Nonexistence result for a semilinear elliptic problem

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