Nash-Moser iteration approach to gradient estimate and Liouville Property of quasilinear elliptic equations on complete Riemannian manifolds

Jie He, Jingchen Hu, Youde Wang

Published 2023-11-05Version 1

In this paper, we employ the Nash-Moser iteration technique to analyze the local and global properties of positive solutions to the equation $$\Delta_pu+a|\nabla u|^qu^r =0$$on a complete Riemannian manifold with Ricci curvature bounded from below, where $p>1$, $q$, $r$ and $a$ are some real constants. Assuming certain conditions on $a,\, p,\, q$ and $r$, we derive a succinct Cheng-Yau type gradient estimate for such solutions. This gradient estimate allows us to obtain a Liouville-type theorem and a Harnack inequality. Our Liouville-type result is novel even in Euclidean space