Some properties of the Yamabe soliton and the related nonlinear elliptic equation

TL;DR

This paper investigates properties of Yamabe solitons and related nonlinear elliptic equations, proving non-existence of certain solutions, analyzing asymptotic behaviors, and deriving curvature properties of associated metrics.

Contribution

It provides new non-existence results, asymptotic analysis of solutions, and explicit curvature calculations for Yamabe solitons, with simplified proofs of existing theorems.

Findings

Non-existence of positive radially symmetric solutions under specified conditions

Asymptotic limits of solutions and scalar curvature at infinity

Explicit formulas for sectional curvature at origin and infinity

Abstract

We will prove the non-existence of positive radially symmetric solution of the nonlinear elliptic equation $\frac{n-1}{m}\Delta v^m+\alpha v+\beta x\cdot\nabla u=0$ in $R^n$ when $n\ge 3$, $0<m\le\frac{n-2}{n}$, $\alpha<0$ and $\beta\le 0$. Let $n\ge 3$ and $g=v^{\frac{4}{n+2}}dx^2$ be a metric on $\R^n$ where $v$ is a radially symmetric solution of the above elliptic equation in $R^n$ with $m=\frac{n-2}{n+2}$, $\alpha=\frac{2\beta+\rho}{1-m}$ and $\rho\in R$. For $n\ge 3$, $m=\frac{n-2}{n+2}$, we will prove that $\lim_{r\to\infty}r^2v^{1-m}(r)=\frac{(n-1)(n-2)}{\rho}$ if $\beta>\frac{\rho}{n-2}>0$, the scalar curvature $R(r)\to\rho$ as $r\to\infty$ if either $\beta>\frac{\rho}{n-2}>0$ or $\rho=0$ and $\alpha>0$ holds, and $\lim_{r\to\infty}R(r)=0$ if $\rho<0$ and $\alpha>0$. We give a simple different proof of a result of P.Daskalopoulos and N.Sesum \cite{DS2} on the positivity of the sectional curvature of rotational symmetric Yamabe solitons $g=v^{\frac{4}{n+2}}dx^2$ with $v$ satisfying the above equation with $m=\frac{n-2}{n+2}$. We will also find the exact value of the sectional curvature of such Yamabe solitons at the origin and at infinity.