Nonexistence result for a semilinear elliptic problem

Abstract

In this paper we prove the nonexistence of nontrivial solution to \begin{equation*} \begin{cases} -Δu =f(u) &\text{in }Ω, \\ u=0 &\text{on } \partial Ω, \end{cases} \end{equation*} being $\Omega \subset {\mathbb{R}}^N$ ($N\in \mathbb{N}$) a bounded domain and $f$ locally Lispchitz with non-positive primitive.