On Elliptic Systems involving critical Hardy-Sobolev exponents

TL;DR

This paper investigates elliptic systems with critical Hardy-Sobolev exponents on unbounded domains, establishing fundamental properties like regularity, symmetry, existence, and extremal functions for associated inequalities.

Contribution

It introduces new results on the existence, symmetry, and extremal functions for elliptic systems involving critical Hardy-Sobolev exponents, especially on cone and unbounded domains.

Findings

Established regularity and symmetry of solutions

Proved existence and multiplicity results

Derived sharp constants and extremal functions for inequalities

Abstract

Let $\Omega\subset \R^N$ ($N\geq 3$) be an open domain which is not necessarily bounded. By using variational methods, we consider the following elliptic systems involving multiple Hardy-Sobolev critical exponents: $$\begin{cases} -\Delta u-\lambda \frac{|u|^{2^*(s_1)-2}u}{|x|^{s_1}}=\kappa\alpha \frac{1}{|x|^{s_2}}|u|^{\alpha-2}u|v|^\beta\quad &\hbox{in}\;\Omega,\\ -\Delta v-\mu \frac{|v|^{2^*(s_1)-2}v}{|x|^{s_1}}=\kappa\beta \frac{1}{|x|^{s_2}}|u|^{\alpha}|v|^{\beta-2}v\quad &\hbox{in}\;\Omega,\\ (u,v)\in \mathscr{D}:=D_{0}^{1,2}(\Omega)\times D_{0}^{1,2}(\Omega), \end{cases}$$ where $s_1,s_2\in (0,2), \alpha>1,\beta>1, \lambda>0,\mu>0,\kappa\neq 0, \alpha+\beta\leq 2^*(s_2)$. Here, $2^*(s):=\frac{2(N-s)}{N-2}$ is the critical Hardy-Sobolev exponent. We mainly study the critical case (i.e., $\alpha+\beta=2^*(s_2)$) when $\Omega$ is a cone (in particular, $\Omega=\R_+^N$ or $\Omega=\R^N$). We will establish a sequence of fundamental results including regularity, symmetry, existence and multiplicity, uniqueness and nonexistence, {\it etc.} In particular, the sharp constant and extremal functions to the following kind of double-variable inequalities $$ S_{\alpha,\beta,\lambda,\mu}(\Omega) \Big(\int_\Omega \big(\lambda \frac{|u|^{2^*(s)}}{|x|^s}+\mu \frac{|v|^{2^*(s)}}{|x|^s}+2^*(s)\kappa \frac{|u|^\alpha |v|^\beta}{|x|^s}\big)dx\Big)^{\frac{2}{2^*(s)}}$$ $$\leq \int_\Omega \big(|\nabla u|^2+|\nabla v|^2\big)dx$$ for $(u,v)\in {\mathscr{D}} $ will be explored. Further results about the sharp constant $S_{\alpha,\beta,\lambda,\mu}(\Omega)$ with its extremal functions when $\Omega$ is a general open domain will be involved.