Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 115, pp. 1-14.

Title: Existence of solutions for Hardy-Sobolev-Maz'ya systems
  
Authors: Jian Wang (Jiangxi Normal Univ., China)
         Xin Wei   (Jiangxi Normal Univ., China)

Abstract:
 The main goal of  this article is to investigate the existence
 of solutions for the Hardy-Sobolev-Maz'ya system
 $$\displaylines{
 -\Delta u-\lambda \frac{u}{|y|^2}=\frac{|v|^{p_t-1}}{|y|^t}v,\quad
 \hbox{in }\Omega,\cr
 -\Delta v-\lambda \frac{v}{|y|^2}=\frac{|u|^{p_s-1}}{|y|^s}u,\quad
 \hbox{in }\Omega,\cr
 u=v=0,\quad  \hbox{on }\partial \Omega
 }$$
 where $0\in\Omega$ which is a bounded, open and smooth subset of
 $\mathbb{R}^k\times \mathbb{R}^{N-k}$, $2\leq k<N$.
 The non-existence of classical positive solutions is
 obtained by a variational identity and the existence result
 by a linking theorem.

Submitted December 26, 2011. Published July 05, 2012.
Math Subject Classifications: 35J47, 35J50, 35J57, 58E05.
Key Words: Variational identity; (PS) condition; linking theorem;
           Hardy-Sobolev-Maz'ya inequality.