Local and Global Log-Gradient estimates of solutions to ${\Delta }_{p}v+b{v}^q+c{v}^r=0$ on manifolds and applications

Abstract

In this paper, we employ the Nash-Moser iteration technique to study local and global properties of positive solutions to the equation $${\Delta }_{p}v+b{v}^q+c{v}^r=0$$ on complete Riemannian manifolds with Ricci curvature bounded from below, where $b,c\in \mathbb{R}$, $p>1$, and $q\le r$ 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.

arXiv admin note: substantial text overlap with arXiv:2311.02568; text overlap with arXiv:2311.13179