Gradient estimates and Liouville theorems for Lichnerowicz-type equation on Riemannian manifolds
Youde Wang, Aiqi Zhang
TL;DR
This work studies gradient estimates and Liouville-type theorems for positive solutions to the Lichnerowicz-type equation $Δv+μ v+ a v^{p+1}+ b v^{-q+1}=0$ on complete Riemannian manifolds with $Ric\ge-(n-1)κ$. It develops a Nash–Moser iteration framework built on differential inequalities for $f=|\nabla u|^2$ with $u=-\ln v$, Bochner calculus, and Saloff-Coste Sobolev embeddings to derive Cheng–Yau-type local gradient bounds under various sign/size conditions on the coefficients. The main results include gradient estimates for both $μ\ge0$ and $μ<0$ (with $a>0,b>0$), Liouville-type nonexistence or constancy conclusions on noncompact manifolds, and specialized results for the Einstein–scalar field Lichnerowicz equation with $p=4/(n-2)$, $q=4(n-1)/(n-2)$. These findings contribute to rigidity and nonexistence phenomena for nonlinear elliptic equations on manifolds and have implications for relativity via the Einstein–scalar field context.
Abstract
In this paper we consider the gradient estimates on positive solutions to the following elliptic (Lichnerowicz) equation defined on a complete Riemannian manifold $(M,\,g)$: $$Δv + μv + a v^{p+1} +b v^{-q+1} =0,$$ where $p\geq-1$, $q\geq1$, $μ$, $a$ and $b$ are real constants. In the case $μ\geq0$ and $b\geq0$ or $μ<0$ , $a>0$ and $b>0$ ($μ$ has a lower bound), we employ the Nash-Moser iteration technique to obtain some refined gradient estimates of the solutions to the above equation, if $(M,\,g)$ satisfies $Ric \geq -(n-1)κ$ , where $n\geq3$ is the dimension of $M$ and $κ$ is a nonnegative constant, and $μ$ , $a$ , $b$ , $p$ and $q$ satisfy some technique conditions. By the obtained gradient estimates we also derive some Liouville type theorems for the above equation under some suitable geometric and analysis conditions. As applications, we can derive some Cheng-Yau's type gradient estimates for solutions to the $n$-dimensional Einstein-scalar field Lichnerowicz equation where $n\geq3$.