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

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 $g=v^{4/(n+2)}d{x}^2$ 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 ${\lim }_{r\to \infty }{r}^{2}v(r{)}^{1-m}=\frac{2(n-1)(n(1-m)-2)}{(1-m)(\alpha (1-m)-2\beta )}$.

12 pages, Section 3 and the proof and statement of Theorem 1.1 is completely rewritten

References in corpus (2)

• On the structure of gradient Yamabe solitons
• Some properties of the Yamabe soliton and the related nonlinear elliptic equation