Opuscula Math. 33, no. 2 (), 373-390
http://dx.doi.org/10.7494/OpMath.2013.33.2.373
Opuscula Mathematica

# 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.
Keywords: existence, compactness, critical Hardy-Sobolev exponent, nonlinear system.
Mathematics Subject Classification: 35J57, 35B33, 35B40.