Electron. J. Diff. Equ., Vol. 2013 (2013), No. 214, pp. 1-12.

Ground state solutions for semilinear problems with a Sobolev-Hardy term

Xiaoli Chen, Weiyang Chen

Abstract:
In this article, we study the existence of solutions to the problem
$$\displaylines{ -\Delta u= \lambda u+\frac{|u|^{2_s^\ast-2}u}{|y|^s}, \quad x\in \Omega,\cr u = 0, \quad x\in \partial \Omega, }$$
where $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$ $(N\geq3)$. We show that there is a ground state solution provided that N=4 and $\lambda_m<\lambda<\lambda_{m+1}$, or that $N\geq 5$ and $\lambda_m\leq\lambda<\lambda_{m+1}$, where $\lambda_m$ is the m'th eigenvalue of $-\Delta$ with Dirichlet boundary conditions.

Submitted April 19, 2013. Published September 26, 2013.
Math Subject Classifications: 35J60, 35J65.
Key Words: Existence; ground state; critical Hardy-Sobolev exponent; semilinear Dirichlet problem.

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Xiaoli Chen
Department of Mathematics, Jiangxi Normal University
Nanchang, Jiangxi 330022, China
email: littleli_chen@163.com
Weiyang Chen
Department of Mathematics, Jiangxi Normal University
Nanchang, Jiangxi 330022, China
email: xiaowei19901207@126.com

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