Generalized Harnack inequality for semilinear elliptic equations

Abstract

This paper is concerned with semilinear equations in divergence form div(A(x)Du) = f(u) where f : R → [0, ∞) is nondecreasing. We introduce a sharp Harnack type inequality for nonnegative solutions which is a quantified version of the condition for strong maximum principle found by Vazquez and Pucci-Serrin in [30, 24] and is closely related to the classical Keller-Osserman condition [15, 22] for the existence of entire solutions.