On Elliptic Systems involving critical Hardy-Sobolev exponents (Part II)

This paper is the second part of a work devoted to the study of elliptic systems involving multiple Hardy-Sobolev critical exponents: $$\begin{cases} -\Delta u-\lambda \frac{|u|^{2^*(s_1)-2}u}{|x|^{s_1}}=\kappa\alpha \frac{1}{|x|^{s_2}}|u|^{\alpha-2}u|v|^\beta\quad &\hbox{in}\;\Omega,\\ -\Delta v-\mu \frac{|v|^{2^*(s_1)-2}v}{|x|^{s_1}}=\kappa\beta \frac{1}{|x|^{s_2}}|u|^{\alpha}|v|^{\beta-2}v\quad &\hbox{in}\;\Omega,\\ \kappa>0,(u,v)\in \mathscr{D}:=D_{0}^{1,2}(\Omega)\times D_{0}^{1,2}(\Omega), \end{cases}$$ where $s_1\neq s_2\in (0,2), \alpha>1,\beta>1, \lambda>0,\mu>0,\kappa>0, \alpha+\beta=2^*(s_2)$. Here, $2^*(s):=\frac{2(N-s)}{N-2}$ is the critical Hardy-Sobolev exponent. When $\Omega$ is a cone (especially $\Omega=\R_+^N$ or $\Omega=\R^N$), we study the existence of positive ground state solution.