\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 35, pp. 1--17.\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-2013/35\hfil Quasilinear elliptic systems] {Existence of solutions for quasilinear elliptic systems involving critical exponents and \\ Hardy terms} \author[D. L\"u \hfil EJDE-2013/35\hfilneg] {Dengfeng L\"u} % in alphabetical order \address{Dengfeng L\"u \newline School of Mathematics and Statistics, Hubei Engineering University, Hubei 432000, China} \email{dengfeng1214@163.com} \thanks{Submitted February 15, 2012. Published January 30, 2013.} \subjclass[2000]{35J92, 35J50, 35B33} \keywords{Quasilinear elliptic system; variational method; critical exponent; \hfill\break\indent Hardy term; multiple solutions} \begin{abstract} Using variational methods, including the Ljusternik-Schnirelmann theory, we prove the existence of solutions for quasilinear elliptic systems with critical Sobolev exponents and Hardy terms. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction and statement of main results} We consider the critical quasilinear elliptic system $$\label{1.1} \begin{gathered} -\Delta_p u-\mu\frac{|u|^{p-2}u}{|x|^p}=\frac{1}{p^{*}}F_{u}(u,v) +G_{u}(u,v), \quad x\in\Omega,\\ -\Delta_p v-\mu\frac{|v|^{p-2}v}{|x|^p}=\frac{1}{p^{*}}F_{v}(u,v) +G_{v}(u,v), \quad x\in\Omega,\\ u=v=0,\quad x\in\partial\Omega, \end{gathered}$$ where $\Omega\subset {\mathbb{R}}^N$ is a bounded domain with smooth boundary $\partial\Omega$, $0\in \Omega$, $\Delta_p u= \operatorname{div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplacian operator, $N\geq p^{2},2\leq p\leq q1,\alpha+\beta=p^{*}$. Using standard tools of the variational theory and the Ljusternik-Schnirelmann category theory, in \cite{9} sufficient conditions on $\lambda, \delta$ are given for \eqref{1.3} to have at least $\operatorname{cat}_{\Omega}(\Omega)$ positive solutions. This result extended the result of Alves and Ding in \cite{2} where the single equation case was studied. Hsu \cite{17} obtained the existence of two positive solutions for \eqref{1.3} including a sublinear perturbation of $10)$ holds for all $(u,v)\in {\mathbb{R}}^{+}\times {\mathbb{R}}^{+}$, \item[(F1)] $F_{u}(0,1)=F_{v}(1,0)=0$, \item[(F2)] $F_{u}(u,v)\geq 0,F_{v}(u,v)\geq 0$ for all $u,v\geq 0$, \item[(F3)] the 1-homogeneous function $(u,v)\mapsto F(u^{\frac{1}{p^{*}}},v^{\frac{1}{p^{*}}})$ is concave for all $(u,v)\in {\mathbb{R}}^{+}\times {\mathbb{R}}^{+}$. \item[(G0)] $G$ is $q$-homogeneous for some $p\leq q 0$; \item[(II)] $q=p$, $0\leq \mu\leq \frac{N^{p-1}(N-p^{2})}{p^p}$ and $\lambda,\delta\in(0,\frac{1}{p}\Lambda_1)$, where $\Lambda_1$ is the first eigenvalue of $(-\Delta_p,W^{1,p}_0(\Omega))$. \end{itemize} Then problem \eqref{1.1} has at least one nonnegative solution. \end{theorem} \begin{theorem}\label{thm2} Suppose {\rm (F0)--(F3), (G0)--(G1)} are satisfied, and one of the following two conditions holds: \begin{itemize} \item[(I)] $\bar{p}0$ such that problem \eqref{1.1} has at least $\operatorname{cat}_{\Omega}(\Omega)$ distinct nonnegative solutions for $\lambda,\delta\in (0,\Lambda)$. \end{theorem} \begin{remark}\label{bl-L2.4}\rm Our Theorem \ref{thm1} is a generalization of \cite[Theorem 1.1]{16} from quasilinear elliptic equations to quasilinear elliptic systems. \end{remark} \begin{remark} \rm %\label{bl-L2.4} Theorem 1 in \cite{9} is the special case of our Theorem \ref{thm2} corresponding to $\mu=0,F(u,v)=2|u|^{\alpha}|v|^{\beta},\alpha+\beta=p^{*}$ and $G(u,v)=\lambda|u|^{q}+\delta|v|^{q}$. In this paper, different from \cite{25}, we can deal with $F(u,v)$ which possesses both coupled and uncoupled terms. For example, let $$F(u,v)=au^{p^{*}}+\sum_{i=1}^{k}b_iu^{\alpha_i}v^{\beta_i}+cv^{p^{*}},$$ where $a,b_i,c \geq 0$, $\alpha_i,\beta_i>1$, $\alpha_i+\beta_i=p^{*}$. $F(u,v)$ obviously satisfies (F0)--(F3). \end{remark} This article is organized as follows. In Section 2, some notation and the mountain pass levels are established and Theorem \ref{thm1} is proven. We present some technical lemmas which are crucial in the proof of Theorem \ref{thm2} in Section 3. Theorem \ref{thm2} is proven in Section 4. \section{Preliminaries and proof of Theorem \ref{thm1}} Throughout this paper, $C,C_i$ will denote various positive constants whose exact values are not important. And $\to$ (respectively $\rightharpoonup$) denotes strong (respectively weak) convergence. $O(\varepsilon^{t})$ denotes $|O(\varepsilon^{t})|/\varepsilon^{t}\leq C,o_{ m}(1)$ denotes $o_{m}(1)\to 0$ as $m\to\infty$. $L^{s}(\Omega), for (1\leq s<+\infty)$, denotes Lebesgue spaces, the norm $L^{s}$ is denoted by $|\cdot|_{s}$ for $1\leq s<+\infty$. Let $B_{r}(x)$ denote a ball centered at $x$ with radius $r$. The dual space of a Banach space $E$ will be denoted by $E^{-1}$. We