Gradient Estimate for Solutions of $Δv+v^r-v^s= 0$ on A Complete Riemannian Manifold
Youde Wang, Aiqi Zhang
TL;DR
$\Delta v+v^r-v^s=0$ on complete Riemannian manifolds with $\mathrm{Ric}\ge-(n-1)\kappa$ is studied to obtain Cheng–Yau type gradient estimates for positive solutions using Nash–Moser iteration after transforming to $u=-\ln v$ with $h=|\nabla u|^2$. A Bochner-based differential inequality for $h$ and Sobolev embeddings drive a Moser iteration that yields a local gradient bound $|\nabla v|^2/v^2 \le c(n,r,s)\,(1+\sqrt{\kappa}R)^2/R^2$ on $B_{R/2}$ under $r\le s$ and subcritical ranges $1<r<\frac{n+3}{n-1}$ or $1<s<\frac{n+3}{n-1}$. The two-dimensional case is handled with a tailored Sobolev framework. As a corollary, on manifolds with nonnegative Ricci curvature the equation admits a unique positive solution $v\equiv1$ when $r<s$ (or the symmetric parameter regime), illustrating Liouville-type rigidity. These results extend gradient estimates to a Lane–Emden type equation with competing powers in a geometric setting.
Abstract
In this paper we consider the gradient estimates on positive solutions to the following elliptic equation defined on a complete Riemannian manifold $(M,\,g)$: $$Δv+v^r-v^s= 0,$$ where $r$ and $s$ are two real constants. When$(M,\,g)$ satisfies $Ric \geq -(n-1)κ$ (where $n\geq2$ is the dimension of $M$ and $κ$ 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 $$1<r<\frac{n+3}{n-1}\quad\quad ~~\mbox{or}~~\quad 1<s<\frac{n+3}{n-1}.$$