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

TL;DR

This paper determines the precise decay rate of solutions to a specific nonlinear elliptic equation related to Yamabe flow, revealing detailed asymptotic behavior of radially symmetric solutions in geometric analysis.

Contribution

It provides the exact decay rate of solutions to a nonlinear elliptic equation associated with Yamabe solitons, extending understanding of their asymptotic properties.

Findings

Derived the exact decay rate of solutions at infinity.

Connected the decay rate to geometric properties of Yamabe solitons.

Confirmed the decay rate for solutions with specific parameter conditions.

Abstract

Let 0<m<(n-2)/n, n>2, $\alpha=(2\beta +\rho)/(1-m)$ and $\beta>m\rho/(n-2-mn)$ for some constant $\rho>0$. Suppose v is a radially symmetric symmetric solution of $\frac{n-1}{m}\Delta v^m+\alpha v+\beta x\cdot\nabla v=0$, 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 $\lim_{r\to\infty}r^2v(r)^{1-m}=\frac{2(n-1)(n(1-m)-2)}{(1-m)(\alpha (1-m)-2\beta)}$.