define the product space $E:= W^{1,p}_0(\Omega)\times W^{1,p}_0(\Omega)$ endowed with the norm $\|(u,v)\|_{E}=\big(\|u\|_{\mu}^p+ \|v\|_{\mu}^p\big)^{1/p}$. In view of (F1), (G1), we can extend the function $F(u,v)$ and $G(u,v)$ to the whole $\mathbb{R}^{2}$ by considering $F(u,v)=F(u^{+},v^{+})$, $G(u,v)=G(u^{+},v^{+})$, where $u^{+}=\max\{u,0\}$ and $v^{+}=\max\{v,0\}$. It is easy to check that $F(u,v)$ and $G(u,v)\in C^{1}(\mathbb{R}^{2})$. Therefore, we always consider $F(u,v)$ and $G(u,v)$ as these extensions. A pair of functions $(u,v)\in E$ is said to be a weak solution of problem \eqref{1.1} if \begin{align*} & \int_{\Omega}(|\nabla u|^{p-2}\nabla u\nabla\varphi_1 -\mu\frac{|u|^{p-2}u\varphi_1}{|x|^p} + |\nabla v|^{p-2}\nabla v\nabla\varphi_2-\mu\frac{|v|^{p-2}v\varphi_2}{|x|^p})dx\\ &- \frac{1}{p^{*}} \int_{\Omega}(F_{u}(u,v)\varphi_1+F_{v}(u,v)\varphi_2)dx - \int_{\Omega}(G_{u}(u,v)\varphi_1+G_{v}(u,v)\varphi_2)dx =0, \end{align*} for all $(\varphi_1,\varphi_2)\in E$. Using (F0)-(G1) and well-known arguments, we know that the weak solutions of \eqref{1.1} are precisely the critical points of the $C^{1}$-functional $I_{\lambda,\delta}: E \to \mathbb{R}$ given by \begin{align*} &I_{\lambda,\delta}(u,v)\\ &=\frac{1}{p}\int_{\Omega}(|\nabla u|^p-\mu\frac{|u|^p}{|x|^p}+|\nabla v|^p-\mu\frac{|v|^p}{|x|^p})dx-\frac{1}{p^{*}} \int_{\Omega}F(u,v)dx-\int_{\Omega} G_{\lambda,\delta}(u,v)dx. \end{align*} We notice that, in the definition of $I_{\lambda,\delta}$, we are denoting $G_{\lambda,\delta}(u,v):= G(u,v)$ for $(u,v)\in \mathbb{R}^{2}$. We shall write $G_{\lambda,\delta}$ instead of $G$ to emphasize that the main theorems depend on the value of the parameters $\lambda$ and $\delta$ defined in \eqref{1.4} and \eqref{1.5}, respectively. The functional $I\in C^{1}(E,{\mathbb{R}})$ is said to satisfy the $(PS)_{c}$ condition if any sequence $\{z_{m}\}\subset E$ such that as $m\to\infty$, $I(z_{m})\to c$, $I'(z_{m})\to 0$ strongly in $E^{-1}$ contains a subsequence converging in $E$ to a critical point of $I$. In this paper, we will take $I = I_{\lambda,\delta}$ and $E = W^{1,p}_0(\Omega)\times W^{1,p}_0(\Omega)$. In this section, we will find the range of $c$ where the $(PS)_{c}$ condition holds for the functional $I_{\lambda,\delta}$. First, let us define $$\label{2.1} S_{F}=\inf_{(u,v)\in E\setminus\{(0,0)\}}\Big\{\frac{\int_{\Omega}|\nabla u|^p-\mu\frac{|u|^p}{|x|^p}+|\nabla v|^p-\mu\frac{|v|^p}{|x|^p}dx}{(\int_{\Omega}F(u,v)dx)^{p/p^*}}: \int_{\Omega}F(u,v)dx>0\Big\}.$$ \begin{lemma}\label{lem2.1} Suppose {\rm (F0)--(F3), (G0)--(G1)} are satisfied, then the functional $I_{\lambda,\delta}$ satisfies the $(PS)_{c}$ condition for all $c<\frac{1}{N}S_{F}^{N/p}$, provided either $p0$ denotes the first eigenvalue of $(-\Delta_p , W^{1,p}_0(\Omega))$. \end{lemma} \begin{proof} Let $\{(u_{m},v_{m})\}\subset E$ such that $I'_{\lambda,\delta}(u_{m},v_{m})\to 0$ and $I_{\lambda,\delta}(u_{m},v_{m})\to c<\frac{1}{N}S_{F}^{N/p}$. Now, we firstly prove that $\{(u_{m},v_{m})\}$ is bounded in $E$. If $p0$ such that \begin{align*} c+C_1\|(u_{m},v_{m})\|_{E}+o_{m}(1) &\geq I_{\lambda,\delta}(u_{m},v_{m})-\frac{1}{q} \langle I'_{\lambda,\delta}(u_{m},v_{m}),(u_{m},v_{m})\rangle\\ &=\Big(\frac{1}{p}-\frac{1}{q}\Big)\|(u_{m},v_{m})\|_{E}^p +\Big(\frac{1}{q}-\frac{1}{p^{*}}\Big)\int_{\Omega}F(u_{m},v_{m})dx\\ &\geq \frac{q-p}{pq}\|(u_{m},v_{m})\|_{E}^p, \end{align*} which implies that $\{(u_{m},v_{m})\}\subset E$ is bounded. When $q=p$, in this case, it follows that $$\int_{\Omega}G_{\lambda,\delta}(u_{m},v_{m})dx \leq \lambda\int_{\Omega}(|u_{m}|^p+|v_{m}|^p)dx\leq \frac{\lambda}{\Lambda_1}\|(u_{m},v_{m})\|_{E}^p,$$ and therefore, \begin{align*} c+C_1\|(u_{m},v_{m})\|_{E}+o_{m}(1) &\geq I_{\lambda,\delta}(u_{m},v_{m})-\frac{1}{p^{*}} \langle I'_{\lambda,\delta}(u_{m},v_{m}),(u_{m},v_{m})\rangle\\ &=\frac{1}{N}\|(u_{m},v_{m})\|_{E}^p - \frac{p}{N}\int_{\Omega}G(u_{m},v_{m})dx\\ &\geq \frac{1}{N}\Big(1-\frac{p\lambda}{\Lambda_1}\Big) \|(u_{m},v_{m})\|_{E}^p. \end{align*} Since $p\lambda<\Lambda_1$, the boundedness of $\{(u_{m},v_{m})\}$ follows as in the first case. So $\{(u_{m},v_{m})\}$ is bounded in $E$. Going if necessary to a subsequence, we can assume that \begin{gather*} (u_{m},v_{m})\rightharpoonup(u,v), \quad \text{in } E,\\ (u_{m},v_{m})\to (u,v), \quad \text{a.e.