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 $g=v^{\frac{4}{n+2}}d{x}^2$ be a metric on ${\mathbb{R}}^n$ 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 ${\lim }_{r\to \infty }{r}^2{v}^{1-m}(r)=\frac{(n-1)(n-2)}{\rho }$ 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 $g=v^{\frac{4}{n+2}}d{x}^2$ 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.

16 pages, proof of Lemma 2.3 is simplified and some typo errors corrected

Cited by in corpus (1)

• Exact decay rate of a nonlinear elliptic equation related to the Yamabe flow