Multiple solutions for asymptotically linear resonant elliptic problems

Abstract

In this paper we establish the existence of multiple solutions for the semilinear elliptic problem \begin{equation}\alignedat 2 -\Delta u&=g(x,u) &\quad&\text{in } \Omega, \\ u&=0 &\quad&\text{on } \partial\Omega, \endalignedat \tag{1.1} \end{equation} where $\Omega \subset {\mathbb R}^N$ is a bounded domain with smooth boundary $\partial \Omega$, a function $g\colon\Omega\times{\mathbb R}\to {\mathbb R}$ is of class $C^1$ such that $g(x,0)=0$ and which is asymptotically linear at infinity. We considered both cases, resonant and nonresonant. We use critical groups to distinguish the critical points.