\ in } \Omega,\\ (u_{m},v_{m})\to (u,v), \quad \text{in } L^{s}(\Omega)\times L^{s}(\Omega),1\leq s 0$, we conclude that$k\geq S_{F}^{N/p}$and therefore $$\frac{1}{N}S_{F}^{N/p}\leq \frac{1}{N}k\leq c<\frac{1}{N}S_{F}^{N/p},$$ which is a contradiction. Hence$k=0$and therefore$(u_{m},v_{m})\to (u,v)$in$E$. \end{proof} For all$\mu\in[0,\bar{\mu})$, we consider the limiting problem $$\label{2.7} \begin{gathered} -\Delta_p U-\mu\frac{U^{p-1}}{|x|^p}= U^{p^{*}-1}, \quad \text{in } \mathbb{R}^N\setminus\{0\},\\ U>0, \quad \text{in } \mathbb{R}^N\setminus\{0\},\\ U\to 0,\quad \text{as } |x|\to+\infty. \end{gathered}$$ From \cite{1}, we know that problem \eqref{2.7} has a ground state$U_{p,\mu}$, which is unique up to scaling. That is, all ground states must be of the form $$\label{2.8} V_{p,\mu,\varepsilon}(x) =\varepsilon^{-\frac{N-p}{p}}U_{p,\mu}\Big(\frac{x}{\varepsilon}\Big) =\varepsilon^{-\frac{N-p}{p}}U_{p,\mu}\Big(\frac{|x|}{\varepsilon}\Big), \quad \varepsilon>0,$$ that satisfy $$\label{2.9} \int_{\mathbb{R}^N}(|\nabla V_{p,\mu,\varepsilon}(x)|^p-\mu\frac{|V_{p,\mu,\varepsilon}(x)|^p}{|x|^p})dx =\int_{\mathbb{R}^N}|V_{p,\mu,\varepsilon}(x)|^{p^{*}}dx =S_{\mu}^{N/p},$$ where$S_{\mu}$is the best Sobolev constant given in \eqref{1.2}. Moreover, the ground state$U_{p,\mu}$is radially symmetric and decreasing, and the following asymptotic properties at the origin and infinity for$U_{p,\mu}(r)$and$U'_{p,\mu}(r)$hold: \begin{gather*} \lim_{r\to 0^{+}}r^{a(\mu)}U_{p,\mu}(r)=c_1>0, \quad \lim_{r\to 0^{+}}r^{a(\mu)+1}|U'_{p,\mu}(r)|=c_1a(\mu)\geq0,\\ \lim_{r\to +\infty}r^{b(\mu)}U_{p,\mu}(r)=c_2>0, \quad \lim_{r\to +\infty}r^{b(\mu)+1}|U'_{p,\mu}(r)|=c_2b(\mu)>0, \end{gather*} where$c_1$and$c_2$are positive constants depending only on$N,p,\mu$, and$a(\mu), b(\mu)$, the zeros of the function$h(t)=(p-1)t^p-(N-p)t^{p-1}+\mu, \ t\geq 0$, which satisfy$0\leq a(\mu)0$for all$t>t_{\rm min}$. That is,$h(t)$is decreasing on the interval$(0, t_{\rm min})$and increasing on the interval$(t_{\rm min},+\infty)$. Thus$0\leq a(\mu)<\frac{N-p}{p}0$such that $$S_{F}=\frac{\|(AV_{p,\mu,\varepsilon},BV_{p,\mu,\varepsilon}\|_{E}^p} {(\int_{\mathbb{R}^N} F(AV_{p,\mu,\varepsilon},BV_{p,\mu,\varepsilon})dx)^{p/p^*}} =\frac{A^p+B^p} {(F(A,B))^{p/p^*}}\cdot\frac{S_{\mu}^{N/p}}{|V_{p,\mu,\varepsilon}|_{p^{*}}^p},$$ from this and \eqref{2.9}, we have $$\label{2.10} S_{F}=\frac{A^p+B^p} {(F(A,B))^{p/p^*}}S_{\mu}.$$ We define a cut-off function$\phi(x)\in C_0^{\infty}(\mathbb{R}^N)$such that$\phi(x)=1$if$|x|\leq R$;$\phi(x)=0$if$|x|\geq 2R$and$0\leq\phi(x)\leq 1$, where$B_{2R}(0)\subset \Omega$and set$u_{\varepsilon}=\frac{\phi(x)V_{p,\mu,\varepsilon}}{|\phi V_{p,\mu,\varepsilon}|_{p^{*}}}, $where$V_{p,\mu,\varepsilon}$was defined in \eqref{2.8}. So,$|u_{\varepsilon}|_{p^{*}}=1$. Thus, we can get the following results from \cite[Lemma 2.2]{26} (or \cite{16}): \begin{gather}\label{2.11} \|u_{\varepsilon}\|_{\mu}^p=S_{\mu}+O(\varepsilon^{pb(\mu)+p-N}), \\ \int_{\Omega}|u_{\varepsilon}|^{\xi}dx\approx \begin{cases} \varepsilon^{(b(\mu)-\frac{N-p}{p})\xi}, & \text{if } 1\leq\xi<\frac{N}{b(\mu)},\\ \varepsilon^{N-\frac{N-p}{p}\xi}|\ln\varepsilon|, & \text{if } \xi=\frac{N}{b(\mu)},\\ \varepsilon^{N-\frac{N-p}{p}\xi}, & \text{if } \frac{N}{b(\mu)}<\xi0$ such that $$\label{2.13} \|(u,v)\|_{E}\geq \rho>0, \ \forall \ (u,v)\in \mathcal{N}_{\lambda,\delta}.$$ It is standard to check that $I_{\lambda,\delta}$ satisfies the mountain pass geometry, so we can use the homogeneity of $F$ and $G$ to prove that $c_{\lambda,\delta}$ can be alternatively characterized by $$\label{2.14} c_{\lambda,\delta}=\inf_{\gamma\in\Gamma}\max_{t\in[0,1]} I_{\lambda,\delta}(\gamma(t))=\inf_{(u,v)\in E\backslash \{(0,0)\}}\max_{t\geq 0}I_{\lambda,\delta}(t(u,v))>0,$$ where $\Gamma=\{\gamma\in C([0,1],E): \gamma(0)=0, I_{\lambda,\delta}(\gamma(1))<0\}$. Moreover, for each $(u,v)\in E \backslash \{(0,0)\}$, there exists a unique $t^{*}> 0$ such that $t^{*}(u,v) \in \mathcal{N}_{\lambda,\delta}$. The maximum of the function $t\mapsto I_{\lambda,\delta}(t(u,v))$, for $t\geq 0$, is achieved at $t=t^{*}$. \begin{lemma}\label{lem2.2} Suppose that $(F_0)-(F_3)$ and $(G_0)-(G_1)$ hold, $\bar{p}0$ when $t$ is close to $0$, there exists $t_{\varepsilon}>0$ such that $$\label{2.15} h(t_{\varepsilon})=\max_{t\geq 0}h(t).$$ Let $$g(t)=\frac{t^p}{p} (A^p+B^p)\|u_{\varepsilon}\|_{\mu}^p-\frac{t^{p^{*}}}{p^{*}}F(A,B), \quad t\geq 0,$$ and notice that the maximum value of $g(t)$ occurs at the point $$\tilde{t}_{\varepsilon} =\Big(\frac{(A^p+B^p)\|u_{\varepsilon}\|_{\mu}^p}{F(A,B)}\Big) ^{\frac{1}{p^{*}-p}}.$$ So, for each $t\geq 0$, $$g(t)\leq g(\tilde{t}_{\varepsilon}) =\frac{1}{N}\Big(\frac{(A^p+B^p)\|u_{\varepsilon}\|_{\mu}^p}{(F(A,B))^{p/p^*}}\Big) ^{N/p},$$ and therefore $$\label{2.16} h(t_{\varepsilon})\leq\frac{1}{N}\Big(\frac{(A^p+B^p) \|u_{\varepsilon}\|_{\mu}^p}{(F(A,B))^{p/p^*}}\Big) ^{N/p}-t_{\varepsilon}^{q}G_{\lambda,\delta}(A,B)|u_{\varepsilon}|_{q}^{q}.$$ We claim that, for some $C_2>0$, there holds $$t_{\varepsilon}^{q}G_{\lambda,\delta}(A,B)\geq C_2.