Keller-Osserman a priori estimates and the Harnack inequality for quasilinear elliptic and parabolic equations with absorption term

Despite of the lack of comparison principle, we prove a priori estimates of Keller-Osserman type. Particularly under some natural assumptions on the function f, for nonnegative solutions of p-Laplace equation with absorption term we prove an estimate of the form ∫0u(x0)f(s)ds≤cr−pup(x0),x0∈Ω,B8r(x0)⊂Ω, with constant c independent of u, using this estimate we give a simple proof of the Harnack inequality. We prove a similar result for the evolution p-Laplace equation with absorption.