\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2018 (2018), No. 86, pp. 1--15.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2018 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2018/86\hfil Schr\"odinger-Newton system with singularity] {Multiple positive solutions for a Schr\"odinger-Newton system with singularity \\ and critical growth} \author[C.-Y. Lei, H.-M. Suo, C.-M. Chu \hfil EJDE-2018/86\hfilneg] {Chun-Yu Lei, Hong-Min Suo, Chang-Mu Chu} \address{Chun-Yu Lei (corresponding author) \newline School of Sciences, Guizhou Minzu University, Guiyang 550025, China} \email{leichygzu@sina.cn} \address{Hong-Min Suo \newline School of Sciences, Guizhou Minzu University, Guiyang 550025, China} \email{11394861@qq.com} \address{Chang-Mu Chu \newline School of Sciences, Guizhou Minzu University, Guiyang 550025, China} \email{372382190@qq.com} \dedicatory{Communicated by Paul H. Rabinowitz} \thanks{Submitted January 16, 2018. Published April 10, 2018.} \subjclass[2010]{35J60, 35J75, 35J30} \keywords{Schr\"odinger-Newton system; critical exponent; singularity} \begin{abstract} In this work, we study a class of Schr\"odinger-Newton systems with singular and critical growth terms in unbounded domains. By using the variational methods and the Br\'ezis-Lieb \cite{BL} classical technique, the existence and multiplicity of positive solutions are established. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \allowdisplaybreaks \section{Introduction and statement of main result} In this work, we are concerned with the existence and multiplicity of positive solutions to the Schr\"odinger-Newton system \begin{equation}\label{1.1} \begin{gathered} -\Delta u= \lambda g(x)u^{-\gamma}+\phi |u|^{2^*-3}u, \quad \text{in } \mathbb{R}^N, \\ -\Delta\phi=|u|^{2^*-1}, \quad \text{in } \mathbb{R}^N, \\ u>0, \quad \text{in } \mathbb{R}^N, \end{gathered} \end{equation} where $N\geq3$, $\gamma\in(0,1)$ and $\lambda>0$ is a real parameter and $g\in L^{\frac{2^*}{2^*+\gamma-1}}(\mathbb{R}^N)$ is a nonnegative function. This system is derived from the Schr\"odinger-Poisson system \begin{equation}\label{1.2} \begin{gathered} -\Delta u+V(x)u+\eta\phi f(u)= h(x,u), \quad \text{in } \mathbb{R}^3, \\ -\Delta\phi=2F(u), \quad \text{in }\ \mathbb{R}^3. \end{gathered} \end{equation} Systems as \eqref{1.2} have been studied extensively by many researchers because \eqref{1.2} has a strong physical meaning, which describes quantum particles interacting with the electromagnetic field generated by the motion. For more details as regards the physical relevance of the Schr\"odinger-Poisson system, we refer to \cite{AR, BF, RD}. System \eqref{1.2} has been extensively studied after the seminal work of Benci and Fortunato \cite{BF}. Many important results concerning existence of positive solutions, ground state solutions and multiplicity of solutions, least energy solutions, and so on, have been reported; see for instance \cite{AD, AV, BM, GG,CG,DM, DV,HC,LS,LY,LZ,LH,LG,SJ,SN,SM,ZJ3,ZQ,ZLZ} and the references therein. There are some references which investigated Schr\"odinger-Poisson systems involving the critical growing nonlocal term, such as \cite{AD,AV,LS,LH}. Precisely, in bounded domains, the system \begin{gather*} -\Delta u+\varepsilon q\phi f(u)= \eta|u|^{p-1}u, \quad \text{in }\Omega, \\ -\Delta\phi=2qF(u), \quad \text{in }\Omega,\\ u=\phi=0,\quad \text{in }\partial\Omega, \end{gather*} was considered in \cite{LS}, where $\Omega\subset\mathbb{R}^N$ ($N\geq3$) is a smooth bounded domain, and the existence and multiplicity results were established when $f$ a subcritical growth condition or the critical growth case by using the methods of a cut-off function and the variational arguments. In \cite{AV}, the following system involving the critical growing nonlocal term was also considered \begin{gather*} -\Delta u=\lambda u+\phi|u|^{2^*-3}u, \quad \text{in }\Omega, \\ -\Delta\phi=|u|^{2^*-1}, \quad \text{in }\Omega,\\ u=\phi=0,\quad \text{in }\partial\Omega. \end{gather*} They proved the existence and nonexistence results of positive solutions when $N=3$ and existence of solutions in both the resonance and the non-resonance case for higher dimensions. Specially, in unbounded domains, Liu \cite{LH} studied the system \begin{gather*} -\Delta u+V(x)u= K(x)\phi|u|^3u+h(x,u), \quad \text{in }\mathbb{R}^3, \\ -\Delta\phi=K(x)|u|^5, \quad \text{in }\mathbb{R}^3, \end{gather*} where $V, K, h$ are asymptotically periodic functions, and a positive solution was obtained by using variational methods. Recently, in a bounded domain, in \cite{ZQ}, the following system involving weak singularity was studied \begin{gather*} -\Delta u+\eta\phi u= \mu u^{-\gamma}, \quad \text{in }\Omega, \\ -\Delta\phi=u^2, \quad \text{in }\Omega, \\ u>0, \quad \text{in }\Omega, \\ u=\phi=0, \quad \text{on }\partial\Omega. \end{gather*} The existence, uniqueness and multiplicity of positive solutions for the above system are obtained in the case when $\eta=\pm1$ by employing the Nehari manifold. In bounded domains, the singular semilinear elliptic problem \begin{gather*} -\Delta u= \lambda f(x)u^{-\gamma}+\mu h(x)u^p, \quad \text{in }\Omega, \\ u>0, \quad \text{in }\Omega,\\ u=0, \quad \text{on }\partial\Omega, \end{gather*} has been extensively studied. For example, Yang \cite{YH} obtained the multiplicity positive solutions by combining variational and sub-supersolution methods when $0<\gamma<10$. When $\Omega=\mathbb{R}^N$, we should mention that semilinear elliptic equations involving singular and subcritical growth terms have been dealt with by a number of authors, see for example, \cite{GS, SYL} and the references therein. Motivated by the above facts, to the best of our knowledge, there are no results on the multiplicity of positive solutions for Schr\"odinger-Newton system involving critical and weak singular nonlinearities on unbounded domains. We shall give a positive answer to this question. Our main result reads as follows. \begin{theorem} \label{thm1.1} Assume that $\gamma\in(0,1)$. Then there exists $\lambda_{\ast}>0$ such that for any $\lambda\in(0,\lambda_{\ast})$, system \eqref{1.1} has at least two positive solutions $(u, \phi_u)\in D^{1,2}(\mathbb{R}^N)\times D^{1,2}(\mathbb{R}^N)$, and one of the solutions is a positive ground state solution. \end{theorem} Throughout this paper, we use the following notation: \begin{itemize} \item The space $D^{1,2}(\mathbb{R}^N)=\{u\in L^{2^*}(\mathbb{R}^N): \frac{\partial u}{\partial x_i}\in L^2(\mathbb{R}^N)\}$ endowed with the norm $\|u\|^2=\int_{\mathbb{R}^N}|\nabla u|^2dx$. The norm in $L^p(\mathbb{R}^N)$ is denoted by $|\cdot|_{p}$; \item $C, C_1, C_2,\dots$ denote various positive constants, which may vary from line to line; \item Let $S$ be the best constant for Sobolev embedding $D^{1,2}(\mathbb{R}^N)\hookrightarrow L^{2^*}(\mathbb{R}^N)$, namely \begin{equation*} S=\inf_{u\in D^{1,2}(\mathbb{R}^N)\backslash\{0\}} \frac{\int_{\mathbb{R}^N}|\nabla u|^2dx} {(\int_{\mathbb{R}^N}|u|^{2^*}dx)^{\frac{2}{2^*}}}. \end{equation*} \end{itemize} \section{Existence of the first positive solution of system \eqref{1.1}} The energy functional associated with system \eqref{1.1} is defined as \begin{align*} I_\lambda(u) &= \frac{1}{2}\|u\|^2-\frac{\lambda}{1-\gamma}\int_{\mathbb{R}^N}g(x) |u|^{1-\gamma}dx-\frac{1}{2(2^*-1)}\int_{\mathbb{R}^N}\phi_{u}|u|^{2^*-1}dx\\ &= \frac{1}{2}\|u\|^2-\frac{\lambda}{1-\gamma}\int_{\mathbb{R}^N}g(x) |u|^{1-\gamma}dx-\frac{1}{2(2^*-1)}\int_{\mathbb{R}^N}|\nabla\phi_{u}|^{2}dx. \end{align*} In general, a function $u\in D^{1,2}(\mathbb{R}^N)$ is called a solution of system \eqref{1.1}, that is $(u, \phi_u)$ is a solution of system \eqref{1.1} and $u>0$ enjoying \begin{equation*} \int_{\mathbb{R}^N}(\nabla u,\nabla v)dx -\lambda\int_{\mathbb{R}^N}g(x)u^{-\gamma}vdx -\int_{\mathbb{R}^N}\phi_u|u|^{2^*-3}uvdx=0, \quad \forall v\in D^{1,2}(\mathbb{R}^N). \end{equation*} It is well known that the singular term leads to the non-differentiability of the functional $I_\lambda$ on $D^{1,2}(\mathbb{R}^N)$, therefore system \eqref{1.1} cannot be considered by using critical point theory directly. In order to obtain the multiple positive solutions of system \eqref{1.1}, we consider a set $$ \mathcal{N}_\lambda=\Big\{u\in D^{1,2}(\mathbb{R}^N) : \|u\|^2-\lambda\int_{\mathbb{R}^N}g(x)|u|^{1-\gamma}dx -\int_{\mathbb{R}^N}\phi_{u}|u|^{2^*-1}dx=0\Big\}, $$ and split $\mathcal{N}_\lambda$ as follows: \begin{gather*} \mathcal{N}_\lambda^+=\left\{u\in\mathcal{N}_\lambda: \psi(u)>0\right\},\\ \mathcal{N}_\lambda^0=\left\{u\in\mathcal{N}_\lambda: \psi(u)=0\right\},\\ \mathcal{N}_\lambda^-=\left\{u\in\mathcal{N}_\lambda: \psi(u)<0\right\}, \end{gather*} where $$ \psi(u)=2\|u\|^2-\lambda(1-\gamma)\int_{\mathbb{R}^N}g(x)|u|^{1-\gamma}dx -2(2^*-1)\int_{\mathbb{R}^N}\phi_{u}|u|^{2^*-1}dx. $$ Before proving our Theorem \ref{thm1.1}, we recall the following lemma (see \cite{AV}). \begin{lemma} \label{lem2.1} For every $u\in D^{1,2}(\mathbb{R}^N)$, there exists a unique $\phi_u\in D^{1,2}(\mathbb{R}^N)$ solution of \begin{equation*} -\Delta\phi=|u|^{2^*-1},~~\mathrm{in }\mathbb{R}^N\,. \end{equation*} Also \begin{itemize} \item[(1)] $\phi_u\geq0$ for $x\in\mathbb{R}^N$. \item[(2)] For each $t\neq0$, $\phi_{tu}=t^{2^*-1}\phi_u$. \item[(3)] $$ \int_{\Omega}\phi_u|u|^{2^*-1}dx=\int_{\Omega}|\nabla\phi_u|^2dx \leq S^{-2^*}\|u\|^{2(2^*-1)}. $$ \item[(4)] Assume that $u_n\rightharpoonup u$ in $D^{1,2}(\mathbb{R}^N)$, then \[ \int_{\mathbb{R}^N}\phi_{u_n}|u_n|^{2^*-1}dx -\int_{\mathbb{R}^N}\phi_{u_n-u}|u_n-u|^{2^*-1}dx =\int_{\mathbb{R}^N}\phi_{u}|u|^{2^*-1}dx+o_n(1). \] \end{itemize} \end{lemma} Set $$ \Lambda_0=|g|^{-1}_{\frac{2^*}{2^*+\gamma-1}}\frac{2(2^*-2) S^{\frac{1-\gamma}{2}}}{2\cdot2^*+\gamma-3} \Big[\frac{(1+\gamma)S^{2^*}}{(2\cdot2^*+\gamma-3)}\Big]^{\frac{1+\gamma}{2(2^*-2)}}. $$ \begin{lemma} \label{lem2.2} Assume $\lambda\in(0, \Lambda_0)$. Then {\rm (1)} $\mathcal{N}_\lambda^{\pm}\neq\emptyset$ and {\rm (2)} $\mathcal{N}_\lambda^0=\{0\}$. \end{lemma} \begin{proof} (i) For each $u\in D^{1,2}(\mathbb{R}^N)\backslash \{0\}$, we have \begin{align*} &t[\frac{d}{dt}I_\lambda(tu)] \\ &= t^2\|u\|^2-\lambda t^{1-\gamma}\int_{\mathbb{R}^N}g(x)|u|^{1-\gamma}dx -t^{2(2^*-1)}\int_{\mathbb{R}^N}\phi_u|u|^{2^*-1}dx\\ &= t^{1-\gamma}\Big[t^{1+\gamma}\|u\|^2-t^{2\cdot2^*+\gamma-3} \int_{\mathbb{R}^N}\phi_u|u|^{2^*-1}dx-\lambda\int_{\mathbb{R}^N} g(x)|u|^{1-\gamma}dx\Big]. \end{align*} Set $$ \Gamma(t)=t^{1+\gamma}\|u\|^2-t^{2\cdot2^*+\gamma-3} \int_{\mathbb{R}^N}\phi_u|u|^{2^*-1}dx,~~t\geq0. $$ We see that $\Gamma(0)=0$ and $\lim_{t\to \infty}\Gamma(t)=-\infty$. Then $\Gamma$ achieves its maximum at $$ t_{\rm max}=\Big[\frac{(1+\gamma)\|u\|^2}{(2\cdot2^*+\gamma-3) \int_{\mathbb{R}^N}\phi_u|u|^{2^*-1}dx}\Big]^{\frac{1}{2(2^*-2)}}, $$ and so, $$ \Gamma(t_{\max})=\frac{2(2^*-2)\|u\|^2}{2\cdot2^*+\gamma-3} \Big[\frac{(1+\gamma)\|u\|^2}{(2\cdot2^*+\gamma-3) \int_{\mathbb{R}^N}\phi_u|u|^{2^*-1}dx}\Big]^{\frac{1+\gamma}{2(2^*-2)}}. $$ Consequently, \begin{equation}\label{1} \begin{aligned} &\Gamma(t_{\max})-\lambda\int_{\mathbb{R}^N}g(x)|u|^{1-\gamma}dx\\ &= \frac{2(2^*-2)\|u\|^2}{2\cdot2^*+\gamma-3} \Big[\frac{(1+\gamma)\|u\|^2}{(2\cdot2^*+\gamma-3) \int_{\mathbb{R}^N}\phi_u|u|^{2^*-1}dx}\Big]^{\frac{1+\gamma}{2(2^*-2)}} -\lambda\int_{\mathbb{R}^N}g(x)|u|^{1-\gamma}dx\\ &\geq \frac{2(2^*-2)\|u\|^2}{2\cdot2^*+\gamma-3} \Big[\frac{(1+\gamma)S^{2^*}\|u\|^2}{(2\cdot2^*+\gamma-3)\|u\|^{2(2^*-1)}} \Big]^{\frac{1+\gamma}{2(2^*-2)}}-\lambda|g|_{\frac{2^*}{2^*+\gamma-1}} S^{-\frac{1-\gamma}{2}}\|u\|^{1-\gamma}\\ &= \Big\{\frac{2(2^*-2)}{2\cdot2^*+\gamma-3} \Big[\frac{(1+\gamma)S^{2^*}}{(2\cdot2^*+\gamma-3)}\Big]^{\frac{1+\gamma}{2(2^*-2)}} -\lambda|g|_{\frac{2^*}{2^*+\gamma-1}}S^{-\frac{1-\gamma}{2}}\Big\} \|u\|^{1-\gamma} >0, \end{aligned} \end{equation} the last inequality holds provided $0<\lambda<\Lambda_0$. Consequently, there exactly exist two points $00>\Gamma'(t^-_u), \end{gather*} which imply that $t^+_uu\in\mathcal{N}_\lambda^+$, $t^-_uu\in\mathcal{N}_\lambda^-$. That is, $\mathcal{N}_\lambda^{\pm}\neq\emptyset$. (ii) We prove (ii) by contradiction, suppose that there exists $u_{0}\neq0$ such that $u_{0}\in\mathcal{N}_\lambda^{0}$, similar to \eqref{1}, it holds that \begin{align*} 0&< \Big\{\frac{2(2^*-2)}{2\cdot2^*+\gamma-3} \Big[\frac{(1+\gamma)S^{2^*}}{(2\cdot2^*+\gamma-3)} \Big]^{\frac{1+\gamma}{2(2^*-2)}}-\lambda|g|_{\frac{2^*}{2^*+\gamma-1}} S^{-\frac{1-\gamma}{2}}\Big\}\|u_0\|^{1-\gamma}\\ &\leq \frac{2(2^*-2)\|u_0\|^2}{2\cdot2^*+\gamma-3}\left[\frac{(1+\gamma)\|u_0\|^2}{(2\cdot2^*+\gamma-3)\int_{\mathbb{R}^N}\phi_{u_0}|u_0|^{2^*-1}dx}\right]^{\frac{1+\gamma}{2(2^*-2)}}\\ &\quad -\lambda\int_{\mathbb{R}^N}g(x)|u_0|^{1-\gamma}dx = 0, \end{align*} this is a contradiction, thereby $\mathcal{N}_\lambda^{0}=\{0\}$ for $\lambda\in(0, \Lambda_0)$. The proof is complete. \end{proof} \begin{lemma} \label{lem2.3} The functional $I_\lambda$ is coercive and bounded below on $\mathcal{N}_\lambda$. \end{lemma} \begin{proof} Suppose $u\in\mathcal{N}_\lambda$, then by Sobolev inequality, \begin{align*} I_\lambda(u) &= \frac{1}{2}\|u\|^2-\frac{\lambda}{1-\gamma} \int_{\mathbb{R}^N}g(x)|u|^{1-\gamma}dx -\frac{1}{2(2^*-1)}\int_{\mathbb{R}^N}\phi_{u}|u|^{2^*-1}dx\\ &= \frac{2^*-2}{2(2^*-1)}\|u\|^{2}-\lambda\Big[\frac{1}{1-\gamma} -\frac{1}{2(2^*-1)}\Big]\int_{\mathbb{R}^N}g(x)|u|^{1-\gamma}dx\\ &\geq \frac{2}{N+2}\|u\|^{2}-\lambda\Big[\frac{1}{1-\gamma} -\frac{1}{2(2^*-1)}\Big]|g|_{\frac{2^*}{2^*+\gamma-1}} S^{-\frac{1-\gamma}{2}}\| u\|^{1-\gamma}, \end{align*} as $0<\gamma<1$, it follows that $I_\lambda$ is coercive and bounded below on $\mathcal{N}_\lambda$. \end{proof} We remark that by Lemma \ref{lem2.2} we have $\mathcal{N}_\lambda=\mathcal{N}_\lambda^+\cup\mathcal{N}_\lambda^-\cup \mathcal{N}_\lambda^0$ for all $\lambda\in(0, \Lambda_0)$. Moreover, we know that $\mathcal{N}_\lambda^+$ and $\mathcal{N}_\lambda^-$ are non-empty and by Lemma \ref{lem2.3} we may define $$ \alpha_\lambda=\inf_{u\in\mathcal{N}_\lambda}I_\lambda(u),\quad \alpha^+_\lambda=\inf_{u\in\mathcal{N}_\lambda^+}I_\lambda(u),\quad \alpha^-_\lambda=\inf_{u\in\mathcal{N}_\lambda^-}I_\lambda(u). $$ \begin{lemma} \label{lem2.4} $\alpha_\lambda\leq\alpha^+_\lambda<0$. \end{lemma} \begin{proof} Assume $u\in\mathcal{N}_\lambda^+$. Then \begin{equation*} \int_{\mathbb{R}^N}\phi_{u}|u|^{2^*-1}dx <\frac{1+\gamma}{2\cdot2^*+\gamma-3}\|u\|^{2}, \end{equation*} so that \begin{align*} I_\lambda(u) &= \frac{1}{2}\|u\|^2-\frac{\lambda}{1-\gamma}\int_{\mathbb{R}^N}g(x) |u|^{1-\gamma}dx-\frac{1}{2(2^*-1)}\int_{\mathbb{R}^N}\phi_{u}|u|^{2^*-1}dx\\ &= \Big(\frac{1}{2}-\frac{1}{1-\gamma}\Big)\|u\|^{2} +\Big(\frac{1}{1-\gamma}-\frac{1}{2(2^*-1)}\Big) \int_{\mathbb{R}^N}\phi_{u}|u|^{2^*-1}dx\\ &< \Big[\Big(\frac{1}{2}-\frac{1}{1-\gamma}\Big) +\Big(\frac{1}{1-\gamma}-\frac{1}{2(2^*-1)}\Big) \frac{1+\gamma}{2\cdot2^*+\gamma-3}\Big]\|u\|^{2}\\ &= \Big[-\frac{1+\gamma}{2(1-\gamma)} +\frac{1+\gamma}{2(2^*-1)(1-\gamma)}\Big]\|u\|^{2}\\ &= -\frac{(2^*-2)(1+\gamma)}{2(2^*-1)(1-\gamma)}\|u\|^{2} <0. \end{align*} By the definitions of $\alpha_\lambda$ and $\alpha_\lambda^+$, one obtains $\alpha_\lambda\leq\alpha_\lambda^+<0$. \end{proof} \begin{lemma} \label{lem2.5} For $u\in\mathcal{N}_\lambda$ (respectively $\mathcal{N}_\lambda^-$), there exist $\varepsilon>0$ and a continuous function $f=f(w)>0, w\in D^{1,2}(\mathbb{R}^N)$, $\|w\|<\varepsilon$ satisfying \[ f(0)=1,\quad f(w)(u+w)\in\mathcal{N}_\lambda\quad \text{(respectively $\mathcal{N}_\lambda^-$)}, \] for all $w\in D^{1,2}(\mathbb{R}^N)$, $\|w\|<\varepsilon$. \end{lemma} \begin{proof} For $u\in\mathcal{N}_\lambda$, define $F: \mathbb{R}\times D^{1,2}(\mathbb{R}^N)\to \mathbb{R}$ by \begin{align*} F(t,w)&= t^2\|u+w\|^2 -t^{2(2^*-1)}\int_{\mathbb{R}^N}\phi_{u+w}|u+w|^{2^*-1}dx \\ &\quad -\lambda t^{1-\gamma}\int_{\mathbb{R}^N}g(x)|u+w|^{1-\gamma}dx. \end{align*} Since $u\in\mathcal{N}_\lambda$, it is easily obtained that $F(1,0)=0$ and $$ F_t(1,0)=2\|u\|^2-2(2^*-1)\int_{\mathbb{R}^N}\phi_{u}|u|^{2^*-1}dx -\lambda(1-\gamma)\int_{\mathbb{R}^N}g(x)|u|^{1-\gamma}dx. $$ As $u\neq0$, by Lemma \ref{lem2.2}, we know that $F_t(1,0)\neq0$. Thus, we can apply the implicit function theorem at the point $(0,1)$, and obtain $\varepsilon>0$ and a continuous function $f: B(0,\varepsilon)\subset D^{1,2}(\mathbb{R}^N)\to \mathbb{R}^+$ satisfying \[ f(0)=1,\quad f(w)>0, \quad f(w)(u+w)\in\mathcal{N}_\lambda, \] for all $w\in D^{1,2}(\mathbb{R}^N)$ with $\|w\|<\varepsilon$. The case $u\in\mathcal{N}_\lambda^-$ can be obtained in the same way. The proof is complete. \end{proof} \begin{lemma} \label{lem2.6} If $\{u_n\}\subset\mathcal{N}_\lambda$ is a minimizing sequence of $I_\lambda$, for each $\phi\in D^{1,2}(\mathbb{R}^N)$ , it holds \begin{equation}\label{2.1} -\frac{|f'_n(0)|\|u_n\|+\|\varphi\|}{n} \leq\langle J'(u_n),\varphi\rangle \leq\frac{|f'_n(0)|\|u_n\|+\|\varphi\|}{n}, \end{equation} where $$ \langle J'(u_n),\varphi\rangle =\int_{\mathbb{R}^N}(\nabla u_n,\nabla\varphi)dx -\int_{\mathbb{R}^N}\phi_{u_n}u_n^{2^*-2}\varphi dx -\lambda\int_{\mathbb{R}^N}g(x)u_n^{-\gamma}\varphi dx. $$ \end{lemma} \begin{proof} According to Lemma \ref{lem2.3}, $I_\lambda$ is coercive on $\mathcal{N}_\lambda$. Applying Ekeland's variational principle, there exists a minimizing sequence $\{u_n\}\subset\mathcal{N}_\lambda$ of $I_\lambda$ such that \begin{equation}\label{2.2} I_\lambda(u_n)<\alpha_\lambda+\frac{1}{n},\quad I_\lambda(v)-I_\lambda(u_n)\geq-\frac{1}{n}\|v-u_n\|,\quad \forall v\in\mathcal{N}_\lambda. \end{equation} Based on $I_\lambda(|u_n|)=I_\lambda(u_n)$, we may assume that $u_n\geq0$ in $\mathbb{R}^N$, and there exist a subsequence (by denoted itself) and $u_*$ in $D^{1,2}(\mathbb{R}^N)$ such that \begin{gather*} u_n\rightharpoonup u_*\quad \text{weakly in } D^{1,2}(\mathbb{R}^N),\\ u_n(x)\to u_*(x)\quad\text{a.e. in }\mathbb{R}^N. \end{gather*} Let $t>0$ small enough, $\varphi\in D^{1,2}(\mathbb{R}^N)$, we set $u=u_n, w=t\varphi\in D^{1,2}(\mathbb{R}^N)$ in Lemma \ref{lem2.5}, then we have $f_n(t)=f_n(t\varphi)$ with $f_n(0)=1, f_n(t)(u_n+t\varphi)\in\mathcal{N}_\lambda$. Note that \begin{equation}\label{2.3} \|u_n\|^2-\lambda\int_{\mathbb{R}^N}g(x)|u_n|^{1-\gamma}dx -\int_{\mathbb{R}^N}\phi_{u_n}|u_n|^{2^*-1}dx=0. \end{equation} From \eqref{2.2}, it follows that \begin{equation}\label{2.4} \begin{aligned} \frac{1}{n}\left[|f_n(t)-1|\cdot\|u_n\|+tf_n(t)\|\varphi\|\right] &\geq \frac{1}{n}\|f_n(t)(u_n+t\varphi)-u_n\|\\ &\geq I_\lambda(u_n)-I_\lambda[f_n(t)(u_n+t\varphi)],\\ \end{aligned} \end{equation} and \begin{align*} &I_\lambda(u_n)-I_\lambda[f_n(t)(u_n+t\varphi)]\\ &= \frac{1-f_{n}^{2}(t)}{2}\|u_n\|^2 +\frac{f_{n}^{2(2^*-1)}(t)-1}{2(2^*-1)} \int_{\mathbb{R}^N}\phi_{(u_n+t\varphi)}|u_n+t\varphi|^{2^*-1}dx\\ &\quad +\lambda\frac{f_{n}^{1-\gamma}(t)-1}{1-\gamma}\int_{\mathbb{R}^N}g(x)|u_n +t\varphi|^{1-\gamma}dx+\frac{f_{n}^{2}(t)}{2}(\|u_n\|^2-\|u_n+t\varphi\|^2)\\ &\quad +\frac{1}{2(2^*-1)}\int_{\mathbb{R}^N}[\phi_{(u_n+t\varphi)} |u_n+t\varphi|^{2^*-1}-\phi_{u_n}|u_n|^{2^*-1}]dx\\ &\quad +\frac{\lambda}{1-\gamma}\int_{\mathbb{R}^N}g(x) ((u_n+t\varphi)^{1-\gamma}-u_n^{1-\gamma})dx. \end{align*} Combined with \eqref{2.3} and \eqref{2.4}, dividing by $t$ and letting $t\to0$, we obtain \begin{align*} &\frac{|f'_n(0)|\|u_n\|+\|\varphi\|}{n}\\ &\geq -f'_{n}(0)\Big\{\|u_n\|^2-\lambda\int_{\mathbb{R}^N} g(x)u_n^{1-\gamma}dx-\int_{\mathbb{R}^N}\phi_{u_n}|u_n|^{2^*-1}dx\Big\}\\ &\quad -\int_{\mathbb{R}^N}(\nabla u_n,\nabla\varphi)dx +\int_{\mathbb{R}^N}\phi_{u_n}|u_n|^{2^*-3}u\varphi dx +\lambda\int_{\mathbb{R}^N}g(x)u_n^{-\gamma}\varphi dx\\ &= -\int_{\mathbb{R}^N}(\nabla u_n,\nabla\varphi)dx +\int_{\mathbb{R}^N}\phi_{u_n}|u_n|^{2^*-3}u\varphi dx +\lambda\int_{\mathbb{R}^N}g(x)u_n^{-\gamma}\varphi dx, \end{align*} so, we obtain that for $\varphi\in D^{1,2}(\mathbb{R}^N)$, $\varphi\geq0$, it holds \begin{equation}\label{2.5} \begin{aligned} &\int_{\mathbb{R}^N}(\nabla u_n,\nabla\varphi)dx -\int_{\mathbb{R}^N}[\phi_{u_n}|u_n|^{2^*-3}u+\lambda g(x)u_n^{-\gamma}]\varphi dx\\ &\geq-\frac{|f'_n(0)|\|u_n\|+\|\varphi\|}{n}. \end{aligned} \end{equation} Since the above inequality holds for $-\varphi$, it follows that \begin{align*} &\int_{\mathbb{R}^N}(\nabla u_n,\nabla\varphi)dx -\int_{\mathbb{R}^N}[\phi_{u_n}|u_n|^{2^*-3}u+\lambda g(x)u_n^{-\gamma}]\varphi dx\\ &\leq\frac{|f'_n(0)|\|u_n\|+\|\varphi\|}{n}. \end{align*} Set \[ \langle J'(u),\varphi\rangle =\int_{\mathbb{R}^N}(\nabla u,\nabla\varphi)dx -\int_{\mathbb{R}^N}\phi_{u_n}|u_n|^{2^*-3}u\varphi dx -\lambda\int_{\mathbb{R}^N}g(x)u_n^{-\gamma}\varphi dx, \] consequently \eqref{2.1} holds. As in \cite{ZQ}, we can prove that $\{f'_n(0)\}$ is bounded for all $n$. The proof is complete. \end{proof} \begin{lemma} \label{lem2.7} Supposes $\{v_n\}\subset\mathcal{N}_\lambda^-$ is a minimizing sequence for $I_\lambda$ with $$ \alpha_\lambda^-<\frac{2}{N+2}S^{\frac{N}{2}}-D\lambda^{\frac{2}{1+\gamma}} \quad\text{where}\quad D=D(N, \gamma, S, |g|_{\frac{2^*}{2^*+\gamma-1}}). $$ Then there exists $v_*\in D^{1,2}(\mathbb{R}^N)$ such that $v_n\to v_*$ in $D^{1,2}(\mathbb{R}^N)$ and \[ \int_{\mathbb{R}^N}\phi_{v_n}|v_n|^{2^*-1}dx \to\int_{\mathbb{R}^N}\phi_{v_*}|v_*|^{2^*-1}dx \] as $n\to\infty$. \end{lemma} \begin{proof} Let $\{v_n\}\subset\mathcal{N}_\lambda^-$ be a minimizing sequence for $I_\lambda$, similarly to the proof of Lemma \ref{lem2.6}, one obtains \begin{equation}\label{2.6} -\frac{|f'_n(0)|\|v_n\|+\|\varphi\|}{n} \leq\langle J'(v_n),\varphi\rangle \leq\frac{|f'_n(0)|\|v_n\|+\|\varphi\|}{n}. \end{equation} Since $\{v_n\}$ is bounded in $D^{1,2}(\mathbb{R}^N)$, there exist a subsequence, still denoted by itself, and a function $v_*\in D^{1,2}(\mathbb{R}^N)$ such that \begin{gather*} v_n\rightharpoonup v_*,\quad\text{weakly in } D^{1,2}(\mathbb{R}^N),\\ v_n(x)\to v_*(x),\quad \text{a.e. in } \mathbb{R}^N \end{gather*} as $n\to\infty$. We firstly claim that $$ \int_{\mathbb{R}^N}g(x)v_n^{1-\gamma}dx\to\int_{\mathbb{R}^N}g(x)v_*^{1-\gamma}dx. $$ In fact, by H\"older's inequality and the boundedness of $\{v_n\}$, it holds that \begin{align*} &\big|\int_{|x|>m}g(x)[v_n^{1-\gamma}-v_*^{1-\gamma}]dx\big|\\ &\leq \int_{|x|>m}g(x)|v_n^{1-\gamma}-v_*^{1-\gamma}|dx\\ &\leq \int_{|x|>m}g(x)(|v_n|^{1-\gamma}+|v_*|^{1-\gamma})dx\\ &= \int_{|x|>m}g(x)|v_n|^{1-\gamma}dx+\int_{|x|>m}g(x)|v_*|^{1-\gamma}dx\\ &\leq \Big(\int_{|x|>m}g(x)^{\frac{2^*}{2^*+\gamma-1}}dx \Big)^{\frac{2^*+\gamma-1}{2^*}}|v_n|_{2^*}^{1-\gamma} +\Big(\int_{|x|>m}g(x)^{\frac{2^*}{2^*+\gamma-1}}dx\Big)^{\frac{2^*+\gamma-1}{2^*}} |v_*|_{2^*}^{1-\gamma}\\ &\leq C\Big(\int_{|x|>m}g(x)^{\frac{2^*}{2^*+\gamma-1}}dx \Big)^{\frac{2^*+\gamma-1}{2^*}}\\ &\to 0,\quad\text{as }m\to\infty, \end{align*} which implies that for any $\varepsilon>0$, there exists $N_1>0$ such that $$ \big|\int_{|x|>m}g(x)[v_n^{1-\gamma}-v_*^{1-\gamma}]dx\big| <\frac{\varepsilon}{2},\quad\text{for each } m>N_1. $$ Let $\mathcal{M}=\{x\in\mathbb{R}^N: |x|\leq N_1+1\}$. Note that $\{v_n\}$ is bounded in $D^{1,2}(\mathbb{R}^N)$, then $\big(\int_{|x|\leq N_1+1}v_n^{2^*}dx\big)^{\frac{1-\gamma}{2^*}}\leq M'$ for some $M'>0$. Moreover, from absolute continuity of the Lebesgue integral, for every $\varepsilon>0$, there exists $\delta'>0$ such that for each $E\subset\mathcal{M}$ with $\operatorname{meas}~E<\delta'$, it holds $$ \int_{E}g(x)^{\frac{2^*}{2^*+\gamma-1}}dx <\Big(\frac{\varepsilon}{M'}\Big)^{\frac{2^*}{2^*+\gamma-1}}. $$ Consequently, \[ \int_{E}g(x)v_n^{1-\gamma}dx \leq \Big(\int_{E}g(x)^{\frac{2^*}{2^*+\gamma-1}}dx\Big)^{\frac{2^*+\gamma-1}{2^*}} \Big(\int_{E}|v_n|^{2^*}dx\Big)^{\frac{1-\gamma}{2^*}} < \varepsilon. \] Hence $\{\int_{|x|\leq N_1+1}g(x)v_n^{1-\gamma}dx, n\in N^+\}$ is equi-absolutely-continuous. It follows easily from Vitali Convergence Theorem that $$ \int_{|x|\leq N_1+1}g(x)v_n^{1-\gamma}dx \to\int_{|x|\leq N_1+1}g(x)v_*^{1-\gamma}dx,\quad \text{as } n\to\infty. $$ That is, there exists $N_2>0$ such that $$ \big|\int_{|x|\leq N_1+1}g(x)[v_n^{1-\gamma}-v_*^{1-\gamma}]dx\big| <\frac{\varepsilon}{2},\quad\text{for each } n>N_2. $$ Therefore, from the above inequalities, it