Periodic parabolic problems with concave and convex nonlinearities

Abstract

Let \(\Omega \subset {\mathbb{R}}^N\) be a smooth bounded domain, let a, b be two functions that are possibly discontinuous and unbounded with a ≥ 0 in \(\Omega\times{\mathbb{R}}\) and b > 0 in a set of positive measure and let 0 Λ. In some cases we also show the existence of a minimal solution for all 0 < λ < Λ and that the solution uλ can be chosen such that λ → uλ is differentiable and increasing. We also give some upper and lower estimates for such a Λ. All results remain true for the analogous elliptic problems.