Solutions for autonomous semilinear elliptic equations

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 $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a smooth bounded domain, $N\ge 1$, $\lambda \in \mathbb{R}$ 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 $N\ge 2$.