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

Abstract:

Let $0<m<\frac {n-2}{n}$, $n\ge 3$, $\alpha =\frac {2\beta +\rho }{1-m}$ and $\beta >\frac {m\rho }{n-2-mn}$ for some constant $\rho >0$. Suppose $v$ is a radially symmetric solution of $\frac {n-1}{m}\Delta v^m+\alpha v+\beta x\cdot \nabla v=0$, $v>0$, in $\mathbb {R}^n$. When $m=\frac {n-2}{n+2}$, the metric $g=v^{\frac {4}{n+2}}dx^2$ corresponds to a locally conformally flat Yamabe shrinking gradient soliton with positive sectional curvature. We prove that the solution $v$ of the above nonlinear elliptic equation has the exact decay rate $\lim _{r\to \infty }r^2v(r)^{1-m}=\frac {2(n-1)(n(1-m)-2)}{(1-m)(\alpha (1-m)-2\beta )}$.