\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2010(2010), No. 116, pp. 1--14.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2010 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2010/116\hfil Asymptotic behavior] {Asymptotic behavior of ground state solution for H\'enon type systems} \author[Y. Wang, J. Yang \hfil EJDE-2010/116\hfilneg] {Ying Wang, Jianfu Yang} % in alphabetical order \address{Ying Wang \newline Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China} \email{yingwang00@126.com} \address{Jianfu Yang \newline Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China} \email{jfyang\_2000@yahoo.com} \thanks{Submitted July 14, 2010. Published August 20, 2010.} \subjclass[2000]{35J50, 35J57, 35J47} \keywords{Asymptotic behavior; H\'enon systems; ground state solution} \begin{abstract} In this article, we investigate the asymptotic behavior of positive ground state solutions, as $\alpha\to\infty$, for the following H\'enon type system \[ -\Delta u=\frac{2p}{p+q}|x|^\alpha u^{p-1}v^q,\quad -\Delta v=\frac{2q}{p+q}|x|^\alpha u^pv^{q-1},\quad \text{in } B_1(0) \] with zero boundary condition, where $B_1(0)\subset\mathbb{R}^N$ ($N\geq3$) is the unit ball centered at the origin, $p,q>1$, $p+q<2^*=2N/(N-2)$. We show that both components of the ground solution pair $(u, v)$ concentrate on the same point on the boundary $\partial B_1(0)$ as $\alpha\to\infty$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \allowdisplaybreaks \section{Introduction} In this article, we investigate the asymptotic behavior of positive ground state solution pairs of the following H\'enon type system \begin{equation}\label{eq:1.1} -\Delta u=\frac{2p}{p+q}|x|^\alpha u^{p-1}v^q,\quad -\Delta v=\frac{2q}{p+q}|x|^\alpha u^pv^{q-1},\quad \text{in } B_1(0) \end{equation} with zero boundary condition, where $B_1(0)\subset\mathbb{R}^N$ ($N\geq3$) is the unit ball centered at the origin, $\alpha>0$, $p,q>1$, $p+q<2^*=2N/(N-2)$. H\'enon \cite{H} considered the so called H\'enon equation \begin{equation}\label{eq:1.2} \begin{gathered} -\Delta u = |x|^\alpha u ^ {p-1}, \quad x \in \Omega, \\ u = 0, \quad x \in \partial \Omega, \end{gathered} \end{equation} which stems from a research of rotating stellar structures. Such a problem enjoys special features. As usual, for arbitrary bounded $\Omega$, critical exponent for problem \eqref{eq:1.2} is $2^*$, while if $\Omega$ is a ball, it was shown in \cite{Ni} that problem \eqref{eq:1.2} has a radially symmetric solution for $p\in (2,\frac {2(N+\alpha)}{N-2})$, the critical exponent $\frac {2(N+\alpha)}{N-2}$ is larger than the critical Sobolev exponent $2^*$. Moreover, even in a ball, problem \eqref{eq:1.2} possesses non-radial solutions under some conditions, see \cite{SSW} and references therein. This can also be seen as in \cite{CP}, where it was shown that the ground state solution of problem \eqref{eq:1.2} has a unique maximum point approaching to a point on $\partial B_1(0)$ provided that $\alpha>0$ fixed, $ p\in(2,2^*)$ and $p\to 2^*$. Similar results for $p\in(2,2^*)$ fixed and $\alpha\to\infty$ can be found in \cite{BW2,BW1,CPY}. For system \eqref{eq:1.1}, we proved in \cite{WY} that there exists $\alpha^*>0$ such that the ground state solution of problem \eqref{eq:1.1} is non-radial if $\alpha>\alpha^*$, $p, q>1$ and $p+q <2^*$; the maximum points of both components $u$ and $v$ of the ground state solution pair $(u,v)$ concentrate at the same point on the boundary $\partial\Omega$ as $p+q\to 2^*$. In this paper, we investigate the asymptotic behavior of the ground state solution pair of problem \eqref{eq:1.1} as $\alpha\to\infty$. Our main result is as follows. \begin{theorem}\label{th1.1} Let $(u_\alpha,v_\alpha)$ be a positive ground state solution of \eqref{eq:1.1} and denote $x_0=(0,\dots,0,1)$. Suppose $x_\alpha, y_\alpha\in B_1(0)$ is a maximum point of $u_\alpha$, $v_\alpha$ respectively. Then \begin{gather*} x_\alpha,y_\alpha\to x\in \partial {B_1(0)},\\ \lim_{\alpha\to+\infty}\alpha(1-|x_\alpha|),\; \lim_{\alpha\to+\infty}\alpha(1-|y_\alpha|)\; \in(0,+\infty),\\ \begin{aligned} &\alpha^{-\frac{(2-N)(p+q)+2N}{p+q-2}}\int_{B_1(0)} (|\nabla (u_\alpha-\alpha^{\frac{2}{p+q-2}}u(\alpha(x-x_0))|^2\\ &+ |\nabla (v_\alpha-\alpha^{\frac{2}{p+q-2}}v(\alpha(x-x_0))|^2)dx \to 0 \end{aligned} \end{gather*} as $\alpha\to+\infty$, where $(u,v)$ is a ground state solution of the system \begin{equation}\label{eq:1.3} -\Delta u=\frac{2p}{p+q}e^{x_N} u^{p-1}v^q,\quad -\Delta v=\frac{2q}{p+q}e^{x_N} u^pv^{q-1},\quad \text{in } \mathbb{R}^N_- \end{equation} with $u = v = 0$ on $\partial\mathbb{R}^N_-$. \end{theorem} The proof of Theorem \ref{th1.1} is inspired by that in \cite{CPY}. In section 2, we prove that \eqref{eq:1.3} has a ground state solution pair. We establish in section 3 an asymptotic estimate for $S_{\alpha,p,q}$ which is defined in the section 3. Then using the blow up argument, we show in section 4 that the maximum points of both components of the ground state solution of \eqref{eq:1.1} concentrate on the same point of the boundary of the domain. The proof of Theorem \ref{th1.1} is also given in section 4. \section{A variational problem} We consider for $\gamma>0$ the variational problem \begin{equation}\label{eq:2.1} m_{\gamma,p,q}=\inf_{0\neq u,v\in D^{1,2}_0(\mathbb{R}^N_-)}\frac{\int_{\mathbb{R}^N_-}(|\nabla u|^2+|\nabla v|^2)dx} {(\int_{\mathbb{R}^N_-}e^{\gamma x_N}|u|^p|v|^qdx)^{2/(p+q)}}, \end{equation} where $\ p+q\in (2,2^*)$. We will show that $m_{\gamma,p,q}$ is achieved. First, we prove that the problem is well defined. For any $u\in C^\infty_0(\mathbb{R}^N_-)$, by H\"older's inequality, \[ |u(x',x_N)| \leq |{x_N}|^{1/2}\Big(\int^0_{-\infty}\big|\frac{\partial u(x',t)}{\partial t}\big|^2\,dt\Big)^{1/2}. \] If $p_1+q_1=2$ and $u, v\in C^\infty_0(\mathbb{R}^N_-)$, we have \begin{align*} &\int_{\mathbb{R}^N_-}e^{\gamma x_N}|u|^{p_1}|v|^{q_1}\,dx \\&\leq\int_{x_N\leq0}|{x_N}|e^{\gamma x_N}\,dx_N\int_{\mathbb{R}^{N-1}}\Big(\int^0_{-\infty}\big|\frac{\partial u(x',t)}{\partial t}\big|^2dt\Big)^{\frac{p_1}{2}} \Big(\int^0_{-\infty}\big|\frac{\partial v(x',t)}{\partial t}\big|^2dt\Big)^{\frac{q_1}{2}}dx' \\& \leq C\Big(\int_{\mathbb{R}^N_-}\big|\frac{\partial u(x',x_N)}{\partial x_N}\big|^2\,dx\Big)^{\frac{p_1}{2}} \Big(\int_{\mathbb{R}^N_-}\big|\frac{\partial v(x',x_N)}{\partial x_N}\big|^2\,dx\Big)^{\frac{q_1}{2}} \\&\leq C\int_{\mathbb{R}^N_-}(|\nabla u|^2+|\nabla v|^2)\,dx. \end{align*} If $p_2+q_2=2^*$, again by H\"older's inequality, \[ \int_{\mathbb{R}^N_-}e^{\gamma x_N}|u|^{p_2}|v|^{q_2}\,dx \leq C\Big(\int_{\mathbb{R}^N_-}(|\nabla u|^2+|\nabla v|^2)\,dx\Big)^{2^*/2}. \] Using interpolation inequality for $p+q\in(2,2^*)$, we have \[ \int_{\mathbb{R}^N_-}e^{\gamma x_N}|u|^{p}|v|^{q}\,dx \leq C\Big(\int_{\mathbb{R}^N_-}(|\nabla u|^2+|\nabla v|^2)\,dx\Big) ^{(p+q)/2}. \] This implies $m_{\gamma,p,q}>0$. Next, for every $ R>0$, \[ \int_{x_N\leq -R}\int_{\mathbb{R}^{N-1}}e^{\gamma x_N}|u|^{p}|v|^{q}\,dx \leq Ce^{-\gamma R/2}\Big(\int_{\mathbb{R}^N_-}(|\nabla u|^2+|\nabla v|^2)\,dx\Big)^{(p+q)/2}. \] Hence, $\int_{x_N\leq -R}e^{\gamma x_N}|u|^{p}|v|^{q}\,dx$ is uniformly decay in the $x_N$-direction. The variational problem $m_{\gamma,p,q}$ is compact in the $x_N$-direction and it is translation invariant in $x_1,\dots,x_{N-1}$. So we may prove as the proof of \cite[Theorem 1.4]{W} the following result. \begin{proposition}\label{prop2.1} Suppose $p+q\in(2,2^*)$, $\gamma>0$, $N\geq3$. Then, $m_{\gamma,p,q}$ is achieved by $(u,v)$ with positive functions $u,v\in D^{1,2}_0(\mathbb{R}^N_-)$. \end{proposition} \section{Estimate for $S_{\alpha,p,q}$} It is known that the problem \[ S_{\alpha,p,q}=\inf_{u,v\in H^1_0(B_1(0))\setminus\{0\}}J_\alpha(u,v) =\inf_{u,v\in H^1_0(B_1(0))\setminus\{0\}}\frac{\int_{B_1(0)}(|\nabla u|^2+|\nabla v|^2)\,dx}{(\int_{B_1(0)} |x|^\alpha|u|^p|v|^q\,dx)^{2/(p+q)}} \] is achieved and the minimizer is a solution of problem \eqref{eq:1.1} up to a constant. Furthermore, we have the following result. \begin{proposition}\label{prop3.1} Let $p+q\geq2$. There is $C>0$ such that $$ C\leq\frac{S_{\alpha,p,q}}{\alpha^{2- N+\frac{2N}{p+q}}} \leq