Gradient Estimate for Solutions of $\Delta v+v^r-v^s= 0$ on A Complete Riemannian Manifold

TL;DR

This paper derives gradient estimates for positive solutions to a nonlinear elliptic equation on complete Riemannian manifolds, revealing conditions under which solutions must be trivial, especially with nonnegative Ricci curvature.

Contribution

It introduces a Cheng-Yau type gradient estimate for solutions of $ riangle v + v^r - v^s=0$ on manifolds with Ricci curvature bounds, extending previous results to this specific nonlinear PDE.

Findings

Gradient estimates established under Ricci curvature bounds.

Nonexistence of positive solutions when Ricci curvature is nonnegative and certain conditions on r and s.

Solutions are trivial (constant) under specified geometric and analytical conditions.

Abstract

In this paper we consider the gradient estimates on positive solutions to the following elliptic equation defined on a complete Riemannian manifold $(M,\,g)$: $$\Delta v+v^r-v^s= 0,$$ where $r$ and $s$ are two real constants. When$(M,\,g)$ satisfies $Ric \geq -(n-1)\kappa$ (where $n\geq2$ is the dimension of $M$ and $\kappa$ is a nonnegative constant), we employ the Nash-Moser iteration technique to derive a Cheng-Yau's type gradient estimate for positive solution to the above equation under some suitable geometric and analysis conditions. Moreover, it is shown that when the Ricci curvature of $M$ is nonnegative, this elliptic equation does not admit any positive solution except for $u\equiv 1$ if $r<s$ and $$1<r<\frac{n+3}{n-1}\quad\quad ~~\mbox{or}~~\quad 1<s<\frac{n+3}{n-1}.$$