Periodic parabolic problems with concave and convex nonlinearities

GODOY, T.; KAUFMANN, U.

NODEA. NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS (PRINTED ED.)

Año: 2007 vol. 14 p. 443 - 453

1021-9722

Let Ω⊂R^{N} be a smooth bounded domain, let a,b be two functions that are possibly discontinuous and unbounded with a≥0 in Ω×R and b>0 in a set of positive measure and let 0<p<1<q. We prove that there exists some 0<Λ<∞ such that the nonlinear Dirichlet periodic parabolic problem Lu=λa(x,t)u^{p}+b(x,t)u^{q} in Ω×R has a positive solution for all 0<λ<Λ and that there is no positive solution if λ>Λ. In some cases we also show the existence of a minimal solution for all 0<λ<Λ and that the solution u_{λ} can be chosen such that λ→u_{λ} 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.