m_{1,p,q}+o(1), $$ where $o(1)\to 0$ as $\alpha\to \infty$. \end{proposition} \begin{proof} We use the idea in \cite{CPY}. We establish the upper bound first. For any $\varepsilon>0$, there exist $w_\varepsilon, h_\varepsilon\in C^\infty_0(\mathbb{R}^N_-)$, $w_\varepsilon, h_\varepsilon \neq 0$, such that $$ J_{\varepsilon,\mathbb{R}^N_-}(w_\varepsilon,h_\varepsilon) = \frac{\int_{\mathbb{R}^N_-}(|\nabla w_\varepsilon|^2+|\nabla h_\varepsilon|^2)dx} {(\int_{\mathbb{R}^N_-}e^{ x_N}|w_\varepsilon|^p|h_\varepsilon|^qdx)^{2/(p+q)}} 0$ is large enough. Denote $\tilde{B}_\alpha=\{y:\alpha^{-1}y+x_0\in B_1(0)\}$, where $x_0=(0,0,\dots,1)$. Then \begin{equation} \label{eq:3.1} \begin{aligned} \int_{B_1(0)}|\nabla u_\alpha|^2dx &=\alpha^{2-N}\int_{\tilde{B}_\alpha}(\sum^{N-1}_{i=1}\Big|D_iw_\varepsilon(x',x_N +\alpha(1+(1-\frac{1}{\alpha^2}|x'|^2)^{1/2})) \\ &\quad +\frac{\alpha^{-1}x_i}{(1-\alpha^{-2}|x'|^2)^{1/2}} D_Nw_\varepsilon(x',x_N +\alpha(1+(1-\frac{1}{\alpha^2}|x'|^2)^{1/2}))\Big|^2 \\ &\quad +|D_Nw_\varepsilon(x',x_N +\alpha(1+(1-\frac{1}{\alpha^2}|x'|^2)^{1/2}))|^2)\,dx. \end{aligned} \end{equation} Let $$y_i=x_i,\ i=1,2,\dots,N-1;\ y_N=x_N+\alpha(1+(1-\frac{1}{\alpha^2}|x'|^2)^{1/2}),$$ then $|det(\frac{\partial y}{\partial x})|=1$. By \eqref{eq:3.1}, \begin{equation} \label{eq:3.2} \begin{aligned} \int_{B_1(0)}|\nabla u_\alpha|^2dx &=\alpha^{2-N}\int_{\mathbb{R}^N_-}(\sum^{N-1}_{i=1}|D_iw_\varepsilon +O(\alpha^{-1})D_Nw_\varepsilon|^2 +|D_Nw_\varepsilon|^2)\ dy \\&=\alpha^{2-N}(\int_{\mathbb{R}^N_-}|\nabla w_\varepsilon|^2\ dy+O(\alpha^{-1})). \end{aligned} \end{equation} Similarly, \begin{equation} \label{eq:3.3} \begin{aligned} \int_{B_1(0)}|\nabla v_\alpha|^2dx =\alpha^{2-N}(\int_{\mathbb{R}^N_-}|\nabla h_\varepsilon|^2\ dy+O(\alpha^{-1})). \end{aligned} \end{equation} For any $x\in sptw_\varepsilon \cap spth_\varepsilon$, we have \[ |\frac{x}{\alpha}+x_0|^\alpha=(1+\frac{2x_N}{\alpha}+O(\alpha^{-2}) )^{\alpha/2}=e^{x_N+O(\alpha^{-1})}. \] Therefore, \begin{equation} \label{eq:3.4} \begin{aligned} &\int_{B_1(0)}|x|^\alpha|u_\alpha|^p|v_\alpha|^q\,dx\\ &=\alpha^{-N}\int_{\widetilde{B}_\alpha}|\frac{x}{\alpha}+x_0|^\alpha|w_\varepsilon(x',x_N+ \alpha(1+(1-\frac{1}{\alpha^2|x'|^2})^{1/2}))|^p\\ & \quad \times |h_\varepsilon(x',x_N+\alpha(1+(1-\frac{1}{\alpha^2|x'|^2})^{1/2}))|^q\,dx \\ &=\alpha^{-N}\int_{\mathbb{R}^N_-}e^{x_N-\alpha(1-(1-\frac{1}{\alpha^2|x'|^2})^{1/2}) +O(\alpha^{-1})}|w_\varepsilon|^p|h_\varepsilon|^q\,dx\\ &=\alpha^{-N}(\int_{\mathbb{R}^N_-}e^{x_N}|w_\varepsilon|^p|h_\varepsilon|^q\ dy+O(\alpha^{-1})). \end{aligned} \end{equation} It follows from \eqref{eq:3.2}, \eqref{eq:3.3} and \eqref{eq:3.4} that \begin{align*} J_\alpha(u_\alpha,v_\alpha) &=\alpha^{2-N+\frac{2N}{p+q}}(J_{\varepsilon,{\mathbb{R}^N_-}}(w_\varepsilon,h_\varepsilon)+O(\alpha^{-1})) \\&<\alpha^{2-N+\frac{2N}{p+q}}(m_{1,p,q}+\varepsilon+O(\alpha^{-1})) \end{align*} and then, \[ \frac{S_{\alpha,p,q}}{\alpha^{2- N+\frac{2N}{p+q}}}\leq m_{1,p,q}+o(1). \] Next, we show the lower bound. Let $r\in (0,1], \ \omega\in S^{N-1}$. For any $u,v\in H^1_0(B_1(0)\setminus\{0\})$, we define $ \varphi(r,\omega)=u(r^\beta,\omega),\psi(r,\omega)=v(r^\beta,\omega)$, where $\beta=\frac{N}{N+\alpha}$. Then \begin{equation} \label{eq:3.5} \int_{B_1(0)}|x|^\alpha|u|^p|v|^qdx =\beta\int^1_0\int_{\omega\in S^{N-1}}|\varphi(r,\omega)|^p|\psi(r,\omega)|^qr^{N-1}\,dr\,d\omega, \end{equation} and \begin{equation} \label{eq:3.6} \begin{aligned} &\int_{B_1(0)}|\nabla u|^2dx\\ &=\beta\int^1_0\int_{\omega\in S^{N-1}}(\frac{1}{\beta^2r^{2(\beta-1)}}|\varphi_r(r,\omega)|^2 +\frac{1}{r^{2\beta}}|\nabla_\omega\varphi(r,\omega)|^2) r^{\beta(N-1)+\beta-1}\, dr\,d\omega\\ &=\frac{1}{\beta}\int^1_0\int_{\omega\in S^{N-1}}(|\varphi_r(r,\omega)|^2+\frac{\beta^2}{r^2} |\nabla_\omega\varphi(r,\omega)|^2))r^{(2-N)(1-\beta)+N-1}\,dr\,d\omega. \end{aligned} \end{equation} Similarly, \begin{equation} \label{eq:3.8} \begin{aligned} &\int_{B_1(0)}|\nabla v|^2dx\\ & =\frac{1}{\beta}\int^1_0\int_{\omega\in S^{N-1}}(|\psi_r(r,\omega)|^2+\frac{\beta^2}{r^2} |\nabla_\omega\psi(r,\omega)|^2)r^{(2-N)(1-\beta)+N-1}\,dr\,d\omega.