Electronic Journal of Differential Equations,
Vol. 2013 (2013), No. 214, pp. 1-12.
Title: Ground state solutions for semilinear problems with a Sobolev-Hardy term
Authors: Xiaoli Chen (Jiangxi Normal Univ., Nanchang, Jiangxi, China)
Weiyang Chen (Jiangxi Normal Univ., Nanchang, Jiangxi, China)
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.