Multiple Solutions for Asymptotically Linear Resonant Elliptic Problems

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.We use critical groups to distinguish the critical points.