On a double-variable inequality and elliptic systems involving critical Hardy-Sobolev exponents

Let $\mathrm{\Omega }\subset {\mathbb{R}}^N$ ($N\ge 3$) be an open domain which is not necessarily bounded. The sharp constant and extremal functions to the following kind of double-variable inequalities $$S_{\alpha ,\beta ,\lambda ,\mu }(\mathrm{\Omega })\left({\int }_{\mathrm{\Omega }}\right(\lambda \frac{\mathrm{\mid }u{\mathrm{\mid }}^{2^{\ast }(s)}}{\mathrm{\mid }x{\mathrm{\mid }}^s}+\mu \frac{\mathrm{\mid }v{\mathrm{\mid }}^{2^{\ast }(s)}}{\mathrm{\mid }x{\mathrm{\mid }}^s}+2^{\ast }(s)\kappa \frac{\mathrm{\mid }u{\mathrm{\mid }}^{\alpha }\mathrm{\mid }v{\mathrm{\mid }}^{\beta }}{\mathrm{\mid }x{\mathrm{\mid }}^s})dx{)}^{\frac{2}{2^{\ast }(s)}}$$ $$\le {\int }_{\mathrm{\Omega }}(\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^2+\mathrm{\mid }\mathrm{\nabla }v{\mathrm{\mid }}^2)dx$$ for $(u,v)\in {\mathscr{D}}:=D_{0}^{1,2}(\Omega)\times D_{0}^{1,2}(\Omega)$ will be explored. Further results about the sharp constant $S_{\alpha ,\beta ,\lambda ,\mu }(\mathrm{\Omega })$ with its extremal functions when $\mathrm{\Omega }$ is a general open domain will be involved. For this goal, we consider the following elliptic systems involving multiple Hardy-Sobolev critical exponents: $$\{\begin{array}{lll}-\mathrm{\Delta }u-\lambda \frac{\mathrm{\mid }u{\mathrm{\mid }}^{2^{\ast }(s_1)-2}u}{\mathrm{\mid }x{\mathrm{\mid }}^{s_1}}=\kappa \alpha \frac{1}{\mathrm{\mid }x{\mathrm{\mid }}^{s_2}}\mathrm{\mid }u{\mathrm{\mid }}^{\alpha -2}u\mathrm{\mid }v{\mathrm{\mid }}^{\beta }& \text{in}\mathrm{\Omega },-\mathrm{\Delta }v-\mu \frac{\mathrm{\mid }v{\mathrm{\mid }}^{2^{\ast }(s_1)-2}v}{\mathrm{\mid }x{\mathrm{\mid }}^{s_1}}=\kappa \beta \frac{1}{\mathrm{\mid }x{\mathrm{\mid }}^{s_2}}\mathrm{\mid }u{\mathrm{\mid }}^{\alpha }\mathrm{\mid }v{\mathrm{\mid }}^{\beta -2}v& \text{in}\mathrm{\Omega },(u,v)\in \mathcal{D}:=D_0^{1,2}(\mathrm{\Omega })\times D_0^{1,2}(\mathrm{\Omega }),\end{array}$$ where $s_1,s_2\in (0,2), \alpha>1,\beta>1, \lambda>0,\mu>0,\kappa\neq 0, \alpha+\beta\leq 2^(s_2)$\mathrm{.}Here,$2^(s):=\frac{2(N-s)}{N-2}$ is the critical Hardy-Sobolev exponent. We mainly study the critical case (i.e., $\alpha +\beta =2^{\ast }(s_2)$) when $\mathrm{\Omega }$ is a cone (in particular, $\mathrm{\Omega }={\mathbb{R}}_{+}^N$ or $\mathrm{\Omega }={\mathbb{R}}^N$). We will establish a sequence of fundamental results including regularity, symmetry, existence and multiplicity, uniqueness and nonexistence, {\it etc.}