
\end{proof}

Define
\begin{equation*}
\phi (u,v)=\langle J'(u,v),(u,v)\rangle .
\end{equation*}
Then, for $(u,v)\in \mathcal{N}$,
\begin{equation} \label{r16}
\begin{split}
\langle \phi '(u,v),(u,v)\rangle
 &= 2\|(u,v)\|_{\mu,a}^2-(2_{\ast })^2P(u,v)-qQ(u,v) \\
&= (2-q)\|(u,v)\|_{\mu,a}^2-2_{\ast }(2_{\ast }-q)P(u,v) \\
&= (2_{\ast }-q)Q(u,v)-(2_{\ast }-2)\|(u,v)\|_{\mu ,a}^2.
\end{split}
\end{equation}
Now, we split $\mathcal{N}$ into three parts:
\begin{gather*}
\mathcal{N}^{+} = \{(u,v)\in \mathcal{N}:\langle \phi '(u,v),(u,v)\rangle >0\}\\
\mathcal{N}^{0} = \{(u,v)\in \mathcal{N}:\langle \phi '(u,v),(u,v)\rangle =0\}\\
\mathcal{N}^{-} = \{(u,v)\in \mathcal{N}:\langle \phi '(u,v),(u,v)\rangle <0\}.
\end{gather*}
We have the following results.

\begin{lemma} \label{lem5}
Suppose that $(u_0,v_0)$ is a local minimizer
for $J$ on $\mathcal{N}$. Then, if $(u_0,v_0)\notin
\mathcal{N}^{0}$, $(u_0,v_0)$ is a critical point of $J$.
\end{lemma}

\begin{proof}
If $(u_0,v_0)$ is a local minimizer for $J$ on $\mathcal{N}$,
then $(u_0,v_0)$ is a solution of the optimization problem
\begin{equation*}
\min_{\{(u,v):\phi (u,v)=0\}}J(u,v).
\end{equation*}
Hence, there exists a Lagrange multipliers $\theta \in\mathbb{R}$ such that
\begin{equation*}
J'(u_0,v_0)=\theta \phi '(u_0,v_0)\text{ in }\mathcal{H}'
\end{equation*}
Thus,
\begin{equation*}
\langle J'(u_0,v_0),(
u_0,v_0)\rangle =\theta \langle \phi '(u_0,v_0),(u_0,v_0)\rangle .
\end{equation*}
But $\langle \phi '(u_0,v_0),(u_0,v_0)\rangle \neq 0$, since
$(u_0,v_0) \notin \mathcal{N}^{0}$. Hence $\theta =0$. This completes the proof.
\end{proof}

\begin{lemma} \label{lem6} There exists a positive number $\Lambda _0$ such that for all
 $ \lambda _1$, $\lambda _2$ satisfying
\begin{equation*}
0<(|\lambda _1||f_1|_{\mathcal{\infty }})^{1/(2-q)}
+(|\lambda_2||f_2|_{\infty })^{1/(2-q)}<\Lambda _0,
\end{equation*}
we have $\mathcal{N}^{0}=\emptyset $.
\end{lemma}

\begin{proof}
By contradiction, suppose $\mathcal{N}^{0}\neq \emptyset $
and that 
$$
0<(|\lambda _1||f_1|_{\mathcal{\infty }})^{1/(2-q)}+(|\lambda _2|
|f_2|_{\infty })^{1/(2-q) }<\Lambda _0.
$$
Then, by \eqref{r16} and for
$(u,v)\in \mathcal{N}^{0}$, we have
\begin{equation} \label{e18}
\|(u,v)\|_{\mu ,a}^2 = 2_{\ast }(2_{\ast }-q)/(2-q)P(u,v)
= ((2_{\ast }-q)/(2_{\ast }-2))Q(u,v)
\end{equation}
Moreover, by the H\"{o}lder inequality and the Sobolev embedding theorem, we
obtain
\begin{equation}
\|(u,v)\|_{\mu ,a}\geq [(S_{\mu
})K(\alpha ,\beta )]^{2_{\ast }/2(2_{\ast
}-2)}[(2-q)/2_{\ast }(2_{\ast }-q)
|h^{+}|_{\infty }]^{-1/(2_{\ast }-2)}  \label{r18'}
\end{equation}
and
\begin{equation}
\|(u,v)\|_{\mu ,a}\leq [(\frac{
2_{\ast }-q}{2_{\ast }-2})^{-1/(2-q)}((
|\lambda _1||f_1|_{\mathcal{
\infty }})^{1/(2-q)}+(|\lambda
_2||f_2|_{\infty })^{1/(2-q)})(C_{a,q})^q].  \label{e19}
\end{equation}
From \eqref{r18'} and \eqref{e19}, we obtain
$(|\lambda _1||f_1|_{\mathcal{\infty }})^{1/(2-q)}+(|\lambda
_2||f_2|_{\infty })^{1/(2-q)}\geq \Lambda _0$, which contradicts an hypothesis.
\end{proof}

