On Elliptic Systems involving critical Hardy-Sobolev exponents (Part II)

Abstract

This paper is the second part of a work devoted to the study of elliptic systems involving multiple Hardy-Sobolev critical exponents: $$\begin{cases} -Δu-λ\frac{|u|^{2^*(s_1)-2}u}{|x|^{s_1}}=κα\frac{1}{|x|^{s_2}}|u|^{α-2}u|v|^β\quad &\hbox{in}\;Ω,\\ -Δv-μ\frac{|v|^{2^*(s_1)-2}v}{|x|^{s_1}}=κβ\frac{1}{|x|^{s_2}}|u|^α|v|^{β-2}v\quad &\hbox{in}\;Ω,\\ κ>0,(u,v)\in \mathscr{D}:=D_{0}^{1,2}(Ω)\times D_{0}^{1,2}(Ω), \end{cases}$$ where $s_1\neq s_2\in (0,2), α>1,β>1, λ>0,μ>0,κ>0, α+β=2^*(s_2)$. Here, $2^*(s):=\frac{2(N-s)}{N-2}$ is the critical Hardy-Sobolev exponent. When $Ω$ is a cone (especially $Ω=\R_+^N$ or $Ω=\R^N$), we study the existence of positive ground state solution.