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: $$\begin{cases} \Delta u+\sum_{i=1}^{l}\lambda_i \frac{u^{2^*(s_i)-1}}{|x|^{s_i}}+u^{2^*-1}=0\;\hbox{in}\;\mathbb{R}^N, u\in D_{0}^{1,2}(\mathbb{R}^N), \end{cases}$$ where $0<s_1<s_2<\cdots<s_l<2, 2^\ast:=\frac{2N}{N-2}, \; 2^\ast(s):=\frac{2(N-s)}{N-2}$ and there exists some $k\in [1, l]$ such that $\lambda_i>0$ 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.