Asymptotically Linear Elliptic Problems in which the Nonlinearity Crosses at Least Two Eigenvalues

In this paper we establish the existence of multiple solutions for the semilinear elliptic problem\begin{eqnarray*}\begin{array}{ccl}-\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 $g(x,0)=0$ andwhich is asymptotically linear at infinity.