On the Harnack inequality for quasilinear elliptic equations with a first order term

TL;DR

This paper establishes a Harnack inequality for weak solutions of certain quasilinear elliptic equations with a first order term, using Moser iteration, leading to a strong comparison principle.

Contribution

It introduces a novel application of Moser iteration to derive Harnack inequalities for equations with first order terms, extending existing results.

Findings

Proved a Harnack comparison inequality for weak solutions.

Derived a strong comparison principle from the inequality.

Extended the applicability of Moser iteration to equations with first order terms.

Abstract

We consider weak solutions to $$-\Delta_pu+a(x,u)|\nabla u|^q=f(x,u),$$ with $p>1$, $q\geq\max\,\{p-1,1\}$. We exploit the Moser iteration technique to prove a Harnack comparison inequality for $C^1$ weak solutions. As a consequence we deduce a strong comparison principle.