$$ Indeed, if this is not the case, we have that $t_{\varepsilon_{m}} \to 0$ for some sequence $\varepsilon_{m}\to 0^{+}$, then $$0N-\frac{N-p}{p}q. Thus, from the above inequality we conclude that, for each \varepsilon>0 small, there holds$$ c_{\lambda,\delta}\leq \sup_{t\geq 0}I_{\lambda,\delta}(tAu_{\varepsilon},tBu_{\varepsilon}) =h(t_{\varepsilon})<\frac{1}{N}S_{F}^{N/p}. $$\noindent\textbf{Case 2:} q=p and 0\leq \mu\leq \frac{N^{p-1}(N-p^{2})}{p^p}. In this case, we have that h'(t)= 0 if and only if$$ (A^p+B^p)\|u_{\varepsilon}\|_{\mu}^p-pG_{\lambda,\delta}(A,B)|u_{\varepsilon}|_p^p =t^{p^{*}-p}F(A,B). Since we suppose \lambda<\frac{1}{p}\Lambda_1, we can use Poincar\'{e} inequality to obtain \begin{align*} pG_{\lambda,\delta}(A,B)|u_{\varepsilon}|_p^p &\leq p\lambda(A^p+B^p)|u_{\varepsilon}|_p^p \\ &<\Lambda_1(A^p+B^p)|u_{\varepsilon}|_p^p \\ &\leq(A^p+B^p)\|u_{\varepsilon}\|_{\mu}^p. \end{align*} Thus, there exists t_{\varepsilon}>0 satisfying \eqref{2.15}. Arguing, as in the first case, we conclude that, from \eqref{2.17}, for \varepsilon>0 small, there holds \begin{align*} h(t_{\varepsilon}) &\leq \frac{1}{N}S_{F}^{N/p}+O(\varepsilon^{pb(\mu)+p-N}) -C_2|u_{\varepsilon}|_p^p\\ &=\begin{cases} \frac{1}{N}S_{F}^{N/p}+O(\varepsilon^{pb(\mu)+p-N}) -O(\varepsilon^p|\ln\varepsilon|), & b(\mu)=\frac{N}{p},\\ \frac{1}{N}S_{F}^{N/p}+O(\varepsilon^{pb(\mu)+p-N}) -O(\varepsilon^p), & b(\mu)>\frac{N}{p}. \end{cases} \end{align*} If b(\mu)=N/p, then pb(\mu)+p-N=p, so \varepsilon^{pb(\mu)+p-N}=o(\varepsilon^p|\ln\varepsilon|). If b(\mu)>N/p, then pb(\mu)+p-N>p, so \varepsilon^{pb(\mu)+p-N}=o(\varepsilon^p). Choosing \varepsilon>0 small enough, we have c_{\lambda,\delta}\leq \sup_{t\geq 0}I_{\lambda,\delta}(tAu_{\varepsilon},tBu_{\varepsilon}) =h(t_{\varepsilon})<\frac{1}{N}S_{F}^{N/p}. $$On the other hand, it is easy to verify that the function$$ g(t)=(p-1)t^p-(N-p)t^{p-1}+\mu, \quad t\geq 0 $$has the only minimal point \bar{t}=\frac{N-p}{p} and is increasing on the interval (\bar{t},+\infty). Thus, for N\geq p^{2} we deduce that$$ \frac{N}{p}\leq b(\mu)\Leftrightarrow g(\frac{N}{p})\leq g(b(\mu))=0\Leftrightarrow 0\leq \mu\leq \frac{N^{p-1}(N-p^{2})}{p^p}. This concludes the proof. \end{proof} Using Lemmas \ref{lem2.1} and \ref{lem2.2}, we can prove our first result. \begin{proof}[Proof of Theorem \ref{thm1}] Since I_{\lambda,\delta} satisfies the geometric conditions of the mountain pass theorem, there exists \{(u_{m},v_{m})\}\subset E such that  I_{\lambda,\delta}(u_{m},v_{m})\to c_{\lambda,\delta}, and I'_{\lambda,\delta}(u_{m},v_{m})\to 0. It follows from Lemmas \ref{lem2.1} and \ref{lem2.2} that \{(u_{m},v_{m})\} converges, along a subsequence, to a nonzero critical point (u,v) \in E of I_{\lambda,\delta}. If we then denote, by u^{-}= \max\{-u,0\} and v^{-}= \max\{-v,0\}, the negative part of u and v, respectively, we obtain \begin{align*} 0&= \langle I'_{\lambda,\delta}(u,v),(u^{-},v^{-})\rangle\\ &=-\|(u^{-},v^{-})\|_{E}^p- \frac{1}{p^{*}}\int_{\Omega}(F_{u}(u,v)u^{-}+F_{v}(u,v)v^{-})dx\\ &\quad -\int_{\Omega}(G_{u}(u,v)u^{-}+G_{v}(u,v)v^{-})dx\\ &\leq-\|(u^{-},v^{-})\|_{E}^p. \end{align*} It thus follows that (u^{-},v^{-})=(0,0). Hence, u, v\geq 0 in \Omega. The theorem \ref{thm1} is thus proven. \end{proof} We finalize this section with the study of the asymptotic behavior of the minimax level c_{\lambda,\delta} as both the parameters \lambda,\delta approach zero. \begin{lemma}\label{lem2.3} \lim_{\lambda,\delta\to 0^{+}} c_{\lambda,\delta}=c_{0,0}=\frac{1}{N}S_{F}^{N/p}. \end{lemma} \begin{proof} We first prove the second equality. It follows from \lambda=\delta= 0 that G_{0,0}\equiv 0. If A, B, u_{\varepsilon}, g_{\varepsilon} and t_{\varepsilon} are the same as those in the proof of Lemma \ref{lem2.2}, we have that (t_{\varepsilon}Au_{\varepsilon},t_{\varepsilon}Bu_{\varepsilon}) \in \mathcal{N}_{0,0}. Thus \begin{align*} c_{0,0}&\leq I_{0,0} (t_{\varepsilon}Au_{\varepsilon},t_{\varepsilon}Bu_{\varepsilon})\\ &=\frac{1}{N}\Big(\frac{A^p+B^p}{(F(A,B))^{p/p^*}} \|u_{\varepsilon}\|_{\mu}^p\Big)^{N/p} \\ &= \frac{1}{N}\Big(\frac{A^p+B^p}{(F(A,B))^{p/p^*}} \Big(S_{\mu}+O\big(\varepsilon^{pb(\mu)+p-N}\big)\Big)\Big)^{N/p}. \end{align*} Taking the limit as \varepsilon\to 0^{+} and using \eqref{2.10}, we conclude that c_{0,0}\leq\frac{1}{N}S_{F}^{N/p}. In order to obtain the reverse inequality, we consider \{(u_{m},v_{m})\}\subset E such that I_{0,0}(u_{m},v_{m})\to c_{0,0} and I'_{0,0}(u_{m},v_{m})\to 0. It is easy to show that the sequence \{(u_{m},v_{m})\} is bounded in E and therefore $\langle I'_{0,0}(u_{m},v_{m}),(u_{m},v_{m})\rangle =\|(u_{m},v_{m})\|_{E}^p-\int_{\Omega}F(u_{m},v_{m})dx= o_{m}(1).