Let $\Omega \subset \mathbb R^N$ a smooth bounded domain. We study
existence and nonexistence of positive solutions for some semilinear
Dirichlet periodic parabolic problems of the form
$Lu = h(x,t,u)$ in $\Omega\times \mathbb R$
for a class of Caratheodory functions
$h : \Omega\times \mathbb R \times [0,\infty) \rightarrow \mathbbR$
such that h (., 0) = 0 and $\lim_{\xi\rightarrow 0^+}\xi^{ −1}h (.,\xi) = 0$
or $-\infty$. All results remain true for the corresponding elliptic
problems.
