On a double-variable inequality and elliptic systems involving critical Hardy-Sobolev exponents

TL;DR

This paper investigates sharp constants and extremal functions for double-variable Hardy-Sobolev inequalities and studies related elliptic systems with critical exponents, focusing on existence, symmetry, and nonexistence results in cone domains.

Contribution

It provides new results on the sharp constants, extremal functions, and solutions of elliptic systems involving critical Hardy-Sobolev exponents in unbounded domains.

Findings

Sharp constant and extremal functions characterized.

Existence and multiplicity of solutions established.

Symmetry and nonexistence results proved.

Abstract

Let $\Omega\subset \mathbb{R}^N$ ($N\geq 3$) be an open domain which is not necessarily bounded. 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 $(u,v)\in {\mathscr{D}}:=D_{0}^{1,2}(\Omega)\times D_{0}^{1,2}(\Omega)$ 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. For this goal, 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, $\Omega=\mathbb{R}_+^N$ or $\Omega=\mathbb{R}^N$). We will establish a sequence of fundamental results including regularity, symmetry, existence and multiplicity, uniqueness and nonexistence, {\it etc.}