$ It follows that \lim_{m\to\infty}\|(u_{m},v_{m})\|_{E}^p=l= \lim_{m\to\infty}\int_{\Omega}F(u_{m},v_{m})dx. Taking the limit in the inequality S_{F}(\int_{\Omega}F(u_{m},v_{m})dx)^{p/p^*}\leq \|(u_{m},v_{m})\|_{E}^p, we conclude, as in the proof of Lemma \ref{lem2.1}, that Nc_{0,0}=l\geq S_{F}^{N/p}. Hence, \begin{align*} c_{0,0}=\lim_{m\to\infty}I_{0,0}(u_{m},v_{m}) &=\lim_{m\to\infty} \Big(\frac{1}{p}\|(u_{m},v_{m})\|_{E}^p -\frac{1}{p^{*}}\int_{\Omega}F(u_{m},v_{m})dx\Big) \\ &=\frac{1}{N}l\geq \frac{1}{N}S_{F}^{N/p}, \end{align*} and therefore c_{0,0}=\frac{1}{N}S_{F}^{N/p}. We proceed now to the calculation of \lim_{\lambda,\delta\to 0^{+}} c_{\lambda,\delta}. Let \{\lambda_{m}\},\{\delta_{m}\}\subset \mathbb{R}^{+} such that \lambda_{m},\delta_{m}\to 0^{+}. Since \delta_{m}, defined in \eqref{1.5}, is positive, we have that G_{\lambda_{m},\delta_{m}}(u,v)\geq 0 whenever (u,v) is nonnegative. Thus, for this kind of function, we have that I_{\lambda_{m},\delta_{m}}(u,v)\leq I_{0,0}(u,v). It follows that \begin{align*} c_{\lambda_{m},\delta_{m}} &=\inf_{(u,v)\neq (0,0)} \max_{t\geq 0}I_{\lambda_{m},\delta_{m}}(t(u,v)) \\ &\leq \inf_{(u,v)\neq (0,0),\, (u,v)\geq 0} \max_{t\geq 0}I_{\lambda_{m},\delta_{m}}(t(u,v)) \\ &\leq \inf_{(u,v)\neq (0,0),\, (u,v)\geq 0} \max_{t\geq 0}I_{0,0}(t(u,v))=c_{0,0}. \end{align*} In the last equality above, we used the infimum c_{0,0}, which can be attained at a nonnegative solution. The above inequality implies that $$\limsup_{m\to\infty}c_{\lambda_{m},\delta_{m}}\leq c_{0,0}. \label{2.18}$$ On the other hand, it follows from Theorem \ref{thm1} that there exists \{(u_{m},v_{m})\}\subset E such that I_{\lambda_{m},\delta_{m}}(u_{m},v_{m})=c_{\lambda_{m},\delta_{m}}, \quad I'_{\lambda_{m},\delta_{m}}(u_{m},v_{m})\to 0. Since c_{\lambda_{m},\delta_{m}} is bounded, the same argument performed in the proof of Lemma \ref{lem2.1} implies that \{(u_{m},v_{m})\} is bounded in E. Since (u_{m},v_{m})\geq 0, we obtain 0\leq \int_{\Omega} G_{\lambda_{m},\delta_{m}}(u_{m},v_{m})dx\leq \lambda_{m}\int_{\Omega}(|u_{m}|^{q} + |v_{m}|^{q})dx, from which it follows that $$\lim_{m\to\infty}\int_{\Omega} G_{\lambda_{m},\delta_{m}}(u_{m},v_{m})dx=0. \label{2.19}$$ Let t_{m}>0 be such that t_{m}(u_{m},v_{m})\in \mathcal{N}_{0,0}. Since (u_{m},v_{m})\in \mathcal{N}_{\lambda_{m},\delta_{m}}, we have that \begin{align*} c_{0,0} &\leq I_{0,0}(t_{m}(u_{m},v_{m}))\\ &=I_{\lambda_{m},\delta_{m}}(t_{m}(u_{m},v_{m}))+t_{m}^{q}\int_{\Omega} G_{\lambda_{m},\delta_{m}}(u_{m},v_{m})dx \\ &\leq I_{\lambda_{m},\delta_{m}}(u_{m},v_{m})+t_{m}^{q}\int_{\Omega} G_{\lambda_{m},\delta_{m}}(u_{m},v_{m})dx \\ &=c_{\lambda_{m},\delta_{m}}+t_{m}^{q}\int_{\Omega} G_{\lambda_{m},\delta_{m}}(u_{m},v_{m})dx. \end{align*} If \{t_{m}\} is bounded, we can use the above estimate and \eqref{2.19} to obtain c_{0,0}\leq \liminf_{m\to\infty}c_{\lambda_{m},\delta_{m}}. $$Using this and \eqref{2.18}, we obtain$$ c_{0,0}\leq \liminf_{m\to\infty}c_{\lambda_{m},\delta_{m}}\leq \limsup_{m\to\infty}c_{\lambda_{m},\delta_{m}}\leq c_{0,0}. Thus, c_{0,0}=\lim_{m\to\infty }c_{\lambda_{m},\delta_{m}}. It remains to check that \{t_{m}\} is bounded. A straightforward calculation shows that $$t_{m}=\Big(\frac{\|(u_{m},v_{m})\|_{E}^p}{\int_{\Omega}F(u_{m},v_{m})dx} \Big)^{\frac{1}{p^{*}-p}}. \label{2.20}$$ Since (u_{m},v_{m})\in \mathcal{N}_{¦Ë\lambda_{m},\delta_{m}}, we obtain \begin{align*} &\|(u_{m},v_{m})\|_{E}^p\\ &=\int_{\Omega}F(u_{m},v_{m})dx+q\int_{\Omega} G_{\lambda_{m},\delta_{m}}(u_{m},v_{m})dx\leq S_{F}^{-\frac{p^{*}}{p}}\|(u_{m},v_{m})\|_{E}^{p^{*}}+o_{m}(1). \end{align*} Hence \|(u_{m},v_{m})\|_{E}^p\geq C_3>0, and therefore from the above expression, it follows that \int_{\Omega}F(u_{m},v_{m})dx\geq C_{4}>0. Thus, the boundedness of \{(u_{m},v_{m})\} and \eqref{2.20} imply that \{t_{m}\} is bounded. This completes the proof. \end{proof} \section{Some technical lemmas} In this section, we will recall and prove some lemmas which are crucial in the proof of Theorem \ref{thm2}. The first lemma is standard, and its proof follows adapting arguments found in \cite{27}. \begin{lemma}\label{lem3.1} Let \{(u_{m},v_{m})\}\subset E such that \int_{\Omega}F(u_{m},v_{m})dx=1 and $\lim_{m\to\infty}\|(u_{m},v_{m})\|_{E}^p=S_{F}.$ Then there exist \{r_{m}\}\subset(0,+\infty) and \{y_{m}\}\subset \mathbb{R}^N such that $$\label{3.1} \omega_{m}(x)=(\omega^{1}_{m}(x),\omega^{2}_{m}(x))=r_{m}^{\frac{N-p}{p}} (u_{m}(r_{m}x+y_{m}),v_{m}(r_{m}x+y_{m}))$$ contains a convergent subsequence, denoted again by \{\omega_{m}\}, such that \omega_{m}\to \omega in \mathcal{D}^{1,p}(\mathbb{R}^N)\times \mathcal{D}^{1,p}(\mathbb{R}^N). Moreover, as m\to\infty, we have r_{m}\to0 and y_{m}\to y\in \overline{\Omega}. \end{lemma} Up to translations, we may assume that 0\in\Omega. Since \Omega is a smooth bounded domain of \mathbb{R}^N, we can choose r>0 small enough such that B_{r}=B_{r}(0)=\{x\in \mathbb{R}^N:d(x,0)r\} are homotopically equivalent to $\Omega$. Let \begin{gather*} W_{0,rad}^{1,p}(B_{r})=\big\{u\in W_0^{1,p}(B_{r}):u \text{ is radial}\big\},\\ E_{\rm rad}(B_{r})=W_{0,{\rm rad}}^{1,p}(B_{r})\times W_{0,{\rm rad}}^{1,p}(B_{r}). \end{gather*} We thus define the functional \begin{align*} I_{B_{r}}(u,v) &=\frac{1}{p}\int_{B_{r}}(|\nabla u|^p-\mu\frac{|u|^p}{|x|^p}+|\nabla v|^p-\mu\frac{|v|^p}{|x|^p})dx\\ &\quad -\frac{1}{p^{*}}\int_{B_{r}} F(u,v)dx-\int_{B_{r}}G_{\lambda,\delta}(u,v)dx \end{align*} for $(u,v)\in E_{\rm rad}(B_{r})$, and set $$m_{\lambda,\delta}=\inf_{(u,v)\in \mathcal{N}^{B_{r}}_{\lambda,\delta}}I_{B_{r}}(u,v),$$ where $$\mathcal{N}^{B_{r}}_{\lambda,\delta}:=\{(u,v)\in E_{\rm rad}(B_{r})\setminus \{(0,0)\}:\langle I'_{B_{r}}(u,v),(u,v)\rangle=0\}.