Some remarks about the Fucik spectrum and application to equations with jumping nonlinearities

Let $L$ be a selfadjoint operator with compact resolvent and $\lambda $ an eigenvalue of $L$. When $\lambda $ is simple, it is well known that the Fucik spectrum $\Sigma $ near $\lambda $ consists of two nonincreasing curves. In this paper, we show that when $\lambda $ is not simple, $\Sigma $ contains two nonincreasing curves such that all points above or under both curves are not in $\Sigma $. After that, we give some existence results of solutions of the equation $Lu=\alpha u^{+}-\beta u^{-}+g(.,u)$ where $u^{\pm }=\max (0,\pm u)$.