Thus $\mathcal{N}=\mathcal{N}^{+}\cup \mathcal{N}^{-}$. Define
\begin{equation*}
c:=\inf_{u\in \mathcal{N}}J(u,v),\quad
c^{+}:=\inf_{u\in \mathcal{N}^{+}}J(u,v), \quad
c^{-}:=\inf_{u\in \mathcal{N}^{-}}J(u,v).
\end{equation*}
In the sequel, we need the following Lemma.

\begin{lemma}\label{lem7}

(i) For all $\lambda _1$, $\lambda _2$ with
$0<(|\lambda _1||f_1|_{\mathcal{\infty }})^{1/(2-q)}+(|\lambda _2||f_2|_{\infty })^{1/(
2-q)}<\Lambda _0$, one has $c\leq c^{+}<0$.

(ii) For all $\lambda _1$, $\lambda _2$ such that $
0<(|\lambda _1||f_1|_{\mathcal{\infty }})^{1/(2-q)}+(|\lambda
_2||f_2|_{\infty })^{1/(2-q)}<(1/2)\Lambda _0$, one has
\begin{align*}
c^{-} &> C_0=C_0(\lambda _1,\lambda _2,S_{\mu
},\|f_1\|_{\mathcal{H}_{\mu }'},
\|f_2\|_{\mathcal{H}_{\mu }'}) \\
&= (\frac{(2_{\ast }-2)}{2_{\ast }2})[\frac{
(2-q)}{2_{\ast }(2_{\ast }-q)|
h^{+}|_{\infty }}]^{2/(2_{\ast }-2)}[
K(\alpha ,\beta )]^{2_{\ast }/(2_{\ast }-2)
}(S_{\mu })^{2_{\ast }/(2_{\ast }-2)} \\
&\quad -(\frac{(2_{\ast }-q)}{2_{\ast }q})((|\lambda _1|
|f_1|_{\mathcal{\infty }})^{1/(2-q)}+(|\lambda _2||
f_2|_{\infty })^{1/(2-q)})(C_{a,q})^q.
\end{align*}
\end{lemma}

\begin{proof}
(i) Let $(u,v)\in \mathcal{N}^{+}$. By \eqref{r16}, we have
\begin{equation*}
[(2-q)/2_{\ast }(2_{\ast }-1)]
\|(u,v)\|_{\mu ,a}^2>P(u,v)
\end{equation*}
and so
\begin{align*}
J(u,v)&= (-1/2)\|(u,v)
\|_{\mu ,a}^2+(2_{\ast }-1)P(u,v)\\
&< -[\frac{2_{\ast }(2_{\ast }-q)-2(2_{\ast
}-1)(2-q)}{2_{\ast }2(2_{\ast }-q)}]
\|(u,v)\|_{\mu ,a}^2.
\end{align*}
We conclude that $c\leq c^{+}<0$.

(ii) Let $(u,v)\in \mathcal{N}^{-}$. By
\eqref{r16}, we obtain
\begin{equation*}
[(2-q)/2_{\ast }(2_{\ast }-q)]
\|(u,v)\|_{\mu ,a}^2<P(u,v).
\end{equation*}
Moreover, by (H1) and Sobolev embedding theorem, we have
\begin{equation*}
P(u,v)\leq [K(\alpha ,\beta )]
^{-2_{\ast }/2}(S_{\mu })^{-2_{\ast }/2}|
h^{+}|_{\infty }\|(u,v)\|_{\mu
,a}^{2_{\ast }}.
\end{equation*}
This implies
\begin{equation}
\|(u,v)\|_{\mu ,a}>[(S_{\mu
})K(\alpha ,\beta )]^{2_{\ast }/2(2_{\ast
}-2)}[\frac{(2-q)}{2_{\ast }(2_{\ast
}-q)|h^{+}|_{\infty }}]^{-1/(
2_{\ast }-2)}
\label{e20'}
\end{equation}
for all $u\in \mathcal{N}^{-}$.
By \eqref{f14}, we obtain
\begin{align*}
J(u,v)&\geq ((2_{\ast }-2)/2_{\ast}2)\|(u,v)\|_{\mu ,a}^2
-(\frac{(2_{\ast }-q)}{2_{\ast }q})\Big((
|\lambda _1||f_1|_{\mathcal{\infty }})^{1/(2-q)} \\
&\quad +(|\lambda_2||f_2|_{\infty })^{1/(2-q)}\Big)(C_{a,p})^q\|(u,v)
\|_{\mu ,a}^q.
\end{align*}
Thus, for all $(\lambda _1,\lambda _2)$ such that
$0<(|\lambda _1||f_1|_{\mathcal{\infty }})^{1/(2-q)}+(|\lambda
_2||f_2|_{\infty })^{1/(2-q)}<(1/2)\Lambda _0$, we have $J(u,v)
\geq C_0$.
\end{proof}

