Non-existence results for an eigenvalue problem involving Lipschitzian nonlinearities with non-positive primitive

Let $Omegasubseteq {f R}^n$ be a bounded domain ($ngeq 2$). In this paper, we prove that if $partialOmega$ has a non-negative mean curvature, or $Omega$ is an annulus, the, for each Lipschitzian function $f:{f R} o {f R}$, with $sup_{\xiin {f R}}int_0^{\xi}f(t)dt=0$, the problem $$cases{-Delta u = f(u) & in $Omega$cr & cr u=0 & on $partialOmega$cr}$$ has no non-zero classical solutions.