\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 $$\label{eSllm} \begin{gathered} -L_{\mu ,a}u= (\alpha +1)|x|^{-2_{\ast}b}h|u|^{\alpha -1}u|v|^{\beta+1} +\lambda _1|x|^{-c}f_1|u| ^{q-2}u\quad \text{in }\Omega \backslash \{0\}\\ -L_{\mu ,a}v= (\beta +1)|x|^{-2_{\ast}b}h|u|^{\alpha +1}|v|^{\beta -1}v+\lambda _2|x|^{-c}f_2|v| ^{q-2}v\quad\text{in }\Omega \backslash \{0\}\\ u=v=0\quad\text{on }\partial \Omega , \end{gathered}$$ 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, $-\infty \varrho _0$. \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 $-\infty 0$, $r>0$ 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 $$\|(u,v)\|_{\mu ,a}^2-2_{\ast }P(u,v)-Q(u,v)=0. \label{e13}$$ 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 $$\label{f14} \begin{split} J(u,v)&= ((2_{\ast }-2)/2_{\ast }2)\|(u,v)\|_{\mu ,a}^2-((2_{\ast}-q)/2_{\ast }q)Q(u,v) \\ &\geq ((2_{\ast }-2)/2_{\ast }2)\|(u,v)\|_{\mu ,a}^2 -(\frac{(2_{\ast }-q)}{2_{\ast }q})\Big((|\lambda _1||f_1|_{\mathcal{ \infty }})^{1/(2-q)}\\ &\quad +(|\lambda_2||f_2|_{\infty })^{1/(2-q)}\Big)(C_{a,p})^q\|(u,v)\|_{\mu ,a}^q. \end{split}$$ 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 $(u,v)\in \mathcal{N}$, $$\label{r16} \begin{split} \langle \phi '(u,v),(u,v)\rangle &= 2\|(u,v)\|_{\mu,a}^2-(2_{\ast })^2P(u,v)-qQ(u,v) \\ &= (2-q)\|(u,v)\|_{\mu,a}^2-2_{\ast }(2_{\ast }-q)P(u,v) \\ &= (2_{\ast }-q)Q(u,v)-(2_{\ast }-2)\|(u,v)\|_{\mu ,a}^2. \end{split}$$ 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 $$\label{e18} \|(u,v)\|_{\mu ,a}^2 = 2_{\ast }(2_{\ast }-q)/(2-q)P(u,v) = ((2_{\ast }-q)/(2_{\ast }-2))Q(u,v)$$ Moreover, by the H\"{o}lder inequality and the Sobolev embedding theorem, we obtain $$\|(u,v)\|_{\mu ,a}\geq [(S_{\mu })K(\alpha ,\beta )]^{2_{\ast }/2(2_{\ast }-2)}[(2-q)/2_{\ast }(2_{\ast }-q) |h^{+}|_{\infty }]^{-1/(2_{\ast }-2)} \label{r18'}$$ and $$\|(u,v)\|_{\mu ,a}\leq [(\frac{ 2_{\ast }-q}{2_{\ast }-2})^{-1/(2-q)}(( |\lambda _1||f_1|_{\mathcal{ \infty }})^{1/(2-q)}+(|\lambda _2||f_2|_{\infty })^{1/(2-q)})(C_{a,q})^q]. \label{e19}$$ 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[(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'} 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 $00, \end{equation*} there exists$t_0^{+}<t^{-}\leq t_0^{-}$such that$J(t_0^{+}u_0^{+},t_0^{+}v_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 $$P(u_{n},v_{n})>A(q)\|(u_{n},v_{n})\|_{\mu ,a}^2, \label{r36}$$ 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 $$P(u_{n},v_{n})>C_1. \label{r36'}$$ 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*} Clearlyg(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$(u,v)\in \mathcal{N}_{\varrho }, then by \eqref{e13}, \eqref{r16} and the Holder inequality, allows us to write \begin{align*} &\langle \phi '(u,v),(u,v) \rangle \\ &\leq \|(u_{n},v_{n})\|_{\mu,a}^2[((|\lambda _1|| f_1|_{\mathcal{\infty }})^{1/(2-q)}+(|\lambda _2||f_2|_{\infty })^{1/(2-q)})B(\varrho ,q)-(2_{\ast }-2)], \end{align*} whereB(\varrho ,q):=(2_{\ast }-1)( C_{a,p})^q\varrho ^{q-2}$. 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 $$\langle \phi '(u,v),(u,v) \rangle <0\text{, for any }(u,v)\in \mathcal{N}_{\varrho }. \label{in13}$$ \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$10$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} \begin{thebibliography}{99} \bibitem{a1} C. O. Alves, D. C. de Morais Filho, M. A. S. Souto; \emph{On systems of elliptic equations involving subcritical or critical Sobolev exponents}, Nonlinear Anal., 42 (2000) 771-787. \bibitem{b1} M. Bouchekif, M. E. O. El Mokhtar; \emph{On nonhomogeneous singular elliptic systems with critical Caffarelli-Kohn-Nirenberg exponent}, preprint Universit\'{e} de Tlemcen, (2011). \bibitem{b2} M. Bouchekif, Y. 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