Solutions for autonomous semilinear elliptic equations

Abstract

We study existence of nontrivial solutions to problem \begin{equation*} \left\lbrace \begin{array}{rcll} -Δu &=& λu+f(u)&\text{ in }Ω,\\ u&=&0&\text{ on }\partial Ω, \end{array}\right. \end{equation*} where $\hat{I}\text{c◯}\subset {\mathbb{R}}^N$ is a smooth bounded domain, $N\ge 1$, $\hat{I}»\in \mathbb{R}$ and $f:\mathbb{R}\to \mathbb{R}$ is any locally Lipschitz function with nonpositive primitive. A complete description is obtained for $N=1$ and partial results for $N\ge 2$.