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

TL;DR

This paper investigates positive solutions of a class of quasilinear elliptic equations with gradient terms, establishing a priori bounds, Liouville theorems, and local Harnack inequalities to understand their qualitative behavior.

Contribution

It introduces new a priori estimates and Liouville type results for quasilinear elliptic equations involving gradient terms, extending previous methods with Bernstein and Keller-Osserman techniques.

Findings

Established several a priori estimates for solutions.

Proved Liouville type theorems under various conditions.

Derived a local Harnack inequality for positive solutions.

Abstract

In this article we study local and global properties of positive solutions of $-\Delta_mu=|u|^{p-1}u+M|\nabla u|^q$ 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.