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