Existence of critical elliptic systems with boundary singularities

Abstract

In this paper, we are concerned with the existence of positive solutions of the following nonlinear elliptic system involving critical Hardy-Sobolev exponent $\begin{equation*}\label{eq:1}(*) \left\{ \begin{array}{lll} -\Delta u= \frac{2\alpha}{\alpha+\beta}\frac{u^{\alpha-1}v^\beta}{|x|^s}-\lambda u^p, & \quad {\rm in}\quad \Omega,\\[2mm] -\Delta v= \frac{2\beta}{\alpha+\beta}\frac{u^\alpha v^{\beta-1}}{|x|^s}-\lambda v^p, & \quad {\rm in}\quad \Omega,\\[2mm] u\gt 0, v\gt 0, &\quad {\rm in}\quad \Omega,\\[2mm] u=v=0, &\quad {\rm on}\quad \partial\Omega, \end{array} \right. \end{equation*}$ where $N\geq 4$ and $\Omega$ is a $C^1$ bounded domain in $\mathbb{R}^N$ with $0\in\partial\Omega$. $0\lt s \lt 2$, $\alpha+\beta=2^*(s)=\frac{2(N-s)}{N-2}$, $\alpha,\beta\gt 1$, $\lambda\gt 0$ and $1 \lt p\lt \frac{N+2}{N-2}$. The case when 0 belongs to the boundary of $\Omega$ is closely related to the mean curvature at the origin on the boundary. We show in this paper that problem $(*)$ possesses at least a positive solution.