\\ \end{aligned} \end{equation} Note that \begin{equation} \label{eq:3.9} |\nabla_\omega\varphi|^2=\sum^{N-1}_{i=1}(\frac{\partial \varphi}{\partial x_i}-\frac{x_i}{(1-|x'|^2)^{1/2}}\frac{\partial \varphi}{\partial x_N})^2 \end{equation} and $ d\omega=(1+|x'|^2)^{-1/2}dx'$. Let $\bar \varphi(r,x')=\varphi(r,\beta x')$, $\bar {\psi}(r,x')=\psi(r,\beta x')$, $S_\beta=\{x\in S^{N-1}:|x'|\leq \eta \beta\}$, where $\eta>0$ is small. Then, we may deduce as in \cite{CPY} that for $\beta>0$ small, \begin{equation} \label{eq:3.12} \begin{aligned} &\int^1_0\int_{S_\beta}(|\varphi_r(r,\omega)|^2+ \frac{\beta^2}{r^2}|\nabla_\omega\varphi(r,\omega)|^2))r^{(2-N)(1-\beta)+N-1}\,dr\,d\omega \\ &\geq C\beta^{N-1}\int_{B_\eta}|\nabla\bar{\varphi}(r,x')|^2r^{(2-N)(1-\beta)}\,dr\,dx' \end{aligned} \end{equation} and \begin{equation} \label{eq:3.13} \begin{aligned} &\int^1_0\int_{S_\beta}(|\psi_r(r,\omega)|^2+\frac{\beta^2}{r^2} |\nabla_\omega\psi(r,\omega)|^2))r^{(2-N)(1-\beta)+N-1}\,dr\,d\omega \\ &\geq C\beta^{N-1}\int_{B_\eta}|\nabla\bar{\psi}(r,x')|^2r^{(2-N)(1-\beta)}\,dr\,dx', \end{aligned} \end{equation} where $B_\eta=\{x\in B_1(0):|x'|\leq \eta\}$. Similarly, \begin{equation} \label{eq:3.14} \begin{aligned} &\int^1_0\int_{S_\beta}|\varphi(r,\omega)|^p|\psi(r,\omega)|^qr^{N-1}\,dr\,d\omega \\ &\leq C\beta^{N-1}\int_{B_\eta}|\bar{\varphi}(r,x')|^p|\bar{\psi}(r,x')|^qr^{N-1}\,dr\,dx'. \end{aligned} \end{equation} Since $\bar{\varphi},\bar{\psi}=0$ on $S^{N-1}$, there exists a constant $C>0$ such that \begin{equation} \label{eq:3.15} (\int_{B_\eta}|\bar{\varphi}(r,x')|^{p+q}r^{N-1}\,dr\,dx') ^{2/(p+q)} \leq C \int_{B_\eta}|\nabla\bar{\varphi}(r,x')|^2r^{(2-N)(1-\beta)} \,dr\,dx' \end{equation} and \begin{equation} \label{eq:3.16} \Big(\int_{B_\eta}|\bar{\psi}(r,x')|^{p+q}r^{N-1}\,dr\,dx'\Big)^{2/(p+q)} \leq C \int_{B_\eta}|\nabla\bar{\psi}(r,x')|^2r^{(2-N)(1-\beta)}\,dr\,dx'. \end{equation} Therefore, by \eqref{eq:3.15} and \eqref{eq:3.16}, \begin{equation} \label{eq:3.17} \begin{aligned} &\Big(\int_{B_\eta}|\bar{\varphi}(r,x')|^p| \bar{\psi}(r,x')|^{q}r^{N-1}\,dr\,dx'\Big)^{2/(p+q)} \\ & \leq C\int_{B_\eta}(|\bar{\varphi}(r,x')|^{p+q}r^{N-1}\,dr\,dx')^{2/(p+q)} +C\int_{B_\eta}(|\bar{\psi}(r,x')|^{p+q}r^{N-1}\,dr\,dx')^{2/(p+q)} \\ & \leq C\int_{B_\eta}(|\nabla\bar{\varphi}(r,x')|^2+ |\nabla\bar{\psi}(r,x')|^2)r^{(2-N)(1-\beta)}\,dr\,dx'. \end{aligned} \end{equation} We derive from \eqref{eq:3.12}-\eqref{eq:3.17} that \begin{equation} \label{eq:3.18} \begin{aligned} &\Big(\int^1_0\int_{S_\beta}|\varphi(r,\omega)|^p|\psi(r,\omega)|^qr^{N-1}\,dr\,d\omega\Big)^{2/(p+q)} \\ &\leq C\beta^{\frac{2(N-1)}{p+q}}(\int_{B_\eta}|\bar{\varphi}(r,x')|^p |\bar{\psi}(r,x')|^qr^{N-1}\,dr\,dx')^\frac{2}{p+q} \\ &\leq C\beta^{\frac{2(N-1)}{p+q}}\int_{B_\eta}(|\nabla\bar{\varphi}(r,x')|^2+ |\nabla\bar{\psi}(r,x')|^2)r^{(2-N)(1-\beta)}\,dr\,dx' \\&\leq C\beta^{1-N+\frac{2(N-1)}{p+q}}\int^1_0\int_{S_\beta}(|\varphi_r(r,\omega)|^2 +\frac{\beta^2}{r^2}|\nabla_\omega\varphi(r,\omega)|^2 \\ &\quad +|\psi_r(r,\omega)|^2+\frac{\beta^2}{r^2}|\nabla_\omega \psi(r,\omega)|^2)r^{(2-N)(1-\beta)+N-1}\,dr\,d\omega \end{aligned} \end{equation} Since \eqref{eq:1.1} is rotation invariant, we may choose $\beta>0$ so that $S^{N-1}$ can be covered by finite number $S_\beta$ up to a rotation, that is $S^{N-1}\subset\cup S_\beta$. Then \begin{equation} \label{eq:3.19} \begin{aligned} &\Big(\int^1_0\int_{\omega\in S^{N-1}}| \varphi(r,\omega)|^p|\psi(r,\omega)|^qr^{N-1}\,dr\,d\omega\Big)^{2/(p+q)} \\ &\leq \sum\Big(\int^1_0\int_{S_\beta}| \varphi(r,\omega)|^p|\psi(r,\omega)|^qr^{N-1}\,dr\,d\omega\Big)^{2/(p+q)} \\ &\leq C\beta^{1-N+\frac{2(N-1)}{p+q}} \int^1_0\int_{S_\beta}(|\varphi_r(r,\omega)|^2 +\frac{\beta^2}{r^2}|\nabla_\omega\varphi(r,\omega)|^2 \\ &\quad +|\psi_r(r,\omega)|^2+\frac{\beta^2}{r^2}| \nabla_\omega\psi(r,\omega)|^2)r^{(2-N)(1-\beta)+N-1}\,dr\,d\omega \\ &\leq C\beta^{1-N+\frac{2(N-1)}{p+q}}\int^1_0\int_{S^{N-1}} (|\varphi_r(r,\omega)|^2 +\frac{\beta^2}{r^2}|\nabla_\omega\varphi(r,\omega)|^2 \\ &\quad +|\psi_r(r,\omega)|^2+\frac{\beta^2}{r^2} |\nabla_\omega\psi(r,\omega)|^2)r^{(2-N)(1-\beta)+N-1}\,dr\,d\omega. \end{aligned} \end{equation} Hence, we deduce from \eqref{eq:3.6}-\eqref{eq:3.8} and \eqref{eq:3.19} that \begin{equation} \label{eq:3.20} \frac{\int_{B_1(0)}(|\nabla u|^2+|\nabla v|^2)\,dx}{(\int_{B_1(0)} |x|^\alpha|u|^p|v|^q\,dx)^{2/(p+q)}} \geq C \beta^{N-2-\frac{2N}{p+q}} =C \alpha^{2-N+\frac{2N}{p+q}}. \end{equation} It yields \begin{equation} \label{eq:3.21} \alpha^{N-2-\frac{2N}{p+q}}\frac{\int_{B_1(0)}(|\nabla u|^2+|\nabla v|^2)\,dx}{(\int_{B_1(0)} |x|^\alpha|u|^p|v|^q\,dx)^{2/(p+q)}}\geq C. \end{equation} The proof is complete since $u$ and $v$ are arbitrary. \end{proof} \section{Asymptotic behavior of ground state solution} Let $(U_\alpha,V_\alpha)$ be a minimizer of $S_{\alpha,p,q}$. Choosing $\lambda_\alpha=(\frac{2}{S_{\alpha,p,q}})^{\frac{1}{2-(p+q)}}$ and defining $u_\alpha=\lambda_\alpha U_\alpha,v_\alpha=\lambda_\alpha V_\alpha$, we see that $(u_\alpha, v_\alpha)$ is a solution pair of \eqref{eq:1.1}, which is also a minimizer of $S_{\alpha,p,q}$. That is, \begin{equation} \label{eq:4.1} \int_{B_1(0)}(|\nabla u_\alpha|^2+|\nabla v_\alpha|^2)\,dx =2\int_{B_1(0)}|x|^\alpha|u_\alpha|^{p}|v_\alpha|^{q}\,dx \end{equation} and \begin{equation} \label{eq:4.2} S_{\alpha,p,q}=\frac{\int_{B_1(0)}(|\nabla u_\alpha|^2+|\nabla v_\alpha|^2)\,dx} {(\int_{B_1(0)}|x|^\alpha|u_\alpha|^{p}|v_\alpha|^{q}\,dx)^{2/(p+q)}}. \end{equation} It yields \begin{equation} \label{eq:4.3} \int_{B_1(0)}(|\nabla u_\alpha|^2+|\nabla v_\alpha|^2)\,dx =2\int_{B_1(0)}|x|^\alpha|u_\alpha|^{p}|v_\alpha|^{q}\,dx= 2^{-\frac{2}{p+q-2}}S_{\alpha,p,q}^{\frac{p+q}{p+q-2}}. \end{equation} By Proposition \ref{prop3.1} \begin{equation} \label{eq:4.4} C\alpha^{\frac{(2-N)(p+q)+2N}{p+q-2}} \leq \int_{B_1(0)}(|\nabla u_\alpha|^2+|\nabla v_\alpha|^2)\,dx \leq C'\alpha^{\frac{(2-N)(p+q)+2N}{p+q-2}}. \end{equation} Let $$\bar{u}_\alpha(x)=\alpha^{-\frac{2}{p+q-2}}u_\alpha(\frac{x}{\alpha}),\ \ \bar{v}_\alpha(x)=\alpha^{-\frac{2}{p+q-2}}v_\alpha(\frac{x}{\alpha}),\ \ x\in B_\alpha(0).$$ Then \begin{equation} \label{eq:4.5} C\leq\int_{B_\alpha(0)}(|\nabla \bar{u}_\alpha|^2+|\nabla \bar{v}_\alpha|^2)\,dx\leq C'. \end{equation} Choose $p_1, q_1 >0$ such that $p>p_1,\, q>q_1$ and $p_1+q_1=2$. \begin{lemma}\label{lemma4.1} As $\alpha\to+\infty$, we have \[ 00$ such that $|\bar{u}_\alpha(x)|\leq C$, $|\bar{v}_\alpha(x)|\leq C$ for $x\in B_\alpha(0)$. \end{lemma} \begin{proof} Since $(u_\alpha,v_\alpha)$ is a solution pair of \eqref{eq:1.1}, then for $x\in B_\alpha(0)$ we have \begin{equation} \label{eq:4.7} -\Delta \bar{u}_\alpha(x) =\frac{2p}{p+q}|\frac{x}{\alpha}|^\alpha \bar{u}^{p-1}_\alpha(x)\bar{v}^{q}_\alpha(x)\leq \frac{2p}{p+q} \bar{u}^{p-1}_\alpha(x)\bar{v}^{q}_\alpha(x) \end{equation} and \[ -\Delta \bar{v}_\alpha(x) =\frac{2q}{p+q}|\frac{x}{\alpha}|^\alpha \bar{u}^{p}_\alpha(x)\bar{v}^{q-1}_\alpha(x)\leq \frac{2q}{p+q} \bar{u}^{p}_\alpha(x)\bar{v}^{q-1}_\alpha(x). \] Now we use the Moser iteration to prove the result. Without confusion, we use $(u,v)$ to denote $(\bar u_\alpha, \bar v_\alpha)$. Let $s\geq 1$. Multiplying \eqref{eq:4.7} by $u^{2s}$ and integrating by parts, we obtain \[ s^{-2}(2s -1)\int_{B_\alpha(0)}|\nabla u^s|^2\,dx \leq \frac{2p}{p+q}\int_{B_\alpha(0)}u^{p-1+2s} v^q\,dx. \] Since $s^{-2}(2s -1)\geq s^{-1}$ if $s\geq 1$, \[ \int_{B_\alpha(0)}|\nabla u^s|^2\,dx \leq \frac{2sp}{p+q}\int_{B_\alpha(0)}u^{p-1+2s} v^q\,dx. \] By Sobolev inequality and H\"older's inequality, we deduce \begin{equation}\label{eq:4.8} \begin{aligned} &\Big(\int_{B_\alpha(0)}u^{2^*s}\,dx\Big)^{2/2^*}\\ &\leq \frac{2sp}{p+q}\int_{B_\alpha(0)}u^{p-1+2s} v^q\,dx \\ &\leq\frac{2sp}{p+q}\Big(\int_{B_\alpha(0)}u^{p+q-1+2s}\,dx\Big)^{\frac{p-1+2s}{p+q-1+2s}}\Big(\int_{B_\alpha(0)}v^{p+q-1+2s}\,dx\Big)^{\frac{q}{p+q-1+2s}}\\ &\leq\frac{sp}{p+q}\int_{B_\alpha(0)}(u^{p+q-1+2s} + v^{p+q-1+2s})\,dx. \end{aligned} \end{equation} Similarly, we have \begin{equation}\label{eq:4.9} \Big(\int_{B_\alpha(0)}v^{2^*s}\,dx\Big)^{2/2^*} \leq\frac{sq}{p+q}\int_{B_\alpha(0)}(u^{p+q-1+2s} + v^{p+q-1+2s})\,dx. \end{equation} Therefore, \begin{equation}\label{eq:4.10} \begin{aligned} \Big(\int_{B_\alpha(0)}(u^{2^*s} + v^{2^*s})\,dx\Big)^{2/2^*} &\leq \Big(\int_{B_\alpha(0)}u^{2^*s}\,dx\Big)^{2/2^*} + \Big(\int_{B_\alpha(0)}v^{2^*s}\,dx\Big)^{2/2^*}\\ &\leq s\int_{B_\alpha(0)}(u^{p+q-1+2s} + v^{p+q-1+2s})\,dx. \end{aligned} \end{equation} Now we define $\{s_j\}$ by induction. Let $p+q-1+2s_0 =2^*$ and $p+q-1+2s_{j+1} =2^*s_j$, $j= 0,1, 2,\dots$. We also define $M_0 = 1$, $M_{j+1} = (s_jM_j)^{\frac {2^*}2}$, $j= 0,1, 2,\dots $. We claim that for all $j\geq 0$, \begin{equation}\label{eq:4.11} \int_{B_\alpha(0)}(u^{p+q-1+2s_{j}} + v^{p+q-1+2s_{j}})\,dx\leq C M_j \end{equation} and \begin{equation}\label{eq:4.12} M_j\leq e^{ms_{j-1}}, \end{equation} where $C, m>0$. \eqref{eq:4.11} and \eqref{eq:4.12} imply \[ \Big(\int_{B_\alpha(0)}(u^{2^*s_{j}} + v^{2^*s_{j}})\,dx\Big) ^{\frac 1{2^*s_j}}\leq C M_j^{\frac 1{2^*s_j}} \leq e^{\frac {ms_{j-1}}{2^*s_j}}\leq C \] for all $j$. The assertion then follows. Now, we show \eqref{eq:4.11}. Obviously, if $j = 0$, \eqref{eq:4.11} holds. Suppose it holds for $j$, we deduce it holds for $j+1$. Indeed, \begin{align*} &\int_{B_\alpha(0)}(u^{p+q-1+2s_{j+1}} + v^{p+q-1+2s_{j+1}})\,dx\\ &= \int_{B_\alpha(0)}(u^{2^*s_{j}} + v^{2^*s_{j}})\,dx\\ &\leq s_j^{2^*/2}\Big(\int_{B_\alpha(0)}(u^{p+q-1+2s_{j}} + v^{p+q-1+2s_{j}})\,dx\Big)^{2^*/2}\\ &\leq \Big(s_jM_j\Big)^{2^*/2} = M_{j+1}. \end{align*} Inequality \eqref{eq:4.12} can be proved, as in \cite{LNT}. \end{proof} Let $M_\alpha=\max_{x\in {B_1(0)}}u_\alpha$, $N_\alpha=\max_{x\in {B_1(0)}}v_\alpha$. \begin{lemma}\label{lemma4.3} There holds $$ C_2\alpha^{\frac{2}{p+q-2}}\leq M_\alpha,N_\alpha\leq C_3\alpha^{\frac{2}{p+q-2}}. $$ \end{lemma} \begin{proof} By Lemmas \ref{lemma4.1} and \ref{lemma4.2}, we have \[ 00 $$ and \[ \int_{B_\alpha(-x_\alpha)}(|\nabla \tilde{u}_\alpha|^2+|\nabla \tilde{v}_\alpha|^2)\,dx\leq C. \] Suppose that $\alpha(1-|x_\alpha|)\to +\infty$, we assume that there are $\tilde{u},\tilde{v}\in D^{1,2}(\mathbb{R}^N)$ such that \begin{gather*} \tilde{u}_\alpha\rightharpoonup \tilde{u},\quad \tilde{v}_\alpha\rightharpoonup \tilde{v},\quad \text{in } D^{1,2}(\mathbb{R}^N)\\ \tilde{u}_\alpha\to \tilde{u},\quad \tilde{v}_\alpha\to \tilde{v},\quad \text{in } C^1_{\rm loc}(\mathbb{R}^N). \end{gather*} Now, we distinguish two cases: (i) $|x_\alpha|\leq l<1$; (ii) $|x_\alpha|\to 1$ as $\alpha\to+\infty$. For any $x$ with $|x|\leq C$, in case (i), we have \[ |\frac{x}{\alpha}+x_\alpha|^\alpha\leq (\frac{C}{\alpha}+|x_\alpha|)^\alpha\leq (\frac{C}{\alpha}+l)^\alpha\leq(l+\varepsilon)^\alpha\to 0, \quad\text{as } \alpha\to+\infty. \] In case (ii), since $\alpha(1-|x_\alpha|)\to +\infty$, \[ |\frac{x}{\alpha}+x_\alpha|^\alpha \leq e^{\alpha \ln (\frac{|x|}{\alpha}+|x_\alpha|-1+1)} =O(e^{\alpha (\frac{|x|}{\alpha}+|x_\alpha|-1)}) =O(e^{|x|+\alpha(|x_\alpha|-1)})\to 0. \] So $\tilde{u}$ satisfies \[ -\Delta \tilde{u}=0, \quad \tilde{u}\in D^{1,2}(\mathbb{R}^N). \] This implies $\tilde{u}=0$, a contradiction to $\tilde{u}(0)=\lim_{\alpha\to+\infty}\tilde{u}_\alpha(0)\geq C>0$. Therefore, $\alpha(1-|x_\alpha|)\to L<+\infty$. Now, we claim $L>0$. Indeed, we have $\tilde{u}(0)=\lim_{\alpha\to+\infty}\tilde{u}_\alpha(0)>0$. Since \eqref{eq:1.1} is invariant under the rotations. After suitably rotating the coordinate system, we may assume that $x_\alpha=(0,\dots,0,x^\alpha_N)$, where $x^\alpha_N\to 1$, as $\alpha\to+\infty$. Then $(\tilde{u},\tilde{v})$ is a positive solution pair of \eqref{eq:1.3} in $\Omega=\mathbb{R}^N_-+(0,\dots,0,L)$ with $\tilde{u}=\tilde{v}=0$ on $\partial\Omega$. If $L=0$, we would have $\Omega=\mathbb{R}^N_-$, and then we obtain $\tilde{u}(0)=0$, a contradiction. The proof is complete. \end{proof} By Lemma \ref{lemma4.4}, we know that $x_\alpha\to x_0\in\partial{B_1(0)}$, $y_\alpha\to y_0\in\partial{B_1(0)}$ if $\alpha\to+\infty$. In the following, we show that $x_0=y_0$. \begin{lemma}\label{lemma4.6} Both $x_\alpha$ and $y_\alpha$ converge to a point $x_0\in \partial B_1(0)$ as $\alpha\to+\infty$. \end{lemma} \begin{proof} We argue by contradiction. Suppose $x_0\neq y_0$, then there is a $\delta>0$ such that $B_\delta(x_0)\cap B_\delta(y_0)=\emptyset$. After suitably rotating the coordinate system, we may assume that $x_0=(0,\dots,0,1)$. Applying the blow up argument for $$ \tilde{u}_\alpha(x)=\alpha^{-\frac{2}{p+q-2}}u_\alpha (\frac{x}{\alpha}+x_0),\quad \tilde{v}_\alpha(x)=\alpha^{-\frac{2}{p+q-2}} v_\alpha(\frac{x}{\alpha}+x_0) $$ in $B_1(0)\cap B_\delta(x_0)$, since $\tilde{u}_\alpha,\tilde{v}_\alpha$ are bounded in $D^{1,2}_0(\mathbb{R}^N_-)$, we may assume that there are $\tilde{u},\tilde{v}\in D^{1,2}_0(\mathbb{R}^N_-)$ such that \begin{gather*} \tilde{u}_\alpha\rightharpoonup \tilde{u},\quad \tilde{v}_\alpha\rightharpoonup \tilde{v},\quad\text{in } D^{1,2}_0(\mathbb{R}^N_-),\\ \tilde{u}_\alpha\to \tilde{u},\quad \tilde{v}_\alpha\to \tilde{v},\quad \text{in } C^1_{\rm loc}(\mathbb{R}^N_-). \end{gather*} Moreover, $(\tilde{u},\tilde{v})$ with $\tilde u,\tilde{v}\in D^{1,2}_0(\mathbb{R}^N_-)$ is a positive solution of \eqref{eq:1.3}. In the same way, we may assume $y_0=(0,\dots,0,1)$. Define $$ \bar{u}_\alpha(x)=\alpha^{-\frac{2}{p+q-2}}u_\alpha(\frac{x}{\alpha} + y_0),\quad \bar{v}_\alpha(x)=\alpha^{-\frac{2}{p+q-2}}v_\alpha(\frac{x}{\alpha}+ y_0). $$ Then \begin{gather*} \bar u_\alpha\rightharpoonup \bar u,\quad \bar v_\alpha\rightharpoonup \bar v, \quad \text{in } D^{1,2}_0(\mathbb{R}^N_-),\\ \bar u_\alpha\to \bar u,\quad \bar v_\alpha\to \bar v,\quad \text{in } C^1_{\rm loc}(\mathbb{R}^N_-), \end{gather*} and $(\bar{u},\bar{v})$ is a positive solution of \eqref{eq:1.3}. It implies \[ \int_{\mathbb{R}^N_-}(|\nabla \tilde{u} |^2+|\nabla \tilde{v} |^2)dx,\quad \int_{\mathbb{R}^N_-}(|\nabla\bar{u} |^2+|\nabla \bar{v} |^2)dx \geq 2^{-\frac{2}{p+q-2}}m^{\frac{p+q}{p+q-2}}_{1,p,q} \] By Proposition \ref{prop3.1}, \begin{align*} I(u_\alpha,v_\alpha) &=(\frac{1}{2}-\frac{1}{p+q})\int_{B_1(0)}(|\nabla u_\alpha|^2+|\nabla v_\alpha |^2)dx\\ &=(\frac{1}{2}-\frac{1}{p+q})2^{-\frac{2}{p+q-2}} S^{\frac{p+q}{p+q-2}}_{\alpha,p,q}\\ & \leq (\frac{1}{2}-\frac{1}{p+q})2^{-\frac{2}{p+q-2}} \alpha^{\frac{(2-N)(p+q)+2N}{p+q-2}} (m^{\frac{p+q}{p+q-2}}_{1,p,q}+o(1)). \end{align*} On the other hand, \begin{align*} I(u_\alpha,v_\alpha) &\geq (\frac{1}{2}-\frac{1}{p+q})\int_{B_1(0)\cap B_\delta(x_0)} (|\nabla u_\alpha|^2+|\nabla v_\alpha |^2)dx \\ &\quad +(\frac{1}{2}-\frac{1}{p+q})\int_{B_1(0)\cap B_\delta( y_0)}(|\nabla u_\alpha|^2+|\nabla v_\alpha |^2)dx \\ &\geq\alpha^{\frac{(2-N)(p+q)+2N}{p+q-2}}(\frac{1}{2}-\frac{1}{p+q}) \int_{B_\alpha(-x_0)\cap B_{\alpha\delta}(0)}(|\nabla \tilde{u}_\alpha |^2 +|\nabla \tilde{v}_\alpha |^2)dx \\ &\quad + \alpha^{\frac{(2-N)(p+q)+2N}{p+q-2}}(\frac{1}{2}-\frac{1}{p+q})\int_{B_\alpha(- y_0)\cap B_{\alpha\delta}(0)}(|\nabla \bar{u}_\alpha |^2+|\nabla \bar{v}_\alpha |^2)dx. \end{align*} So we obtain \begin{align*} &\int_{B_\alpha(-x_0)\cap B_{\alpha\delta}(0)}(|\nabla \tilde{u}_\alpha |^2+|\nabla \tilde{v}_\alpha |^2)dx + \int_{B_\alpha(-\theta y_0)\cap B_{\alpha\delta}(0)}(|\nabla \bar{u}_\alpha |^2+|\nabla \bar{v}_\alpha |^2)dx \\&\leq 2^{-\frac{2}{p+q-2}}(m^{\frac{p+q}{p+q-2}}_{1,p,q}+o(1)). \end{align*} Therefore, \begin{align*} 2^{-\frac{2}{p+q-2}}(m^{\frac{p+q}{p+q-2}}_{1,p,q}+o(1)) &\geq \int_{\mathbb{R}^N_-}(|\nabla \tilde{u} |^2+|\nabla \tilde{v} |^2)dx + \int_{\mathbb{R}^N_+}(|\nabla \bar{u} |^2+|\nabla \bar{v} |^2)dx \\ &\geq 2^{-\frac{2}{p+q-2}}(m^{\frac{p+q}{p+q-2}}_{1,p,q}+M^{\frac{p+q}{p+q-2}}_{1,p,q}), \end{align*} which is impossible. The proof is complete. \end{proof} Now, we may assume that $x_0=(0,\dots,0,1)$. Let \begin{equation} \label{eq:4.13} \hat{u}_\alpha(x)=\alpha^{-\frac{2}{p+q-2}}u_\alpha(\frac{x}{\alpha}+x_0),\ \ \hat{v}_\alpha(x)=\alpha^{-\frac{2}{p+q-2}}v_\alpha(\frac{x}{\alpha}+x_0), \end{equation} which, as before, satisfies \begin{gather*} \hat u_\alpha\rightharpoonup \hat u,\quad \hat v_\alpha\rightharpoonup \hat v, \quad \text{in } D^{1,2}_0(\mathbb{R}^N_-), \\\ \hat u_\alpha\to \hat u,\quad \hat v_\alpha\to \hat v,\quad \text{in } C^1_{\rm