For each $(u,v)\in \mathcal{H}$ with
$\int_{\Omega }|x|^{-2_{\ast }b}h|u|^{\alpha +1}|v|^{\beta +1}dx>0$, we write
\begin{equation*}
t_{m}:=t_{\rm max }(u,v)=[\frac{(2-q)\|
(u,v)\|_{\mu ,a}^2}{2_{\ast }(2_{\ast
}-q)\int_{\Omega }|x|^{-2_{\ast }b}h|
u|^{\alpha +1}|v|^{\beta +1}dx}]
^{(2-q)/2_{\ast }(2_{\ast }-q)}>0.
\end{equation*}

\begin{lemma}\label{lem8}
Let $\lambda _1$, $\lambda _2$ real parameters such that
$0<|\lambda _1|\|f_1\|_{\mathcal{H}_{\mu }'}+|\lambda _2|\|
f_2\|_{\mathcal{H}_{\mu }'}<\Lambda _0$. For each
$(u,v)\in \mathcal{H}$ with $\int_{\Omega }|x|^{-2_{\ast }b}h|u|^{\alpha +1}|
v|^{\beta +1}dx>0$, one has the following:

(i) If  $Q(u,v)\leq 0$, then there exists a
unique $t^{-}>t_{m}$ such that $(t^{-}u,t^{-}v)\in \mathcal{N}
^{-}$ and
\begin{equation*}
J(t^{-}u,t^{-}v)=\sup_{t\geq 0}(tu,tv).
\end{equation*}

(ii) If  $Q(u,v)>0$, then there exist unique $t^{+}$ and $t^{-}$ such that
 $0<t^{+}<t_{m}<t^{-}$, $(t^{+}u,t^{+}v)\in \mathcal{N}^{+}$,
$(t^{-}u,t^{-}v)\in \mathcal{N}^{-}$,
\begin{equation*}
J(t^{+}u,t^{+}v)=\inf_{0\leq t\leq t_{m}}J(
tu,tv)\text{ and }J(t^{-}u,t^{-}v)=\sup_{t\geq
0}J(tu,tv).
\end{equation*}
\end{lemma}

The proof of the above lemma is the same as in \cite{b3}, 
with minor modifications.


\begin{proposition}[\cite{b3}] \label{prop1}

(i) For all $\lambda _1$, $\lambda _2$ such that
$$
0<(|\lambda _1||f_1|_{\mathcal{\infty }})^{1/(2-q)}+(|\lambda
_2||f_2|_{\infty })^{1/(2-q)}<\Lambda _0,
$$
there exists a $(PS)_{c^{+}}$ sequence in $\mathcal{N}^{+}$.

(ii) For all $\lambda _1$, $\lambda _2$ such that $
0<(|\lambda _1||f_1|_{\mathcal{\infty }})^{1/(2-q)}+(|\lambda
_2||f_2|_{\infty })^{1/(2-q)}<(1/2)\Lambda _0$, there exists a
 $(PS)_{c^{-}}$ sequence in $\mathcal{N}^{-}$.
\end{proposition}

\section{Proof of Theorem \ref{thm1}}

Now, taking as a starting point the work of Tarantello \cite{t1},
we establish the existence of a local minimum for $J$ on $\mathcal{N}^{+}$.

\begin{proposition}\label{prop2} 
For all $\lambda _1$, $\lambda _2$ with
 $0<(|\lambda _1||f_1|_{\mathcal{\infty }})^{1/(2-q)}
+(|\lambda_2||f_2|_{\infty })^{1/(2-q)}<\Lambda _0$, the functional $J$ has a minimizer
$(u_0^{+},v_0^{+})\in \mathcal{N}^{+}$ and it satisfies:
\begin{itemize}
\item[(i)]  $J(u_0^{+},v_0^{+})=c=c^{+}$,

