Multiple solutions for elliptic problems with asymmetric nonlinearity

In this paper we establish the existence of multiple solutions for the semilinear elliptic problem\begin{eqnarray*}\begin{array}{ccccc}-\Delta u = g(x,u) & {\rm in} & \Omega\\u = 0\ \ & {\rm on} & \partial \Omega,\end{array}\end{eqnarray*}where $\Omega \subset \mathbb{R}^N$ is a bounded domain with smooth boundary $\partial \Omega$, $g:\Omega \times\mathbb{R}\to \mathbb{R}$ is a function of class $C^1$ such that $g(x,0)=0$ and which is asymptotically linear at infinity with jumping nonlinearities. We considered both cases resonant and nonresonant with respect to Fu$\check{\rm c}$ik Spectrum. We use critical groups to distinguish the critical points.