Existence results for strongly indefinite elliptic systems

In this paper, we show the existence of solutions for the strongly indefinite elliptic system $$displaylines{ -Delta u=lambda u+f(x,v) quadhbox{in }Omega, cr -Delta v=lambda v+g(x,u) quadhbox{in }Omega, cr u=v=0, quadhbox{on }partialOmega, }$$ where $Omega$ is a bounded domain in $mathbb{R}^N; (Ngeq 3)$ with smooth boundary, $lambda_{k_0} Keywords Strongly indefinite elliptic system --- existence