\item[(ii)] $(u_0^{+},v_0^{+})$ is a nontrivial
solution of \eqref{eSllm}.
\end{itemize}
\end{proposition}

\begin{proof}
If $0<(|\lambda _1||f_1|_{\mathcal{\infty }})^{1/(2-q)}+(|
\lambda _2||f_2|_{\infty })^{1/(2-q)}<\Lambda _0$, then by Proposition
\ref{prop1} (i), there exists a $(u_{n},v_{n})_{n}$ $(PS)_{c^{+}}$ sequence in
 $\mathcal{N}^{+}$, thus it bounded by Lemma \ref{lem4}. Then, there exists
$(u_0^{+},v_0^{+})\in \mathcal{H}$ and we can extract a subsequence which will
denoted by $(u_{n},v_{n})_{n}$ such that
\begin{equation} \label{e32}
\begin{gathered}
(u_{n},v_{n})\rightharpoonup (u_0^{+},v_0^{+})\quad \text{weakly in }\mathcal{H}   \\
(u_{n},v_{n}) \rightharpoonup (u_0^{+},v_0^{+})\quad \text{weakly in }(L^{2_{\ast }}(
\Omega ,|x|^{-2_{\ast }b}))^2   \\
(u_{n},v_{n}) \to (u_0^{+},v_0^{+}) \quad
\text{strongly in }(L^q(\Omega ,|x|^{-c}))^2   \\
u_{n} \to u_0^{+}\quad \text{a.e in }\Omega,\\
v_{n}\to v_0^{+}\quad \text{a.e in }\Omega .
\end{gathered}
\end{equation}
Thus, by \eqref{e32}, $(u_0^{+},v_0^{+})$
is a weak nontrivial solution of \eqref{eSllm}. Now, we show that $(u_{n},v_{n})$
converges to $(u_0^{+},v_0^{+})$ strongly in $\mathcal{H}$.
Suppose otherwise. By the lower semi-continuity of the norm, then either
$\|u_0^{+}\|_{\mu ,a}<\liminf_{n\to \infty }\|u_{n}\|_{\mu ,a}$ or
$\|v_0^{+}\|_{\mu ,a}<\liminf_{n\to \infty }\|v_{n}\|_{\mu ,a}$ and we obtain
\begin{align*}
c &\leq J(u_0^{+},v_0^{+})=((2_{\ast
}-2)/2_{\ast }2)\|(u_0^{+},v_0^{+})
\|_{\mu ,a}^2-((2_{\ast }-q)/2_{\ast}q)Q(u_0^{+},v_0^{+})\\
&<\liminf_{n\to \infty }J(u_{n},v_{n})=c.
\end{align*}
We obtain a contradiction. Therefore, $(u_{n},v_{n})$ converge to
$(u_0^{+},v_0^{+})$ strongly in $\mathcal{H}$. Moreover, we
have $(u_0^{+},v_0^{+})\in \mathcal{N}^{+}$. If not, then
by Lemma \ref{lem8}, there are two numbers $t_0^{+}$ and $t_0^{-}$,
uniquely defined so that $(t_0^{+}u_0^{+},t_0^{+}v_0^{+})\in \mathcal{N}^{+}$ and
$(t^{-}u_0^{+},t^{-}v_0^{+})\in \mathcal{N}^{-}$. In
particular, we have $t_0^{+}$ $<$ $t_0^{-}=1$. Since
\begin{equation*}
\frac{d}{dt}J(tu_0^{+},tv_0^{+})_{\downharpoonleft
t=t_0^{+}}=0\quad \text{and}\quad \frac{d^2}{dt^2}J(
tu_0^{+},tv_0^{+})_{\downharpoonleft t=t_0^{+}}>0,
\end{equation*}
there exists $t_0^{+}<$ $t^{-}\leq $ $t_0^{-}$ such that
$J(t_0^{+}u_0^{+},t_0^{+}v_0^{+})<J(t^{-}u_0^{+},t^{-}v_0^{+})$.
By Lemma \ref{lem8}, we obtain
\begin{equation*}
J(t_0^{+}u_0^{+},t_0^{+}v_0^{+})<J(
t^{-}u_0^{+},t^{-}v_0^{+})<J(
t_0^{-}u_0^{+},t_0^{-}v_0^{+})=J(
u_0^{+},v_0^{+}),
\end{equation*}
which contradicts the fact that $J(u_0^{+},v_0^{+})=c^{+}$.
Since $J(u_0^{+},v_0^{+})=J(|u_0^{+}|,|v_0^{+}|)$ and
 $(|u_0^{+}|,|v_0^{+}|)\in \mathcal{N}^{+}$, then by Lemma $\ref{lem5}$,
 we may assume that $(u_0^{+},v_0^{+})$ is a nontrivial nonnegative solution of
\eqref{eSllm}. By the Harnack
inequality, we conclude that $u_0^{+}>0$ and $v_0^{+}>0$, see for
example \cite{d3}.
\end{proof}

