Electronic Journal of Differential Equations,
Vol. 2008(2008), No. 81, pp. 1-11.
Title: Existence results for strongly indefinite elliptic systems
Authors: Jianfu Yang (Normal Univ., Nanchang, China)
Ying Ye (Jiangxi Normal Univ., Nanchang, China)
Xiaohui Yu (Central Univ. Finance and Economics, Beijing, China)
Abstract:
In this paper, we show the existence of solutions for
the strongly indefinite elliptic system
$$\displaylines{
-\Delta u=\lambda u+f(x,v) \quad\hbox{in }\Omega, \cr
-\Delta v=\lambda v+g(x,u) \quad\hbox{in }\Omega, \cr
u=v=0, \quad\hbox{on }\partial\Omega,
}$$
where $\Omega$ is a bounded domain in $\mathbb{R}^N\; (N\geq 3)$ with smooth
boundary, $\lambda_{k_0}<\lambda<\lambda_{k_0+1}$, where $\lambda_k$ is
the $k$th eigenvalue of $-\Delta$ in $\Omega$ with zero Dirichlet
boundary condition. Both cases when $f,g$ being superlinear and
asymptotically linear at infinity are considered.
Submitted April l7, 2008. Published May 28, 2008.
Math Subject Classifications: 35J20,3 5J25.
Key Words: Strongly indefinite elliptic system; existence.