Electron. J. Diff. Equ., Vol. 2012 (2012), No. 115, pp. 1-14.

Existence of solutions for Hardy-Sobolev-Maz'ya systems

Jian Wang, Xin Wei

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 5, 2012.
Math Subject Classifications: 35J47, 35J50, 35J57, 58E05.
Key Words: Variational identity; (PS) condition; linking theorem; Hardy-Sobolev-Maz'ya inequality.

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  Jian Wang
Department of Mathematics, Jiangxi Normal University
Nanchang, Jiangxi 330022, China
email: jianwang2007@126.com
Xin Wei
Department of Finance, Jiangxi Normal University
Nanchang, Jiangxi 330022, China
email: wxgrat@sohu.com

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