\section{Proof of Theorem \ref{thm2}}

Next, we establish the existence of a local minimum for $J$ on $\mathcal{N}^{-}$.
For this, we require the following Lemma.

\begin{lemma}\label{lem9}
For all $\lambda _1$, $\lambda _2$ such that
$0<(|\lambda _1||f_1|_{\mathcal{\infty }})^{1/(2-q)}+(|\lambda_2||f_2|_{\infty })^{1/(
2-q)}<(1/2)\Lambda _0$, the functional $J$ has a
minimizer $(u_0^{-},v_0^{-})$ in $\mathcal{N}^{-}$ and it
satisfies:
\begin{itemize}
\item[(i)] $J(u_0^{-},v_0^{-})=c^{-}>0$,

\item[(ii)] $(u_0^{-},v_0^{-})$ is a nontrivial
solution of \eqref{eSllm} in $\mathcal{H}$.
\end{itemize}
\end{lemma}

\begin{proof}
If $0<(|\lambda _1||f_1|_{\mathcal{\infty }})^{1/(2-q)}+(|
\lambda _2||f_2|_{\infty })^{1/(2-q)}<(1/2)\Lambda _0$, then by Proposition
\ref{prop1} (ii) there exists a $(u_{n},v_{n})_{n} $, $(PS)_{c^{-}}$ sequence
in $\mathcal{N}^{-}$, thus it
bounded by Lemma \ref{lem4}. Then, there exists $(u_0^{-},v_0^{-})\in \mathcal{H}$
and we can extract a subsequence
which will denoted by $(u_{n},v_{n})_{n}$ such that
\begin{gather*}
(u_{n},v_{n}) \rightharpoonup (u_0^{-},v_0^{-})\quad \text{weakly in }\mathcal{H} \\
(u_{n},v_{n}) \rightharpoonup (u_0^{-},v_0^{-})\quad \text{weakly in }
(L^{2_{\ast }}(\Omega ,|y|^{-2_{\ast }b}))^2 \\
(u_{n},v_{n}) \to (u_0^{-},v_0^{-}) \quad \text{strongly in }(L^q(\Omega ,|x|
^{-c}))^2 \\
u_{n} \to u_0^{-}\quad\text{a.e in }\Omega, \\
v_{n}\to v_0^{-}\quad \text{a.e in }\Omega .
\end{gather*}
This implies
$P(u_{n},v_{n})\to P(u_0^{-},v_0^{-})$, as $n\to \infty$.
Moreover, by (H2) and \eqref{r16} we obtain
\begin{equation}
P(u_{n},v_{n})>A(q)\|(u_{n},v_{n})\|_{\mu ,a}^2,  \label{r36}
\end{equation}
where, $A(q):=(2-q)/2_{\ast }(2_{\ast}-q)$. By \eqref{r18'} and \eqref{r36}
there exists a positive number
\begin{equation*}
C_1:=[A(q)K(\alpha ,\beta )]
^{2_{\ast }/(2_{\ast }-2)}(S_{\mu })^{2_{\ast}/(2_{\ast }-2)},
\end{equation*}
such that
\begin{equation}
P(u_{n},v_{n})>C_1.  \label{r36'}
\end{equation}
This implies $P(u_0^{-},v_0^{-})\geq C_1$.

