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

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$\beta\le 0$\mathrm{.}Let$n\ge 3$and$g=v^{\frac{4}{n+2}}dx^2$beametricon$\R^n$where$v$isaradiallysymmetricsolutionoftheaboveellipticequationin$R^n$with$m=\frac{n-2}{n+2}$,$\alpha=\frac{2\beta+\rho}{1-m}$and$\rho\in R$\mathrm{.}For$n\ge 3$,$m=\frac{n-2}{n+2}$,wewillprovethat$\lim_{r\to\infty}r^2v^{1-m}®=\frac{(n-1)(n-2)}{\rho}$if$\beta>\frac{\rho}{n-2}>0$,thescalarcurvature$R®\to\rho$as$r\to\infty$ifeither$\beta>\frac{\rho}{n-2}>0$or$\rho=0$and$\alpha>0$holds,and$\lim_{r\to\infty}R®=0$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.