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

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.