Now, we prove that $(u_{n},v_{n})_{n}$ converges to
$(u_0^{-},v_0^{-})$ strongly in $\mathcal{H}$. Suppose otherwise.
Then, either $\|u_0^{-}\|_{\mu
,a}<\liminf_{n\to \infty }\|u_{n}\|
_{\mu ,a}$ or $\|v_0^{-}\|_{\mu ,a}<\liminf_{n\to \infty }\|v_{n}\|
_{\mu ,a}$. By Lemma $\ref{lem8}$ there is a unique $t_0^{-}$ such that
$(t_0^{-}u_0^{-},t_0^{-}v_0^{-})\in \mathcal{N}^{-}$.
Since
\begin{equation*}
(u_{n},v_{n})\in \mathcal{N}^{-},J(
u_{n},v_{n})\geq J(tu_{n},tv_{n}),\quad \text{for all }t\geq 0,
\end{equation*}
we have
\begin{equation*}
J(t_0^{-}u_0^{-},t_0^{-}v_0^{-})
<\lim_{n\to \infty }J(
t_0^{-}u_{n},t_0^{-}v_{n})\leq \lim_{n\to\infty }J(u_{n},v_{n})=c^{-},
\end{equation*}
and this is a contradiction. Hence,
$(u_{n},v_{n})_{n}\to (u_0^{-},v_0^{-})$  strongly in $\mathcal{H}$.
Thus,
\begin{equation*}
J(u_{n},v_{n})\text{ converges to }J(
u_0^{-},v_0^{-})=c^{-}\text{ as }n\to +\infty .
\end{equation*}
Since $J(u_0^{-},v_0^{-})=J(|u_0^{-}|,|v_0^{-}|)$ and
$(u_0^{-},v_0^{-})\in \mathcal{N}^{-}$, then by \eqref{r36'}
 and Lemma \ref{lem5}, we may assume that $(u_0^{-},v_0^{-})$
is a nontrivial nonnegative solution of \eqref{eSllm}.
 By the maximum principle, we conclude that $u_0^{-}>0$ and $v_0^{-}>0$.
\end{proof}

Now, we complete the proof of Theorem \ref{thm2}. By Propositions \ref{prop2}
and Lemma \ref{lem9}, we obtain that \eqref{eSllm} has two positive solutions
$(u_0^{+},v_0^{+})\in \mathcal{N}^{+}$ and
$(u_0^{-},v_0^{-})\in \mathcal{N}^{-}$. Since
$\mathcal{N}^{+}\cap \mathcal{N}^{-}=\emptyset $, this implies that
$(u_0^{+},v_0^{+})$ and $(u_0^{-},v_0^{-})$ are distinct.

\section{Proof of Theorem \ref{thm3}}

In this section, we consider the  Nehari submanifold of $\mathcal{N}$
\begin{equation*}
\mathcal{N}_{\varrho }=\{(u,v)\in \mathcal{H}\backslash
\{0,0\}:\langle J'(u,v),(u,v)\rangle =0\text{ and }\|(u,v)
\|_{\mu ,a}\geq \varrho >0\}.
\end{equation*}
Thus, $(u,v)\in \mathcal{N}_{\varrho }$ if and only if
\begin{equation*}
\|(u,v)\|_{\mu ,a}^2-2_{\ast }P(
u,v)-Q(u,v)=0\text{ and }\|(u,v)\|_{\mu ,a}\geq \varrho >0.
\end{equation*}

Firstly, we need the following Lemmas.

\begin{lemma} \label{lem1}
 Under the hypothesis of theorem \ref{thm3}, there exist
$\varrho _0$, $\Lambda _2>0$ such that $\mathcal{N}_{\varrho }$ is nonempty for
any $\lambda \in (0,\Lambda _2)$ and $\varrho \in (0,\varrho _0)$.
\end{lemma}