$$ Clearly, $m_{\lambda,\delta}$ is nonincreasing in $\lambda,\delta$. Note that $m_{\lambda,\delta}>0$ for all $\lambda,\delta>0$. Arguing, as in the proof of Lemma \ref{lem2.3} and Theorem \ref{thm1}, we obtain the following result. \begin{lemma}\label{lem3.2} Suppose {\rm (F0)-(F3), (G0)--(G1)} are satisfied. Then the infimum $m_{\lambda,\delta}$ is attained by a positive radial function $(u_{\lambda,\delta},v_{\lambda,\delta})\in E_{\rm rad}$ whenever $\bar{p}0$, or $q=p,0\leq \mu\leq \frac{N^{p-1}(N-p^{2})}{p^p}$ and $\lambda,\delta \in(0, \frac{1}{p}\Lambda_{1,rad})$, and where $\Lambda_{1,rad}> 0$ is the first eigenvalue of the operator $(-\Delta _pu, W^{1,p}_{0,rad}(B_{r}))$. Moreover, $$m_{\lambda,\delta}<\frac{1}{N}S_{F}^{N/p}, \quad \lim_{\lambda,\delta\to 0^{+}}m_{\lambda,\delta}=\frac{1}{N}S_{F}^{N/p}.$$ \end{lemma} We introduce the barycenter map $\beta:\mathcal{N}_{\lambda,\delta}\to \mathbb{R}^N$ as $$\beta(u,v)=S_{F}^{-N/p}\int_{\Omega}F(u,v)x\,dx.$$ This map has the following property. \begin{lemma}\label{lem3.3} If {\rm (F0)--(F3), (G0)--(G1)}, then there exists $\lambda^{*}>0$ such that $\beta(u,v)\in \Omega_{ r}^{+}$ whenever $(u,v)\in \mathcal{N}_{\lambda,\delta},\lambda,\delta\in(0,\lambda^{*})$ and $I_{\lambda,\delta}(u,v)\leq m_{\lambda,\delta}$. \end{lemma} \begin{proof} Arguing by contradiction, we suppose that there exist $\{\lambda_{m}\},\{\delta_{m}\}\subset \mathbb{R}^{+}$ and $\{(u_{m},v_{m})\}\subset \mathcal{N}_{\lambda_{m},\delta_{m}}$ such that $\lambda_{m},\delta_{m}\to 0^{+}$ as $m\to\infty, I_{\lambda_{m},\delta_{m}}(u_{m},v_{m})\leq m_{\lambda_{m},\delta_{m}}$, but $\beta(u_{m},v_{m})\not\in\Omega_{r}^{+}$. From $\{(u_{m},v_{m})\}\subset \mathcal{N}_{\lambda_{m},\delta_{m}}$ and $I_{\lambda_{m},\delta_{m}}(u_{m},v_{m})\leq m_{\lambda_{m},\delta_{m}}$, it follows that $\{(u_{m},v_{m})\}$ is bounded in $E$. Moreover, \begin{align*} 0&=\langle I'_{\lambda_{m},\delta_{m}}(u_{m},v_{m}),(u_{m},v_{m})\rangle\\ &= \|(u_{m},v_{m})\|_{E}^p-\int_{\Omega}F(u_{m},v_{m})dx -q\int_{\Omega}G_{\lambda_{m},\delta_{m}} (u_{m},v_{m})dx. \end{align*} Since $\lambda_{m}\to 0$, we can use the boundedness of $\{(u_{m},v_{m})\}$ to get $$0\leq \int_{\Omega}G_{\lambda_{m},\delta_{m}}(u_{m},v_{m})dx\leq \lambda_{m}\int_{\Omega}(|u_{m}|^{q}+|v_{m}|^{q})dx\to 0,$$ from which it follows that $$\lim_{m\to\infty}\|(u_{m},v_{m})\|_{E}^p = \lim_{m\to\infty}\int_{\Omega}F(u_{m},v_{m})dx=k\geq 0.$$ Notice that \begin{align*} c_{\lambda_{m},\delta_{m}} &\leq I_{\lambda_{m},\delta_{m}}(u_{m},v_{m})\\ &= \frac{1}{p}\|(u_{m},v_{m})\|_{E}^p-\frac{1}{p^{*}} \int_{\Omega}F(u_{m},v_{m})dx- \int_{\Omega}G_{\lambda_{m},\delta_{m}}(u_{m},v_{m})dx \\ &\leq m_{\lambda_{m},\delta_{m}}. \end{align*} Recalling that $c_{\lambda_{m},\delta_{m}}$ and $m_{\lambda_{m},\delta_{m}}$ both converge to $\frac{1}{N}S_{F}^{N/p}$, we can use the above expression and $\int_{\Omega}G_{\lambda_{m},\delta_{m}}(u_{m},v_{m})dx\to 0$ again to conclude that $k=S_{F}^{N/p}$. That is, $$\lim_{m\to\infty}\|(u_{m},v_{m})\|_{E}^p=S_{F}^{N/p}= \lim_{m\to\infty}\int_{\Omega}F(u_{m},v_{m})dx.\label{3.2}$$ Let $t_{m}=(\int_{\Omega}F(u_{m},v_{m})dx)^{-1/p^*}>0$ and notice that $t_{m}(u_{m},v_{m})$ satisfies the hypotheses of Lemma \ref{lem3.1}. Using Lemma \ref{lem3.1}, there exist sequences $\{r_{m}\}\subset (0,+\infty)$ and $\{y_{m}\}\subset \mathbb{R}^N$ satisfying $r_{m}\to 0, y_{m}\to y \in \overline{\Omega}$. We thus have that $\omega_{m} \to \omega$ in $\mathcal{D}^{1,p}(\mathbb{R}^N)\times \mathcal{D}^{1,p}(\mathbb{R}^N)$. The definition of $\beta(u,v)$, \eqref{3.2}, the strong convergence of $\{\omega_{m}\}$, and Lebesgue's Theorem provide \begin{align*} \beta(u_{m},v_{m}) &=t_{m}^{-p^{*}}S_{F}^{-N/p}\int_{\Omega}F(t_{m}(u_{m},v_{m}))xdx \\ &=(1+o_{m}(1))\int_{\Omega}F(t_{m}u_{m},t_{m}v_{m})xdx \\ &=(1+o_{m}(1))\int_{\Omega}F(\omega_{m})(r_{m}x+y_{m})dx \\ &=(1+o_{m}(1))\Big(\int_{\Omega}F(\omega)\bar{y}dx+o_{m}(1)\Big). \end{align*} Since $\bar{y}\in \overline{\Omega}$ and $\int_{\Omega}F(\omega)dx=1$, the above expression implies that $$\lim_{m\to\infty}\operatorname{dist} \ (\beta(u_{m},v_{m}),\overline{\Omega})=0.