On Elliptic Equations and Systems involving critical Hardy-Sobolev exponents (non-limit case)

TL;DR

This paper investigates the existence and nonexistence of solutions for elliptic systems involving multiple critical Hardy-Sobolev exponents on domains with boundary singularities, using variational methods and addressing open problems in the field.

Contribution

It introduces new results on ground state solutions for elliptic systems with multiple Hardy-Sobolev critical exponents, extending previous work and partially solving a problem posed by Li and Lin.

Findings

Established conditions for existence of ground state solutions.

Proved nonexistence results under certain parameter regimes.

Provided partial answers to an open problem by Li and Lin.

Abstract

Let $\Omega\subset \R^N$ ($N\geq 3$) be an open domain (may be unbounded) with $0\in \partial\Omega$ and $\partial\Omega$ be of $C^2$ at $0$ with the negative mean curvature $H(0)$. By using variational methods, we consider the following elliptic systems involving multiple Hardy-Sobolev critical exponents, $$\begin{cases} -\Delta u+\lambda^*\frac{u}{|x|^{\sigma_0}}-\lambda_1 \frac{|u|^{2^*(s_1)-2}u}{|x|^{s_1}}=\lambda \frac{1}{|x|^{s_2}}|u|^{\alpha-2}u|v|^\beta\quad &\hbox{in}\;\Omega,\\ -\Delta v+\mu^*\frac{v}{|x|^{\eta_0}}-\mu_1 \frac{|v|^{2^*(s_1)-2}v}{|x|^{s_1}}=\mu \frac{1}{|x|^{s_2}}|u|^{\alpha}|v|^{\beta-2}v\quad &\hbox{in}\;\Omega,\\ (u,v)\in D_{0}^{1,2}(\Omega)\times D_{0}^{1,2}(\Omega), \end{cases}$$ where $ 0\leq \sigma_0, \eta_0, s_2<2, s_1\in (0,2);$ the parameters $ \lambda^*\neq 0, \mu^*\neq 0, \lambda_1>0, \mu_1>0, \lambda\mu>0$; $\alpha,\beta>1$ satisfying $\alpha+\beta \leq 2^*(s_2)$. Here, $2^*(s):=\frac{2(N-s)}{N-2}$ is the critical Hardy-Sobolev exponent. We obtain the existence and nonexistence of ground state solution under different specific assumptions. As the by-product, we study \be\lab{zou=a1} \begin{cases} &\Delta u+\lambda \frac{u^p}{|x|^{s_1}}+\frac{u^{2^*(s_2)-1}}{|x|^{s_2}}=0\;\quad \hbox{in}\;\Omega,\\ &u(x)>0\;\hbox{in}\;\Omega,\\ & u(x)=0\;\hbox{on}\;\partial\Omega, \end{cases} \ee we also obtain the existence and nonexistence of solution under different hypotheses. In particular, we give a partial answers to a generalized open problem proposed by Y. Y. Li and C. S. Lin (ARMA, 2012). Around the above two types of equation or systems, we systematically study the elliptic equations which have multiple singular terms and are defined on any open domain. We establish some fundamental results. \vskip0.23in {\it Key words:} Elliptic system, Ground state, Hardy-Sobolev exponent.