\begin{proof}
Fix $(u_0,v_0)\in \mathcal{H}\backslash \{0,0\}$ and let
\begin{align*}
g(t)&= \langle J'(tu_0,tv_0)
,(tu_0,tv_0)\rangle \\
&= t^2\|(u_0,v_0)\|_{\mu ,a}^2-2_{\ast }t^{2_{\ast }}P(u_0,v_0)-tQ(
u_0,v_0).
\end{align*}
Clearly $g(0)=0$ and $g(t)\to -\infty $
as $n\to +\infty $. Moreover, we have
\begin{align*}
g(1)&= \|(u_0,v_0)\|_{\mu
,a}^2-2_{\ast }P(u_0,v_0)-Q(u_0,v_0)\\
&\geq [\|(u_0,v_0)\|_{\mu
,a}^2-2_{\ast }[K(\alpha ,\beta )]^{-2_{\ast
}/2}(S_{\mu })^{-2_{\ast }/2}|h^{+}|
_{\infty }\|(u_0,v_0)\|_{\mu
,a}^{2_{\ast }}] \\
&\quad -((|\lambda _1||f_1|_{\mathcal{\infty }
})^{1/(2-q)}+(|\lambda _2||f_2|_{\infty })^{1/(2-q)})
\|(u_0,v_0)\|_{\mu ,a}.
\end{align*}
If $\|(u_0,v_0)\|_{\mu ,a}\geq \varrho>0 $ for
$$
0<\varrho <\varrho _0=(|h^{+}|_{\infty }2_{\ast }(2_{\ast }-1))^{-1/(2_{\ast
}-2)}([K(\alpha ,\beta )]S_{\mu
})^{2_{\ast }/2(2_{\ast }-2)},
$$
 $| h^{+}|_{\infty }\in (0,\alpha _0)$ for
 $\alpha _0=([K(\alpha ,\beta )]S_{\mu })
^{2_{\ast }/2}/(2_{\ast }(2_{\ast }-1))^{(
2_{\ast }-1)/2_{\ast }}$, then there exists
\begin{equation*}
\Lambda _2:=[(|h^{+}|_{\infty }2_{\ast
}(2_{\ast }-1))([K(\alpha ,\beta
)]S_{\mu })^{-2_{\ast }/2}]^{-1/(2_{\ast
}-2)}-\Theta \times \Phi ,
\end{equation*}
where
\begin{gather*}
\Theta :=(2_{\ast }(2_{\ast }-1))^{2_{\ast
}-1}((|h^{+}|_{\infty })^{2_{\ast
}/2}[K(\alpha ,\beta )]S_{\mu })^{-(2_{\ast })^2/2},
\\
\Phi :=[(|h^{+}|_{\infty }2_{\ast }(
2_{\ast }-1))([K(\alpha ,\beta )
]S_{\mu })^{-2_{\ast }/2}]^{-1/(2_{\ast}-2)}
\end{gather*}
and there exists $t_0>0$ such that $g(t_0)=0$. Thus,
$(t_0u_0,t_0v_0)\in \mathcal{N}_{\varrho }$ and
$\mathcal{N}_{\varrho }$ is nonempty for any
$\lambda \in (0,\Lambda_2)$.
\end{proof}

\begin{lemma}\label{lem10} There exist $M$, $\Lambda _1$ positive reals such that
\[
\langle \phi '(u,v),(u,v)
\rangle <-M<0
\]
 for $(u,v)\in \mathcal{N}_{\varrho }$
and any $\lambda _1,\lambda _2$ satisfying
\begin{equation*}
0<(|\lambda _1||f_1|_{
\mathcal{\infty }})^{1/(2-q)}+(|\lambda
_2||f_2|_{\mathcal{\infty }})
^{1/(2-q)}<\min ((1/2)\Lambda _0,\Lambda_1).
\end{equation*}
\end{lemma}

\begin{proof}
Let $(u,v)\in \mathcal{N}_{\varrho }$, then by \eqref{e13}, \eqref{r16}
 and the Holder inequality, allows us to write
\begin{align*}
&\langle \phi '(u,v),(u,v) \rangle \\
&\leq \|(u_{n},v_{n})\|_{\mu,a}^2[((|\lambda _1||
f_1|_{\mathcal{\infty }})^{1/(2-q)}+(|\lambda _2||f_2|_{\infty
})^{1/(2-q)})B(\varrho ,q)-(2_{\ast }-2)],
\end{align*}
where $B(\varrho ,q):=(2_{\ast }-1)(
C_{a,p})^q\varrho ^{q-2}$. Thus, if
\begin{equation*}
0<(|\lambda _1||f_1|_{\mathcal{\infty }})^{1/(2-q)}+(|\lambda
_2||f_2|_{\mathcal{\infty }})^{1/(2-q)}<\Lambda _{3}=[(2_{\ast }-2)
/B(\varrho ,q)],
\end{equation*}
and choosing $\Lambda _1:=\min (\Lambda _2,\Lambda _{3})$
with $\Lambda _2$ defined in Lemma \ref{lem1}, then we obtain that
\begin{equation}
\langle \phi '(u,v),(u,v)
\rangle <0\text{, for any }(u,v)\in \mathcal{N}_{\varrho }.  \label{in13}
\end{equation}
\end{proof}

\begin{lemma}\label{lem11}
Suppose $N\geq \max (3,6(a-b+1))$ and $\int_{\Omega }|x|^{-2_{\ast
}b}h|u|^{\alpha +1}|v|^{\beta+1}dx>0$. Then, there exist $r$ and $\eta $
positive constants such that
\begin{itemize}
\item[(i)] we have
\begin{equation*}
J(u,v)\geq \eta >0\text{ \ for }\|(u,v)
\|_{\mu ,a}=r.
\end{equation*}

