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

Abstract

In this article we study quasilinear equations model of which are −∑i=1n(|uxi|pi−2uxi)xi+f(u)=0,u≥0,∂u∂t−∑i=1n(u(mi−1)(pi−1)|uxi|pi−2uxi)xi+f(u)=0,u≥0.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.