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

In this article we study quasilinear equations model of which are -Sigma(n)(i=1)(vertical bar u(xi vertical bar)(pi-2)u(xi))(xi) + f(u) = 0, u >= 0, partial derivative u/partial derivative t-Sigma(n)(i=1)(u((mi-1)(pi-1))vertical bar u(xi)vertical bar(pi-2)u(xi))(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 integral(u(x0))(0) f(s)ds <= cr(-p)u(p)(x(0)), x(0) is an element of Omega, B-8r(x(0)) subset of Omega, 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. (C) 2017 Elsevier Ltd. All rights reserved.