A Nonlinear Elliptic PDE with Two Sobolev-Hardy Critical Exponents

In this paper, we consider the following PDE involving two Sobolev-Hardy critical exponents, \label{0.1} {& \Delta u + \lambda\frac{u^{2^(s_1)-1}}{|x|^{s_1}} + \frac{u^{2^(s_2)-1}}{|x|^{s_2}} =0 $\text{in}$ \Omega, & u=0 \qquad $\text{on}$ \Omega, where $0\le s_2<s_1\le 2$, $0\mathrm{\ne }\lambda \in \mathbb{R}$ and $0\in \mathrm{\partial }\mathrm{\Omega }$. The existence (or nonexistence) for least-energy solutions has been extensively studied when $s_1=0$ or $s_2=0$. In this paper, we prove that if $0<s_2<s_1<2$ and the mean curvature of $\mathrm{\partial }\mathrm{\Omega }$ at 0 $H(0)<0$, then \eqref{0.1} has a least-energy solution. Therefore, this paper has completed the study of \eqref{0.1} for the least-energy solutions. We also prove existence or nonexistence of positive entire solutions of \eqref{0.1} with $\mathrm{\Omega }=\text{rn}$ under different situations of $s_1,s_2$ and $\lambda$.