Electron. J. Diff. Eqns., Vol. 2008(2008), No. 81, pp. 1-11.

Existence results for strongly indefinite elliptic systems

Jianfu Yang, Ying Ye, Xiaohui Yu

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.

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Jianfu Yang
Department of Mathematics, Jiangxi Normal University
Nanchang, Jiangxi 330022, China
email: jfyang_2000@yahoo.com
Ying Ye
Department of Mathematics, Jiangxi Normal University
Nanchang, Jiangxi 330022, China
email: yeying19851985@163.com
Xiaohui Yu
China Institute for Advanced Study
Central University of Finance and Economics
Beijing 100081, China
email: yuxiao_211@163.com

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