\item[(ii)] there exists $(\sigma ,\omega )\in \mathcal{N}_{\varrho }$
when $\|(\sigma ,\omega )\|_{\mu ,a}>r$,
with $r=\|(u,v)\|_{\mu ,a}$, such that $J(\sigma ,\omega )\leq 0$.
\end{itemize}
\end{lemma}


\begin{proof}
We assume that the minima of $J$ are realized by $(u_0^{+},v_0^{+})$ and
$(u_0^{-},v_0^{-})$. The
geometric conditions of the mountain pass theorem are satisfied. Indeed, we
have

(i) By \eqref{r16}, \eqref{in13} and 
 $P(u,v)\leq [K(\alpha ,\beta )
]^{-2_{\ast }/2}(S_{\mu })^{-2_{\ast }/2}|
h^{+}|_{\infty }\|(u,v)\|_{\mu,a}^{2_{\ast }}$, we obtain
\begin{equation*}
J(u,v)\geq [(1/2)-(2_{\ast }-2)
/(2_{\ast }-q)q]\|(u,v)\|
_{\mu ,a}^2-C_2\|(u,v)\|_{\mu,a}^{2_{\ast }},
\end{equation*}
where $C_2=[K(\alpha ,\beta )]^{-2_{\ast
}/2}(S_{\mu })^{-2_{\ast }/2}|h^{+}|_{\infty }$
Using the function $l(x)=x(2_{\ast }-x)$ and
if $N\geq \max (3,6(a-b+1))$, we obtain that
 $[(1/2)-(2_{\ast }-2)/(2_{\ast}-q)q]>0$ for $1<q<2$. Thus, there exist $\eta $,
 $r>0$ such that
\begin{equation*}
J(u,v)\geq \eta >0\quad \text{when }r=\|(u,v)\|_{\mu ,a}\text{ is small.}
\end{equation*}

(ii) Let $t>0$. Then for all $(\phi ,\psi )\in \mathcal{N}_{\varrho }$
\begin{equation*}
J(t\phi ,t\psi ):=(t^2/2)\|(\phi,\psi )\|_{\mu }^2-(t^{2_{\ast }})P(
\phi ,\psi )-(t^q/q)Q(\phi ,\psi ).
\end{equation*}
Letting $(\sigma ,\omega )=(t\phi ,t\psi )$ for $t$ large enough. Since
\begin{equation*}
P(\phi ,\psi ):=\int_{\Omega }|x|^{-2_{\ast }b}h|\phi |^{\alpha +1}|\psi
|^{\beta +1}dx>0,
\end{equation*}
we obtain $J(\sigma ,\omega )\leq 0$. For $t$ large enough we can
ensure $\|(\sigma ,\omega )\|_{\mu ,a}>r$.
\end{proof}

Let $\Gamma $ and $c$ defined by
\begin{gather*}
\Gamma :=\{\gamma :[0,1]\to \mathcal{N}
_{\varrho }:\gamma (0)=(u_0^{-},v_0^{-}),\quad \gamma (1)=(u_0^{+},v_0^{+})\},\\
c:=\inf_{\gamma \in \Pi }\max_{t\in [0,1]}(J(\gamma (t))).
\end{gather*}

\begin{proof}[Proof of Theorem \ref{thm3}]
If
\begin{equation*}
(|\lambda _1||f_1|_{\mathcal{\infty }})^{1/(2-q)}+(|\lambda
_2||f_2|_{\mathcal{\infty }})^{1/(2-q)}<\min ((1/2)\Lambda _0,\Lambda _1),
\end{equation*}
then, by the Lemma \ref{lem4} and Proposition \ref{prop1} (ii), 
the function $J$ satisfying the Palais-Smale condition on $\mathcal{N}_{\varrho }$.
Moreover, from the Lemmas \ref{lem5}, \ref{lem10} and \ref{lem11}, there
exists $(u_{c},v_{c})$ such that
\begin{equation*}
J(u_{c},v_{c})=c\quad \text{and}\quad (u_{c},v_{c})\in \mathcal{N}_{\varrho }.
\end{equation*}
Thus $(u_{c},v_{c})$ is the third solution of our system such
that $(u_{c},v_{c})\neq (u_0^{+},v_0^{+})$ and
$(u_{c},v_{c})\neq (u_0^{-},v_0^{-})$. 
Since $(\mathcal{S}_{\lambda _1,\lambda _2,\mu })$ is odd with
respect $(u,v)$, we obtain that $(-u_{c},-v_{c})$
is also a solution of \eqref{eSllm}.
\end{proof}

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\end{thebibliography}

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