Gradient estimate for solutions of the equation $\Delta_p v+av^{q}=0$ on a complete Riemannian manifold

TL;DR

This paper derives gradient estimates and Liouville theorems for positive solutions of a nonlinear p-Laplacian equation on complete Riemannian manifolds using Nash-Moser iteration, under specific conditions on parameters.

Contribution

It introduces new gradient bounds and Liouville theorems for nonlinear elliptic equations on Riemannian manifolds, expanding understanding of solution behaviors.

Findings

Established gradient estimates for solutions.

Proved Liouville type theorems under certain conditions.

Applied Nash-Moser iteration to nonlinear elliptic equations.

Abstract

In this paper, we use Nash-Moser iteration method to study the local and global behaviours of positive solutions to the nonlinear elliptic equation $\Delta_pv +av^{q}=0$ defined on a complete Riemannian manifolds $(M,g)$ where $p>1$, $a$ and $q$ are constants. Under some assumptions on $a$, $p$ and $q$, we derive gradient estimates and Liouville type theorems for such positive solutions.