follows that \begin{align*} &\big|\int_{\mathbb{R}^N}g(x)v_n^{1-\gamma}dx-\int_{\mathbb{R}^N} g(x)v_*^{1-\gamma}dx\big|\\ &= \big|\int_{|x|\leq N_1+1}g(x)[v_n^{1-\gamma}-v_*^{1-\gamma}]dx +\int_{|x|> N_1+1}h(x)[v_n^{1-\gamma}-v_*^{1-\gamma}]dx\big|\\ &\leq \big|\int_{|x|\leq N_1+1}g(x)[v_n^{1-\gamma}-v_*^{1-\gamma}]dx\big| +\big|\int_{|x|> N_1+1}g(x)[v_n^{1-\gamma}-v_*^{1-\gamma}]dx\big| < \varepsilon \end{align*} for $n>N_2$, which implies $$ \int_{\mathbb{R}^N}g(x)v_n^{1-\gamma}dx \to\int_{\mathbb{R}^N}g(x)v_*^{1-\gamma}dx,\quad\text{as } n\to\infty. $$ Now, set $w_n=v_n-v_*$, then $\|w_n\|\to0$. Otherwise, there exists a subsequence (still denoted by $w_n$) such that $$ \lim_{n\to\infty}\|w_n\|=l>0. $$ From \eqref{2.6}, letting $n\to\infty$, for every $\varphi\in D^{1,2}(\mathbb{R}^N)$, it follows \begin{equation}\label{2.7} \int_{\mathbb{R}^N}(\nabla v_*, \nabla\varphi)dx -\lambda\int_{\mathbb{R}^N}g(x)v_*^{-\gamma}\varphi dx -\int_{\mathbb{R}^N}\phi_{v_*}v_*^{2^*-2}\varphi dx=0. \end{equation} Taking the test function $\varphi=v_*$ in \eqref{2.7}, it follows that \begin{equation}\label{2.8} \|v_*\|^2-\lambda\int_{\mathbb{R}^N}g(x)v_*^{1-\gamma}dx -\int_{\mathbb{R}^N}\phi_{v_*}v_*^{2^*-1}dx=0. \end{equation} Putting $\varphi=v_n$ in \eqref{2.6}, by the Br\'ezis-Lieb's lemma (see \cite{BL}) and Lemma \ref{lem2.1}, it follows that \begin{equation}\label{2.9} \begin{aligned} &\|w_n\|^2+\|v_*\|^2-\int_{\mathbb{R}^N}[\phi_{w_n}|w_n|^{2^*-1} +\phi_{v_*}|v_*|^{2^*-1}]dx\\ &-\lambda\int_{\mathbb{R}^N}g(x)v_*^{1-\gamma}dx=o(1). \end{aligned} \end{equation} It follows from \eqref{2.8} and \eqref{2.9} that \begin{equation}\label{2.10} \|w_n\|^2-\int_{\mathbb{R}^N}\phi_{w_n}|w_n|^{2^*-1}dx=o(1). \end{equation} Note that $\int_{\mathbb{R}^N}\phi_{w_n}|w_n|^{2^*-1}dx \leq S^{-2^*}\|w_n\|^{2(2^*-1)}$; then $$ l\geq S^{\frac{2^*}{2(2^*-2)}},\quad l>0. $$ On the one hand, from \eqref{2.8}, by the Young inequality, \begin{align*} I_{\lambda}(v_*) &= \frac{1}{2}\|v_*\|^2-\frac{1}{2(2^*-1)} \int_{\mathbb{R}^N}\phi_{v_*}v_*^{2^*-1}dx -\frac{\lambda}{1-\gamma}\int_{\mathbb{R}^N}g(x) v_*^{1-\gamma}dx\\ &= \frac{2^*-2}{2(2^*-1)}\|v_*\|^2 -\lambda\big[\frac{1}{1-\gamma}-\frac{1}{2(2^*-1)}\big] \int_{\mathbb{R}^N}g(x)v_*^{1-\gamma}dx\\ &\geq \frac{2}{N+2}\|v_*\|^2 -\lambda\big[\frac{1}{1-\gamma}-\frac{1}{2(2^*-1)}\big] |g|_{\frac{2^*}{2^*+\gamma-1}}S^{-\frac{1-\gamma}{2}}\| v_*\|^{1-\gamma}\\ &\geq -D\lambda^{\frac{2}{1+\gamma}}, \end{align*} where $D=D(N, \gamma, S, |g|_{\frac{2^*}{2^*+\gamma-1}})>0$ is a constant (independent of $\lambda$). On the other hand, from \eqref{2.10}, \begin{align*} I_{\lambda}(v_*) &= I_{\lambda}(v_n)-\frac{1}{2}\|w_n\|^2 +\frac{1}{2(2^*-1)}\int_{\mathbb{R}^N}\phi_{w_n}|w_n|^{2^*-1}dx+o(1)\\ &= I_{\lambda}(v_n)-\frac{2^*-2}{2(2^*-1)}\|w_n\|^2+o(1)\\ &\leq \alpha_\lambda^--\frac{2}{N+2}l^2\\ &< \frac{2}{N+2}S^{\frac{N}{2}}-D\lambda^{\frac{2}{1+\gamma}} -\frac{2}{N+2}S^{\frac{N}{2}}\\ &= -D\lambda^{\frac{2}{1+\gamma}}. \end{align*} This is a contradiction. Therefore, $l=0$, it implies that $v_n\to v_*$ in $D^{1,2}(\mathbb{R}^N)$. Note that \begin{align*} 0&\leq \int_{\mathbb{R}^N}\phi_{v_n}v_n^{2^*-1}dx -\int_{\mathbb{R}^N}\phi_{v_*}v_*^{2^*-1}dx\\ &= \int_{\mathbb{R}^N}\phi_{w_n}w_n^{2^*-1}dx+o(1)\\ &\leq S^{-2^*}\|w_n\|^{2(2^*-1)}+o(1) \to 0, \end{align*} which implies that $\int_{\mathbb{R}^N}\phi_{v_n}v_n^{2^*-1}dx \to\int_{\mathbb{R}^N}\phi_{v_*}v_*^{2^*-1}dx$ as $n\to\infty$. The proof is complete. \end{proof} \begin{theorem} \label{thm2.8} Under the assumptions of Theorem \ref{thm1.1}, system \eqref{1.1} has a positive ground state solution $(u_\lambda, \phi_{u_\lambda})\in D^{1,2}(\mathbb{R}^N)\times D^{1,2}(\mathbb{R}^N)$ with $I_\lambda(u_\lambda)<0$. \end{theorem} \begin{proof} There exists a constant $\delta>0$ such that $\frac{2}{N+2}S^{\frac{N}{2}}-D\lambda^{\frac{2}{1+\gamma}}>0$ for $\lambda<\delta$. Set $\Lambda_1=\min\{\Lambda_0, \delta\}$, then Lemmas \ref{lem2.1}--\ref{lem2.7} hold for all $0<\lambda<\Lambda_1$. Therefore, there exist a bounded minimizing sequence $\{u_n\}\subset\mathcal{N}_\lambda$ of $I_\lambda$ and $u_\lambda\in D^{1,2}(\mathbb{R}^N)$ such that \begin{gather*} u_n\rightharpoonup u_\lambda,\quad\text{weakly in } D^{1,2}(\mathbb{R}^N),\\ u_n(x)\to u_\lambda(x),\quad\text{a.e. in }\mathbb{R}^N, \end{gather*} as $n\to\infty$. Now we will prove that $u_\lambda$ is a positive ground state solution of system \eqref{1.1}. Indeed, by Lemmas \ref{lem2.4}--\ref{lem2.7}, we can deduce that \[ \lim_{n\to\infty}\int_{\mathbb{R}^N}\phi_{u_n}u_n^{2^*-1}dx =\int_{\mathbb{R}^N}\phi_{u_\lambda}u_\lambda^{2^*-1}dx. \] By Fatou's lemma, \begin{equation}\label{2.11} \begin{aligned} 0&= \lim_{n\to\infty}\Big\{\|u_n\|^2-\lambda\int_{\mathbb{R}^N}g(x) u_n^{1-\gamma}dx-\int_{\mathbb{R}^N}\phi_{u_n}u_n^{2^*-1}dx\Big\}\\ &\geq \|u_\lambda\|^2-\lambda\int_{\mathbb{R}^N}g(x)u_\lambda^{1-\gamma}dx -\int_{\mathbb{R}^N}\phi_{u_\lambda}u_\lambda^{2^*-1}dx. \end{aligned} \end{equation} Letting $n\to\infty$ in \eqref{2.1} and using the Fatou's lemma again, for each $\varphi\in D^{1,2}(\mathbb{R}^N)$, $\varphi\geq0$, it holds \begin{equation}\label{2.12} \int_{\mathbb{R}^N}(\nabla u_\lambda,\nabla\varphi)dx -\lambda\int_{\mathbb{R}^N}g(x)u_\lambda^{-\gamma}\varphi dx -\int_{\mathbb{R}^N}\phi_{u_\lambda}u_\lambda^{2^*-2}\varphi dx\geq0. \end{equation} Now, for any $v\in D^{1,2}(\mathbb{R}^N)$, we set $\Psi=(u_\lambda+\varepsilon v)^{+}$, it follows from \eqref{2.11} and \eqref{2.12} that \begin{equation}\label{2.13} \begin{aligned} 0&\leq \int_{\mathbb{R}^N}[(\nabla u_\lambda,\nabla\Psi) -\phi_{u_\lambda}u_\lambda^{2^*-2}\Psi -\lambda g(x)u_\lambda^{-\gamma}\Psi]dx\\ &= \int_{\{u_\lambda+\varepsilon v>0\}}\big[(\nabla u_\lambda,\nabla(u_\lambda+\varepsilon v)) -\phi_{u_\lambda}u_\lambda^{2^*-2}(u_\lambda+\varepsilon v)\\ &\quad -\lambda g(x)u_\lambda^{-\gamma}(u_\lambda+\varepsilon v)\big]dx\\ &= \Big(\int_{\mathbb{R}^N}-\int_{\{u_\lambda+\varepsilon v\leq0\}}\Big) [(\nabla u_\lambda,\nabla(u_\lambda+\varepsilon v)\\ &\quad -\phi_{u_\lambda}u_\lambda^{2^*-2}(u_\lambda+\varepsilon v) -\lambda g(x)u_\lambda^{-\gamma}(u_\lambda+\varepsilon v)]dx\\ &\leq \|u_\lambda\|^2-\int_{\mathbb{R}^N}\phi_{u_\lambda}u_\lambda^{2^*-1}dx -\lambda\int_{\mathbb{R}^N}g(x)u_\lambda^{1-\gamma}dx\\ &\quad +\varepsilon\int_{\mathbb{R}^N}\left[(\nabla u_\lambda,\nabla v) -\phi_{u_\lambda}u_\lambda^{2^*-2}v-\lambda g(x)u_\lambda^{-\gamma}v\right]dx\\ &\quad -\int_{\{u_\lambda+\varepsilon v\leq0\}}(\nabla u_\lambda,\nabla(u_\lambda +\varepsilon v))dx\\ &\quad +\int_{\{u_\lambda+\varepsilon v\leq0\}} \left[\phi_{u_\lambda}u_\lambda^{2^*-2}(u_\lambda+\varepsilon v) +\lambda g(x)u_\lambda^{-\gamma}(u_\lambda+\varepsilon v)\right]dx\\ &\leq \varepsilon\int_{\mathbb{R}^N}\left[(\nabla u_\lambda,\nabla v) -\phi_{u_\lambda}u_\lambda^{2^*-2}v-\lambda g(x)u_\lambda^{-\gamma}v\right]dx\\ &\quad -\varepsilon\int_{\{u_\lambda+\varepsilon v\leq0\}} (\nabla u_\lambda,\nabla v)dx. \end{aligned} \end{equation} Since $\nabla u_\lambda=0$ for a.e.\ $x\in\mathbb{R}^3$ with $u_\lambda(x)=0$ and $$ \operatorname{meas}\{x| u_\lambda(x)+\varepsilon v(x)<0, u_\lambda(x)>0\}\to 0 \quad\text{as } \varepsilon\to0, $$ then, we have $$ \big|\int_{\{u_\lambda+\varepsilon v<0\}}(\nabla u_\lambda,\nabla v)dx\big| =\int_{\{u_\lambda+\varepsilon v<0, u_\lambda>0\}}(\nabla u_\lambda,\nabla v)dx \to 0\quad \text{as } \varepsilon\to0. $$ Therefore, dividing by $\varepsilon$ and letting $\varepsilon\to0$ in \eqref{2.13}, one gets \begin{equation*} \int_{\mathbb{R}^N}(\nabla u_\lambda,\nabla v)dx -\int_{\mathbb{R}^N}\phi_{u_\lambda}u_\lambda^{2^*-2}vdx -\lambda\int_{\mathbb{R}^N}g(x)u_\lambda^{-\gamma}vdx\geq0. \end{equation*} As $v$ is arbitrarily, consequently, $u_\lambda$ is a nonzero negative solution of system \eqref{1.1}. Note that $u_\lambda\in\mathcal{N}_\lambda$ and $\alpha_\lambda<0$ (by Lemma \ref{lem2.4}), then \begin{align*} \big[\frac{1}{1-\gamma}-\frac{1}{2(2^*-1)}\big] \lambda\int_{\mathbb{R}^N}g(x)u_\lambda^{1-\gamma}dx &= \frac{2}{N+2}\|u_\lambda\|^2-I_\lambda(u_\lambda)\\ &\geq \frac{2}{N+2}\|u_\lambda\|^2-\alpha_\lambda > 0, \end{align*} which implies that $u_\lambda\not\equiv0$. Note that $u_\lambda\geq0$ in $\mathbb{R}^N$. By standard arguments as in DiBenedetto \cite{DE} and Tolksdorf \cite{TP}, we have that $u_\lambda\in L^\infty(\mathbb{R}^N)$ and $u_\lambda\in C^{1,\alpha}_{loc}(\mathbb{R}^N)$ with $0<\alpha<1$. Furthermore, by Harnack's inequality (see Trudinger \cite{TN}), $u_\lambda>0$ for any $x\in\mathbb{R}^N$. Furthermore, we have \begin{equation}\label{2.14} \alpha_\lambda=\lim_{n\to\infty}I_\lambda(u_n)=I_\lambda(u_\lambda). \end{equation} Next, we claim that $u_\lambda\in\mathcal{N}_\lambda^+$. On the contrary, assume that $u_\lambda\in\mathcal{N}_\lambda^-$ $(\mathcal{N}_\lambda^0=\{0\}$ for $\lambda\in(0, \Lambda_0))$, then by Lemma \ref{lem2.2}, there exist positive numbers $t^+0$ such that for each $\lambda\in(0, \Lambda_3)$, it holds \begin{equation}\label{3.1} \sup_{t\geq0}I_\lambda(t\Phi)<\frac{2}{N+2}S^{\frac{N}{2}} -D\lambda^{\frac{2}{1+\gamma}}. \end{equation} \end{lemma} \begin{proof} We are going to give an estimate of the value of $I_\lambda$. Observe that, multiplying the second equation of system \eqref{1.1} by $|u|$ and integrating, one has $$ |u|_{2^*}^{2^*}=\int_{\mathbb{R}^N}\nabla\phi_u|\nabla|u||dx \leq\frac{1}{2}|\nabla\phi_u|_2^2+\frac{1}{2}|\nabla|u||_2^2. $$ So, if we introduce the new functional $J_\lambda: D^{1,2}(\mathbb{R}^N)\to\mathbb{R}$ defined in the following way $$ J_\lambda(u)=\frac{N}{N+2}\|u\|^2-\frac{1}{2^*-1}\int_{\mathbb{R}^N}|u|^{2^*}dx -\frac{\lambda}{1-\gamma}\int_{\mathbb{R}^N}g(x)|u|^{1-\gamma}dx. $$ Then, we have $I_\lambda(u)\leq J_\lambda(u)$ for any $u\in D^{1,2}(\mathbb{R}^N)$. For $t\geq0$, set \begin{align*} h(t) &= \frac{Nt^2}{N+2}\|\Phi\|^2-\frac{t^{2^*}}{2^*-1}\int_{\mathbb{R}^N}\Phi^{2^*}dx\\ &= \frac{Nt^2}{N+2}S^{\frac{N}{2}}-\frac{t^{2^*}}{2^*-1}S^{\frac{N}{2}}. \end{align*} Then \[ \sup_{t\geq0}h(t) = \sup_{t\geq0}\Big\{\frac{Nt^2}{N+2}S^{\frac{N}{2}} -\frac{t^{2^*}}{2^*-1}S^{\frac{N}{2}}\Big\} = \frac{2}{N+2}S^{\frac{N}{2}}. \] When $\lambda\in(0, \delta)$, we have $\frac{2}{N+2}S^{\frac{N}{2}}-D\lambda^{\frac{2}{1+\gamma}}>0$, which implies that there exists $t_0>0$ small such that $$ \sup_{0\leq t\leq t_0}I_\lambda(t\Phi)<\frac{2}{N+2}S^{\frac{N}{2}} -D\lambda^{\frac{2}{1+\gamma}}\quad\text{for each }\lambda\in(0, \delta). $$ We next consider the case where $t>t_0$. Since $\frac{2}{1+\gamma}>1$, there exists $\Lambda_2>0$ such that $$ -\lambda\frac{t_0^{1-\gamma}}{1-\gamma}\int_{\mathbb{R}^N}g(x) \Phi^{1-\gamma}dx<-D\lambda^{\frac{2}{1 +\gamma}} \quad\text{for each } \lambda\in(0,\Lambda_2). $$ Then, for each $\lambda\in(0,\Lambda_2)$, it follows \begin{align*} \sup_{t\geq t_0}I_\lambda(t\Phi) &\leq \frac{2}{N+2}S^{\frac{N}{2}}-\lambda\frac{t^{1-\gamma}}{1-\gamma} \int_{\mathbb{R}^N}g(x)\Phi^{1-\gamma}dx\\ &\leq \frac{2}{N+2}S^{\frac{N}{2}} -\lambda\frac{t_0^{1-\gamma}}{1-\gamma}\int_{\mathbb{R}^N}g(x)\Phi^{1-\gamma}dx\\ &< \frac{2}{N+2}S^{\frac{N}{2}}-D\lambda^{\frac{2}{1+\gamma}}. \end{align*} Set $\Lambda_3=\min\{\delta, \Lambda_2\}$. From the above information, it holds that $$ \sup_{t\geq0}I_\lambda(t\Phi)<\frac{2}{N+2}S^{\frac{N}{2}} -D\lambda^{\frac{2}{1+\gamma}}\quad\text{for each }\lambda\in(0, \Lambda_3). $$ Therefore, \eqref{3.1} holds true when $\lambda<\Lambda_3$. The proof is complete. \end{proof} \begin{theorem} \label{thm3.2} There exists $\lambda_*>0$ such that problem \eqref{1.1} has a positive solution $v_\lambda$ with $v_\lambda\in\mathcal{N}_\lambda^-$ for each $0<\lambda<\lambda_*$. \end{theorem} \begin{proof} Let $\lambda_*=\min\{\Lambda_0, \Lambda_3\}$. Since $I_\lambda$ is also coercive on $\mathcal{N}_\lambda^-$, we apply the Ekeland's variational principle to the minimization problem $\alpha_\lambda^{-}=\inf_{v\in\mathcal{N}_\lambda^-}I_\lambda(v)$, there exists a minimizing sequence $\{v_{n}\}\subset\mathcal{N}_\lambda^-$ of $I_\lambda$ with the following properties: \begin{itemize} \item[(i)] $I_\lambda(v_{n})<\alpha_\lambda^{-}+\frac{1}{n}$; \item[(ii)] $I_\lambda(u)\geq I_\lambda(v_{n})-\frac{1}{n}\|u-v_{n}\|$ for all $u\in\mathcal{N}_\lambda^-$. \end{itemize} Since $\{v_{n}\}$ is bounded in $D^{1,2}(\mathbb{R}^N)$, up to a subsequence if necessary, there exists $v_\lambda\in D^{1,2}(\mathbb{R}^N)$ such that \begin{gather*} v_n\rightharpoonup v_\lambda,\quad\text{weakly in }D^{1,2}(\mathbb{R}^N), \\ v_n(x)\to v_\lambda(x),\quad\text{a.e. in} \mathbb{R}^N\text{ as } n\to\infty. \end{gather*} Using Lemmas \ref{lem2.5}--\ref{lem2.7} and Lemma \ref{lem3.1}, similarly, we can get that $v_\lambda$ is a non-negative solution of system \eqref{1.1}. Now, we prove that $v_\lambda>0$ in $\mathbb{R}^N$. Since $v_{n}\in\mathcal{N}_\lambda^-$, it holds \begin{align*} (1+\gamma)\|v_{n}\|^{2} &< (2\cdot2^*+\gamma-3)\int_{\mathbb{R}^N}\phi_{v_n}v_{n}^{2^*-1}dx\\ &< (2\cdot2^*+\gamma-3)S^{-2^*}\|v_n\|^{2(2^*-1)}, \end{align*} so that \begin{equation*} \|v_{n}\|>\Big(\frac{(1+\gamma)S^{2^*}}{2\cdot2^*+\gamma-3}\Big)^{\frac{1}{2(2^*-2)}} \quad \forall v_{n}\in\mathcal{N}_\lambda^-, \end{equation*} which implies that $v_\lambda\not\equiv0$. Similarly, by Harnack¡¯s inequality, we also obtain $v_\lambda>0$ for any $x\in\mathbb{R}^N$. Next, we prove that $v_\lambda\in\mathcal{N}_\lambda^-$, it suffices to prove that $\mathcal{N}_\lambda^-$ is closed. Indeed, by Lemma \ref{lem2.7} and Lemma \ref{lem3.1}, for $\{v_n\}\subset\mathcal{N}_\lambda^-$, it holds \begin{equation*} \lim_{n\to\infty}\int_{\mathbb{R}^N}\phi_{v_n}v_{n}^{2^*-1}dx =\int_{\mathbb{R}^N}\phi_{v_\lambda}v_\lambda^{2^*-1}dx. \end{equation*} By the definition of $\mathcal{N}_\lambda^-$, it holds that \begin{equation*} 2\|v_{n}\|^{2}-(2^*-1)\int_{\mathbb{R}^N}\phi_{v_n}v_{n}^{2^*-1}dx -\lambda(1-\gamma)\int_{\mathbb{R}^N}g(x)v_{n}^{1-\gamma}dx<0, \end{equation*} thus \begin{equation*} 2\|v_\lambda\|^{2}-(2^*-1)\int_{\mathbb{R}^N}\phi_{v_\lambda}v_\lambda^{2^*-1}dx -\lambda(1-\gamma)\int_{\mathbb{R}^N}g(x)v_\lambda^{1-\gamma}dx\leq0, \end{equation*} which implies that $v_\lambda\in\mathcal{N}_\lambda^{0}\cup\mathcal{N}_\lambda^-$. If $\mathcal{N}_\lambda^-$ is not closed, then we have $v_\lambda\in\mathcal{N}_\lambda^{0}$, by Lemma \ref{lem2.2}, it follows that $v_\lambda=0$, this contradicts $v_\lambda>0$. Consequently, $v_\lambda\in\mathcal{N}_\lambda^-$. 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