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: $$\{\begin{array}{ll}-\mathrm{\Delta }u-\lambda \frac{\mathrm{\mid }u{\mathrm{\mid }}^{2^{\ast }(s_1)-2}u}{\mathrm{\mid }x{\mathrm{\mid }}^{s_1}}=\kappa \alpha \frac{1}{\mathrm{\mid }x{\mathrm{\mid }}^{s_2}}\mathrm{\mid }u{\mathrm{\mid }}^{\alpha -2}u\mathrm{\mid }v{\mathrm{\mid }}^{\beta }& \text{in}\mathrm{\Omega },\\ -\mathrm{\Delta }v-\mu \frac{\mathrm{\mid }v{\mathrm{\mid }}^{2^{\ast }(s_1)-2}v}{\mathrm{\mid }x{\mathrm{\mid }}^{s_1}}=\kappa \beta \frac{1}{\mathrm{\mid }x{\mathrm{\mid }}^{s_2}}\mathrm{\mid }u{\mathrm{\mid }}^{\alpha }\mathrm{\mid }v{\mathrm{\mid }}^{\beta -2}v& \text{in}\mathrm{\Omega },\\ \kappa >0,(u,v)\in \mathcal{D}:=D_0^{1,2}(\mathrm{\Omega })\times D_0^{1,2}(\mathrm{\Omega }),\end{array}$$ where $s_1\neq s_2\in (0,2), \alpha>1,\beta>1, \lambda>0,\mu>0,\kappa>0, \alpha+\beta=2^(s_2)$\mathrm{.}Here,$2^(s):=\frac{2(N-s)}{N-2}$isthecriticalHardy-Sobolevexponent\mathrm{.}When$\Omega$isacone(especially$\Omega=\R_+^N$or$\Omega=\R^N$), we study the existence of positive ground state solution.