$$ Such contradicts $\beta(u_{m},v_{m})\not\in \Omega_{r}^{+}$. \end{proof} According to Lemma \ref{lem3.2}, for each $\lambda,\delta>0$ small, the infimum $m_{\lambda,\delta}$ is attained by a nonnegative radial function $\sigma_{\lambda,\delta}=(u_{\lambda,\delta},v_{\lambda,\delta})\in \mathcal{N}^{B_{r}}_{\lambda,\delta}$. We consider $$I_{\lambda,\delta}^{m_{\lambda,\delta}}=\{(u,v)\in E:I(u,v)\leq m_{\lambda,\delta}\}$$ and define the function $\gamma:\Omega_{r}^{-}\to I_{\lambda,\delta}^{m_{\lambda,\delta}}$ by setting, for each $y\in \Omega_{r}^{-}$, $$\gamma(y)= \begin{cases} \sigma_{\lambda,\delta}(x-y), & \text{if } x\in B_{r}(y),\\ 0, & \text{otherwise}. \end{cases}\label{3.3}$$ A change of variables and straightforward calculations show that the map $\gamma$ is well defined. Since $\sigma_{\lambda,\delta}$ is radial, we have that $\int_{B_{r}} F(u_{\lambda,\delta},v_{\lambda,\delta})xdx=0$. Hence, for each $y\in \Omega_{r}^{-}$, we obtain \begin{align*} (\beta\circ\gamma)(y) &=S_{F}^{-N/p}\int_{\Omega} F(u_{\lambda,\delta}(x-y),v_{\lambda,\delta}(x-y))xdx \\ &=S_{F}^{-N/p}\int_{\Omega} F(u_{\lambda,\delta}(t),v_{\lambda,\delta}(t))(t+y)dt \\ &=S_{F}^{-N/p}\int_{\Omega} F(u_{\lambda,\delta}(t),v_{\lambda,\delta}(t))ydt = y\alpha_{\lambda,\delta}, \end{align*} where $\alpha_{\lambda,\delta}=S_{F}^{-N/p} \int_{\Omega} F(u_{\lambda,\delta}(t),v_{\lambda,\delta}(t))dt$. Along the way of proving Lemma \ref{lem3.3}, we can check easily the following. \begin{lemma}\label{lem3.4} If $\lambda,\delta\to 0^{+}$, then $\alpha_{\lambda,\delta}\to 1$. \end{lemma} \begin{proof} By Lemma \ref{lem3.2}, we have \begin{align*} m_{\lambda,\delta} &= \frac{1}{p}\int_{B_{r}}\Big(|\nabla u_{\lambda,\delta}|^p+|\nabla v_{\lambda,\delta}|^p-\mu\frac{|u_{\lambda,\delta}|^p+|v_{\lambda,\delta}|^p}{|x|^p}\Big)dx \\ &\quad -\frac{1}{p^{*}}\int_{B_{r}}F(u_{\lambda,\delta}, v_{\lambda,\delta})dx- \int_{B_{r}}G_{\lambda,\delta}(u_{\lambda,\delta}, v_{\lambda,\delta})dx \\ & < \frac{1}{N}S_{F}^{N/p}. \end{align*} As before, $\int_{B_{r}}G_{\lambda,\delta}(u_{\lambda,\delta}, v_{\lambda,\delta})dx\to 0$. Thus, $I'_{B_{r}}(u_{\lambda,\delta}, v_{\lambda,\delta})=0$, and the above expression and the same arguments used in the proof of Lemma \ref{lem3.2} imply that $$\int_{\Omega}F(u_{\lambda,\delta}, v_{\lambda,\delta})dx\to S_{F}^{N/p}.$$ The above equality and the definition of $\alpha_{\lambda,\delta}$ imply that $\alpha_{\lambda,\delta}\to 1$. The lemma is thus proven. \end{proof} Next we define $H_{\lambda,\delta}:[0,1]\times (\mathcal{N}_{\lambda,\delta}\cap I_{\lambda,\delta}^{m_{\lambda,\delta}})\to \mathbb{R}^N$ by $$H_{\lambda,\delta}(t,(u,v))=\Big(t+\frac{1-t}{\alpha_{\lambda,\delta}}\Big) \beta(u,v).$$ \begin{lemma}\label{lem3.5} Suppose {\rm (F0)--(F3), (G0)--(G1)} are satisfied. There then exists $\lambda^{**}>0$ such that $$H_{\lambda,\delta}\big([0,1]\times (\mathcal{N}_{\lambda,\delta}\cap I_{\lambda,\delta}^{m_{\lambda,\delta}})\big)\subset \Omega_{r}^{+} \label{3.4}$$ for all $\lambda,\delta\in (0,\lambda^{**})$. \end{lemma} \begin{proof} Arguing by contradiction, we suppose that there exist sequences $\{\lambda_{m}\}$, $\{\delta_{m}\}\subset \mathbb{R}^{+}$ and $t_{m}\in [0,1], (u_{m},v_{m})\in (\mathcal{N}_{\lambda,\delta}\cap I_{\lambda,\delta}^{m_{\lambda,\delta}})$ such that $\lambda_{m},\delta_{m}\to 0^{+}$ as $m\to\infty$ and $H_{\lambda_{m},\delta_{m}}(t_{m},(u_{m},v_{m}))\not\in\Omega_{r}^{+}$ for all $m$, up to a subsequence $t_{m}\to t_0\in[0,1]$. Moreover, the compactness of $\overline{\Omega}$ and Lemma \ref{lem3.3} imply that, up to a subsequence, $\beta(u_{m},v_{m})\to y\in \overline{\Omega}$. From Lemma \ref{lem3.4} $\alpha_{\lambda_{m},\delta_{m}}\to 1$, so we can use the definition of $H_{\lambda,\delta}$ to conclude that $H_{\lambda_{m},\delta_{m}}(t_{m},(u_{m},v_{m}))\to y\in \overline{\Omega}$, which is a contradiction. The lemma is proven. \end{proof} \section{Proof of Theorem \ref{thm2}} We begin with the following lemma. \begin{lemma}\label{lem4.1} If $(u,v)$ is a critical point of $I_{\lambda,\delta}$ on $\mathcal{N}_{\lambda,\delta}$, then it is a critical point of $I_{\lambda,\delta}$ in $E$. \end{lemma} \begin{proof} The proof is almost the same as \cite[Lemma 3.2]{22} and is thus omitted here. \end{proof} \begin{lemma}\label{lem4.2} Suppose {(F0)--(F3), (G0)--(G1)} are satisfied. Then any sequence $\{(u_{m},v_{m})\}\subset \mathcal{N}_{\lambda,\delta}$ such that $I_{\lambda,\delta}(u_{m},v_{m})\to c<\frac{1}{N}S_{F}^{N/p}$ and $I'_{\lambda,\delta}(u_{m},v_{m})\to 0$ contains a convergent subsequence for $\lambda,\delta>0$ if $q>p$ and $\lambda,\delta\in (0,\lambda^{*})$ if $q=p$ for some small $\lambda^{*}>0$. \end{lemma} \begin{proof} By hypothesis, there exists a sequence $\theta_{m}\in \mathbb{R}$ such that $\|I'_{\lambda,\delta}(u_{m},v_{m}) -\theta_{m}J'_{\lambda,\delta}(u_{m},v_{m})\|_{E} \to 0$ as $m\to\infty$, where $J_{\lambda,\delta}(u,v)=\int_{\Omega}(|\nabla u|^p-\mu\frac{|u|^p}{|x|^p}+|\nabla v|^p-\mu\frac{|v|^p}{|x|^p})dx-\int_{\Omega}F(u,v)dx -q\int_{\Omega}G_{\lambda,\delta}(u,v)dx.