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

• Published in: Nonlinear differential equations and applications Vol. 32; no. 4
• Main Author: Vuono, Domenico
• Format: Journal Article
• Language: English
• Published: Heidelberg Springer Nature B.V 01.07.2025
• Subjects: Elliptic functions; Linearization; Maximum principle
• ISSN: 1021-9722; 1420-9004
• DOI: 10.1007/s00030-025-01071-5

We consider weak solutions of the equation $$\begin{aligned} -\Delta _p^H u+a(x,u)H^q(\nabla u)=f(x,u) \quad \text {in } \Omega , \end{aligned}$$ - Δ p H u + a ( x , u ) H q ( ∇ u ) = f ( x , u ) in Ω , where H is in some cases called Finsler norm, $$\Omega $$ Ω is a domain of $${\mathbb {R}}^N$$ R N , $$p>1$$ p > 1 , $$q\ge \max \{p-1,1\}$$ q ≥ max { p - 1 , 1 } , and $$a(\cdot ,u)$$ a ( · , u ) , $$f(\cdot ,u)$$ f ( · , 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.