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

Abstract

Let 0 2, $α=(2β+ρ)/(1-m)$ and $β>mρ/(n-2-mn)$ for some constant $ρ>0$. Suppose v is a radially symmetric symmetric solution of $\frac{n-1}{m}Δv^m+αv+βx\cdot\nabla v=0$, v>0, in $R^n$. When m=(n-2)/(n+2), the metric $g=v^{4/(n+2)}dx^2$ 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 $\lim_{r\to\infty}r^2v(r)^{1-m}=\frac{2(n-1)(n(1-m)-2)}{(1-m)(α(1-m)-2β)}$.