loc}(\mathbb{R}^N_-) \end{gather*} and $(\hat u,\hat v)\neq (0,0)$ is a positive solution of \eqref{eq:1.3}. Finally, we have following result. \begin{proposition}\label{prop4.1} The pair $(\hat u,\hat v)$ is a minimizer of $m_{1,p,q}$, which satisfies $$ \int_{R^N_-}(|\nabla (\hat{u}_\alpha-\hat u )|^2+ |\nabla (\hat{v}_\alpha-\hat v)|^2)dx\to 0,\quad\text{as } \alpha\to+\infty. $$ \end{proposition} \begin{proof} By \eqref{eq:1.3}, we have \[ \int_{\mathbb{R}^N_-}(|\nabla \hat u|^2+|\nabla \hat v|^2)dx=2\int_{\mathbb{R}^N_-}e^{x_N}\hat u^p\hat v^qdx, \] and $$ m_{1,p,q}\leq\frac{\int_{\mathbb{R}^N_-}(|\nabla u|^2+|\nabla v|^2)dx} {(\int_{\mathbb{R}^N_-}e^{ x_N}|u|^p|v|^qdx)^{2/(p+q)}}. $$ So we obtain \[ \int_{\mathbb{R}^N_-}(|\nabla u|^2+|\nabla v|^2)dx \geq 2^{-\frac{2}{p+q-2}}m^{\frac{p+q}{p+q-2}}_{1,p,q}. \] For $R>0$ define $$ B_{R,\alpha}=\{x:\frac{x}{\alpha}+x_0\in B_{\frac{R}{\alpha}(x_0)}\cap B_1(0)\},\ \ \Omega_{\alpha}=\{x:\frac{x}{\alpha}+x_0\in B_1(0)\}. $$ By Proposition \ref{prop3.1}, $(u_\alpha,v_\alpha)$ is a minimizer of $S_{\alpha,p,q}$ and satisfies \eqref{eq:1.1}, then \begin{equation} \label{eq:4.14} \int_{B_1(0)}(|\nabla u_\alpha|^2+|\nabla v_\alpha|^2)dx \leq 2^{-\frac{2}{p+q-2}}\alpha^{\frac{(2-N)(p+q)+2N}{p+q-2}} (m^{\frac{p+q}{p+q-2}}_{1,p,q}+o(1)). \end{equation} Moreover, \begin{equation} \label{eq:4.15} \begin{aligned} &\int_{B_1(0)}(|\nabla u_\alpha|^2+|\nabla v_\alpha|^2)dx\\ &=\int_{B_{\frac{R}{\alpha}(x_0)}\cap B_1(0)}(|\nabla u_\alpha|^2+|\nabla v_\alpha|^2)dx +\int_{ B_1(0)/{B_{\frac{R}{\alpha}(x_0)}}} (|\nabla u_\alpha|^2+|\nabla v_\alpha|^2)dx\\ &=\alpha^{\frac{(2-N)(p+q)+2N}{p+q-2}}\Big(\int_{B_{R,\alpha}} (|\nabla \hat{u}_\alpha|^2+|\nabla \hat{v}_\alpha|^2)dx+ \int_{\Omega_\alpha/{B_{R,\alpha}}}(|\nabla \hat{u}_\alpha|^2+ |\nabla \hat{v}_\alpha|^2)dx\Big)\\ &\geq \alpha^{\frac{(2-N)(p+q)+2N}{p+q-2}}\Big(\int_{\mathbb{R}^N_-\cap B_R(0)} (|\nabla \hat{u}_\alpha|^2+|\nabla \hat{v}_\alpha|^2)dx+o(1)\Big) \\ &\geq \alpha^{\frac{(2-N)(p+q)+2N}{p+q-2}}\Big(\int_{\mathbb{R}^N_-\cap B_R(0)} (|\nabla u|^2+|\nabla v|^2)dx+o(1)\Big) \\ &\geq \alpha^{\frac{(2-N)(p+q)+2N}{p+q-2}}(2^{-\frac{2}{p+q-2}} m^{\frac{p+q}{p+q-2}}_{1,p,q}+o(1)) \\ &= 2^{-\frac{2}{p+q-2}}\alpha^{\frac{(2-N)(p+q)+2N}{p+q-2}} ((m^{\frac{p+q}{p+q-2}}_{1,p,q}+o(1)). \end{aligned} \end{equation} By \eqref{eq:4.9} and \eqref{eq:4.15}, \[ \int_{\Omega_\alpha/{B_{R,\alpha}}}(|\nabla \hat{u}_\alpha|^2+ |\nabla \hat{v}_\alpha|^2)dx=o(1)+o_R(1), \] \begin{align*} \int_{B_{R,\alpha}}(|\nabla \hat{u}_\alpha|^2+|\nabla \hat{v}_\alpha|^2)dx &=\int_{\mathbb{R}^N_-\cap B_R(0)}(|\nabla u|^2+|\nabla v|^2)dx+o(1) \\&=2^{-\frac{2}{p+q-2}}m^{\frac{p+q}{p+q-2}}_{1,p,q}+o(1)+o_R(1). \end{align*} Let $R\to+\infty$, the above equation yields \begin{equation} \label{eq:4.17} \int_{\mathbb{R}^N_-}(|\nabla u|^2+|\nabla v|^2)dx =2^{-\frac{2}{p+q-2}}m^{\frac{p+q}{p+q-2}}_{1,p,q}. \end{equation} An application of the Brezis-Lieb's Lemma gives \[ \int_{\mathbb{R}^N_-}(|\nabla (\hat{u}_\alpha-u)|^2+|\nabla (\hat{v}_\alpha-v)|^2)dx\to 0 \] as $\alpha\to+\infty$. On the other hand, by \eqref{eq:1.3} and \eqref{eq:4.17}, \[ \frac{\int_{\mathbb{R}^N_-}(|\nabla u|^2+|\nabla v|^2)dx} {(\int_{\mathbb{R}^N_-}e^{ x_N}|u|^p|v|^qdx)^{2/(p+q)}} =2^{2/(p+q)}(\int_{\mathbb{R}^N_-}(|\nabla u|^2+|\nabla v|^2)dx)^{\frac{p+q-2}{p+q}} =m_{1,p,q}. \] This implies that $(u,v)$ achieves $m_{1,p,q}$. As a consequence, we have \begin{align*} &\alpha^{-\frac{(2-N)(p+q)+2N}{p+q-2}}\int_{B_1(0)} (|\nabla (u_\alpha-\alpha^{\frac{2}{p+q-2}}u(\alpha(x-x_0))|^2\\ &+ |\nabla (v_\alpha-\alpha^{\frac{2}{p+q-2}} v(\alpha(x-x_0))|^2)dx\to 0, \end{align*} as $\alpha\to+\infty$. \end{proof} Now the the proof of Theorem \ref{th1.1} is completed by Lemmas \ref{lemma4.4}, \ref{lemma4.6} and Proposition \ref{prop4.1}. \subsection*{Acknowledgements} This research was partially supported by grants N10961016 and 10631030 from the NNSF of China, and 2009GZS0011 from NSF of Jiangxi. \begin{thebibliography}{99} \bibitem{BW2} J. Byeon, Z.-Q. 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