Harnack inequalities for quasilinear anisotropic elliptic equations with a first order term

Abstract

We consider weak solutions of the equation $$-{\Delta }_p^{H}u+a(x,u)H^q(\nabla u)=f(x,u)\text{in }\Omega ,$$ where $H$ is in some cases called Finsler norm, $\Omega$ is a domain of ${\mathbb{R}}^N$, $p>1$, $q\ge \max \{p-1,1\}$, and $a(\cdot ,u)$, $f(\cdot ,u)$ are functions satisfying suitable assumptions. We exploit the Moser iteration technique to prove a Harnack type comparison inequality for solutions of the equation and a Harnack type inequality for solutions of the linearized operator. As a consequence, we deduce a Strong Comparison Principle for solutions of the equation and a strong Maximum Principle for solutions of the linearized operator.