$ Thus, $$I'_{\lambda,\delta}(u_{m},v_{m}) =\theta_{m}J'_{\lambda,\delta}(u_{m},v_{m})+o_{m}(1).$$ Recall that for all $(u_{m},v_{m})\in \mathcal{N}_{\lambda,\delta}$, $$\langle J'_{\lambda,\delta}(u_{m},v_{m}),(u_{m},v_{m})\rangle= (p-p^{*})\int_{\Omega}F(u_{m},v_{m})dx+ (p-q)\int_{\Omega}G_{\lambda,\delta}(u_{m},v_{m})dx\leq 0.$$ If $\langle J'_{\lambda,\delta}(u_{m},v_{m}),(u_{m},v_{m})\rangle\to 0$, we have $$\int_{\Omega}F(u_{m},v_{m})dx\to 0, \quad \int_{\Omega}G_{\lambda,\delta}(u_{m},v_{m})dx\to 0.$$ Consequently, $\|(u_{m},v_{m})\|_{E}\to 0$. On the other hand, if $(u_{m},v_{m})\in \mathcal{N}_{\lambda,\delta}$, it follows that $$1\leq C(\lambda\|(u_{m},v_{m})\|_{E}^{q-p}+\delta\|(u_{m},v_{m})\|_{E}^{q-p}+ \|(u_{m},v_{m})\|_{E}^{p^{*}-p})$$ for some $C>0$. Hence we arrive at a contradiction if $\lambda,\delta> 0$ and $q>p$ or $\lambda,\delta\in (0,\lambda^{*})$ for small $\lambda^{*}> 0$ when $q=p$. We may thus assume that $\langle J'_{\lambda,\delta}(u_{m},v_{m}),(u_{m},v_{m})\rangle\to \ell< 0$. Since $\langle I'_{\lambda,\delta}(u_{m},v_{m}),(u_{m},v_{m})\rangle=0$, we conclude that $\theta_{m}=0$ and, consequently, $I'_{\lambda,\delta}(u_{m},v_{m})\to 0$. Using this fact, we have $$I'_{\lambda,\delta}(u_{m},v_{m})\to c<\frac{1}{N}S_{F}^{N/p} \quad \text{and} \quad I'_{\lambda,\delta}(u_{m},v_{m})\to 0.$$ By Lemma \ref{lem2.1} the proof is completed. \end{proof} Hereafter, we denote the restriction of $I_{\lambda,\delta}$ on $\mathcal{N}_{\lambda,\delta}$ by $I_{\mathcal{N}_{\lambda,\delta}}$. \begin{lemma}\label{lem4.3} If {\rm (F0)--(F3), (G0)--(G1)} are satisfied. Let $\Lambda=\min\{\lambda^{*},\lambda^{**}\}>0$, $\lambda,\delta\in (0,\Lambda)$. Then $\operatorname{cat}_{I_{\mathcal{N}_{\lambda,\delta}}^{m_{\lambda,\delta}}} (I_{\mathcal{N}_{\lambda,\delta}}^{m_{\lambda,\delta}})\geq \operatorname{cat}_{\Omega}(\Omega)$, where $\lambda^{*},\lambda^{**}$ are given by Lemma \ref{lem3.3} and \ref{lem3.5}, respectively. \end{lemma} \begin{proof} Suppose that $I_{\mathcal{N}_{\lambda,\delta}}^{m_{\lambda,\delta}} =A_1\cup A_2\cup\cdots\cup A_{m}$, where $A_j, j=1,2,\cdots,m$, are closed and contractible sets in $I_{\mathcal{N}_{\lambda,\delta}}^{m_{\lambda,\delta}}$, this means that there exists $h_j\in C([0,1]\times A_j,I_{\mathcal{N}_{\lambda,\delta}} ^{m_{\lambda,\delta}})$ such that $$h_j(0,z)=z, \quad h_j(1,z)=\vartheta, \quad \text{for all } z\in A_j,$$ where $\vartheta\in A_j$ is fixed. Consider $B_j=\gamma^{-1}(A_j)$, $1\leq j\leq m$. The sets $B_j$ are closed and $$\Omega_{r}^{-}=B_1\cup B_2\cup\cdots\cup B_{m}.$$ We define the deformation $g_j:[0,1]\times B_j$ by setting $$g_j(t,y)=H_{\lambda,\delta}(t,h_j(t,\gamma(y)))$$ for $\lambda,\delta\in (0,\Lambda)$. Note that $$g_j(0,y)=H_{\lambda,\delta}(0,h_j(0,\gamma(y)))= \frac{(\beta\circ\gamma)(y)}{\alpha_{\lambda,\delta}}$$ implies $$g_j(0,y)=\frac{\alpha_{\lambda,\delta}y}{\alpha_{\lambda,\delta}}=y, \quad \text{for all } y\in B_j,$$ and $g_j(1,y)=H_{\lambda,\delta}(1,h_j(1,\gamma(y))) =\beta(h_j(1,\gamma(y)))$ implies $$g_j(1,y)=\beta(\vartheta)\in \Omega_{r}^{+}.$$ Thus $B_j$ are contractible in $\Omega_{r}^{+}$. Hence $\operatorname{cat}_{\Omega}(\Omega) =\operatorname{cat}_{\Omega_{r}^{+}}(\Omega_{r}^{+})\leq m$.\end{proof} \begin{proof}[Proof of Theorem \ref{thm2}] Using Lemma \ref{lem2.1}, Lemma \ref{lem2.2}, and Lemma \ref{lem3.2} we know that $c_{\lambda,\delta}, m_{\lambda,\delta}<\frac{1}{N}S_{F}^{N/p}$ for $\lambda,\delta\in (0,\Lambda)$. Moreover, by Lemma \ref{lem4.2}, $I_{\mathcal{N}_{\lambda,\delta}}$ satisfies the $(PS)_{c}$ condition for all $c<\frac{1}{N}S_{F}^{N/p}$. Therefore, by Lemma \ref{lem4.3}, a standard deformation argument implies that for $\lambda,\delta\in (0,\Lambda)$, $I_{\mathcal{N}_{\lambda,\delta}}$ contains at least $\operatorname{cat}_{\Omega}(\Omega)$ critical points of the restriction of $I_{\lambda,\delta}$ on $\mathcal{N}_{\lambda,\delta}$. Now, Lemma \ref{lem4.1} implies that $I_{\lambda,\delta}$ has at least $\operatorname{cat}_{\Omega}(\Omega)$ critical points, and therefore at least $\operatorname{cat}_{\Omega}(\Omega)$ nontrivial solutions of \eqref{1.1}. As Theorem \ref{thm1}, the obtained solutions are nonnegative in $\Omega$. The proof is completed. \end{proof} \subsection*{Acknowledgments} The author is indebted to the anonymous referee for the valuable comments and suggestions. 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