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

TLDR: In this article, a priori estimates of Keller-Osserman type of quasilinear equations with absorption term were proved for nonnegative solutions of p-Laplace equation with absorption.

Abstract: In this article we study quasilinear equations model of which are − ∑ i = 1 n ( | u x i | p i − 2 u x i ) x i + f ( u ) = 0 , u ≥ 0 , ∂ u ∂ t − ∑ i = 1 n ( u ( m i − 1 ) ( p i − 1 ) | u x i | p i − 2 u x i ) x i + 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 ∫ 0 u ( x 0 ) f ( s ) d s ≤ c r − p u p ( x 0 ) , x 0 ∈ Ω , B 8 r ( x 0 ) ⊂ 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.