Positive solutions for sublinear periodic parabolic problems Original Research Article

Let Ω be a bounded domain in View the MathML source. We characterize the set of positive principal eigenvalues for the Dirichlet periodic parabolic problem Lu=λg(x,t,u) in View the MathML source, under the assumptions that g is a function such that ξ→g(x,t,ξ)/ξ is continuously differentiable and nonincreasing in [0,∞) a.e. View the MathML source satisfying some integrability and positivity conditions. We also prove the uniqueness of the positive solution. As a consequence we obtain a necessary and sufficient condition for the existence of a (unique) positive solution for the periodic parabolic logistic equation with unbounded weights. Existence of positive solutions for a generalization of the logistic equation is also shown.