Opuscula Mathematica
Opuscula Math. 33, no. 2 (), 373-390
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.
Cite this article as:
Jianfu Yang, Yimin Zhou, Existence of critical elliptic systems with boundary singularities, Opuscula Math. 33, no. 2 (2013), 373-390, http://dx.doi.org/10.7494/OpMath.2013.33.2.373
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ISSN 1232−9274, e-ISSN 2300−6919, DOI https://doi.org/10.7494/OpMath
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