A priori estimates and Liouville type results for quasilinear elliptic equations involving gradient terms

Abstract

In this article we study local and global properties of positive solutions of $-{\Delta }_{m}u=|u{|}^{p-1}u+M|\nabla u{|}^q$ in a domain $\Omega$ of ${\mathbb{R}}^N$, with $m>1$, $p,q>0$ and $M\in \mathbb{R}$. Following some ideas used in \cite{BV,Vron1}, and by using a direct Bernstein method combined with Keller-Osserman's estimate, we obtain several a priori estimates as well as Liouville type theorems. Moreover, we prove a local Harnack inequality with the help of Serrin's classical results.