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Let $\Omega\subset \R^N$ ($N\geq 3$) be an open domain (may be unbounded)with $0\in \partial\Omega$ and $\partial\Omega$ be of $C^2$ at $0$ with thenegative mean curvature $H(0)$. By using variational methods, we consider thefollowing elliptic systems involving multiple Hardy-Sobolev critical exponents,$$\begin{cases} -\Delta u+\lambda^*\frac{u}{|x|^{\sigma_0}}-\lambda_1\frac{|u|^{2^*(s_1)-2}u}{|x|^{s_1}}=\lambda\frac{1}{|x|^{s_2}}|u|^{\alpha-2}u|v|^\beta\quad &\hbox{in}\;\Omega,\\ -\Deltav+\mu^*\frac{v}{|x|^{\eta_0}}-\mu_1 \frac{|v|^{2^*(s_1)-2}v}{|x|^{s_1}}=\mu\frac{1}{|x|^{s_2}}|u|^{\alpha}|v|^{\beta-2}v\quad &\hbox{in}\;\Omega,\\(u,v)\in D_{0}^{1,2}(\Omega)\times D_{0}^{1,2}(\Omega), \end{cases}$$ where $0\leq \sigma_0, \eta_0, s_2<2, s_1\in (0,2);$ the parameters $ \lambda^*\neq 0,\mu^*\neq 0, \lambda_1>0, \mu_1>0, \lambda\mu>0$; $\alpha,\beta>1$ satisfying$\alpha+\beta \leq 2^*(s_2)$. Here, $2^*(s):=\frac{2(N-s)}{N-2}$ is thecritical Hardy-Sobolev exponent. We obtain the existence and nonexistence ofground state solution under different specific assumptions. As the by-product,we study \be\lab{zou=a1} \begin{cases} &\Delta u+\lambda\frac{u^p}{|x|^{s_1}}+\frac{u^{2^*(s_2)-1}}{|x|^{s_2}}=0\;\quad\hbox{in}\;\Omega,\\ &u(x)>0\;\hbox{in}\;\Omega,\\ &u(x)=0\;\hbox{on}\;\partial\Omega, \end{cases} \ee we also obtain the existenceand nonexistence of solution under different hypotheses. In particular, we givea partial answers to a generalized open problem proposed by Y. Y. Li and C. S.Lin (ARMA, 2012). Around the above two types of equation or systems, wesystematically study the elliptic equations which have multiple singular termsand are defined on any open domain. We establish some fundamental results.\vskip0.23in  {\it Key words:} Elliptic system, Ground state, Hardy-Sobolev exponent.

On Elliptic Equations and Systems involving critical Hardy-Sobolev exponents (non-limit case)