Local and Global Log-Gradient estimates of solutions to $\Delta_pv+bv^q+cv^r =0$ on manifolds and applications

TL;DR

This paper derives sharp local and global gradient estimates for positive solutions to a nonlinear p-Laplace type equation on manifolds, leading to new Liouville theorems, Harnack inequalities, and applications to reaction-diffusion equations.

Contribution

It introduces novel sharp gradient estimates for solutions of a nonlinear PDE on manifolds, extending Liouville theorems and applications to classical equations.

Findings

Sharp Cheng-Yau type gradient estimates derived

Liouville-type theorems established in Euclidean spaces

Global gradient estimates obtained for solutions

Abstract

In this paper, we employ the Nash-Moser iteration technique to study local and global properties of positive solutions to the equation $$\Delta_pv+bv^q+cv^r =0$$ on complete Riemannian manifolds with Ricci curvature bounded from below, where $b, c\in\mathbb R$, $p>1$, and $q\leq r$ are some real constants. Assuming certain conditions on $b,\, c,\, p,\, q$ and $r$, we derive succinct Cheng-Yau type gradient estimates for positive solutions, which is of sharp form. These gradient estimates allow us to obtain some Liouville-type theorems and Harnack inequalities. Our Liouville-type results are novel even in Euclidean spaces. Based on the local gradient estimates and a trick of Sung and Wang, we also obtain the global gradient estimates for such solutions. As applications we show the uniqueness of positive solutions to some generalized Allen-Cahn equation and Fisher-KPP equation.