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

TL;DR

This paper establishes Harnack inequalities and related principles for solutions of anisotropic quasilinear elliptic equations involving a first order term, extending classical results to more general Finsler-type operators.

Contribution

It introduces new Harnack inequalities for anisotropic elliptic equations with first order terms, using Moser iteration, and derives strong comparison and maximum principles.

Findings

Proved Harnack comparison inequality for solutions.

Established Harnack inequality for the linearized operator.

Derived strong comparison and maximum principles.

Abstract

We consider weak solutions of the equation $$-\Delta_p^H u+a(x,u)H^q(\nabla u)=f(x,u) \quad \text{in } \Omega,$$ 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.