A nonlinear elliptic PDE with multiple Hardy-Sobolev critical exponents in ${\mathbb{R}}^N$

In this paper, we will study the following PDE in ${\mathbb{R}}^N$ involving multiple Hardy-Sobolev critical exponents: $$\{\begin{array}{l}\mathrm{\Delta }u+{\sum }_{i=1}^l{\lambda }_i\frac{u^{2^{\ast }(s_i)-1}}{\mathrm{\mid }x{\mathrm{\mid }}^{s_i}}+u^{2^{\ast }-1}=0\text{in}{\mathbb{R}}^N,u\in D_0^{1,2}({\mathbb{R}}^N),\end{array}$$ where $0<s_1<s_2<\cdots<s_l<2, 2^\ast:=\frac{2N}{N-2}, ; 2^\ast(s):=\frac{2(N-s)}{N-2}$andthereexistssome$k\in [1, l]$suchthat$\lambda_i>0$for$1\leq i\leq k$;$\lambda_i<0$for$k+1\leq i\leq l$. We develop an interesting way to study this class of equations involving mixed sign parameters. We prove the existence and non-existence of the positive ground state solution. The regularity of the least-energy solution are also investigated.