Periodic parabolic problems with nonlinearities indefinite in sign

GODOY, T.; KAUFMANN, U.

PUBLICACIONS MATEMATIQUES

Año: 2007 vol. 51 p. 45 - 57

0214-1493

Let Ω⊂R^{N} be a smooth bounded domain. We give sufficient conditions (which are also necessary in many cases) on two nonnegative functions a,b that are possibly discontinuous and unbounded for the existence of nonnegative solutions for semilinear Dirichlet periodic parabolic problems of the form Lu=λa(x,t)u^{p}-b(x,t)u^{q} in Ω×R, where 0<p,q<1 and λ>0. In some cases we also show the existence of solutions u_{λ} in the interior of the positive cone and that u_{λ} can be chosen such that λ→u_{λ} is differentiable and increasing. A uniqueness theorem is also given in the case p≤q. All results remain valid for the corresponding elliptic problems.