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


Author: Shu-Yu Hsu
Journal: Proc. Amer. Math. Soc. 142 (2014), 4239-4249
MSC (2010): Primary 35J70, 35B40; Secondary 58J37, 58J05
DOI: https://doi.org/10.1090/S0002-9939-2014-12152-6
Published electronically: August 6, 2014
MathSciNet review: 3266992
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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 )}$.


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Additional Information

Shu-Yu Hsu
Affiliation: Department of Mathematics, National Chung Cheng University, 168 University Road, Min-Hsiung, Chia-Yi 621, Taiwan, Republic of China
Email: syhsu@math.ccu.edu.tw

DOI: https://doi.org/10.1090/S0002-9939-2014-12152-6
Keywords: Nonlinear elliptic equation, Yamabe soliton, exact decay rate
Received by editor(s): November 14, 2012
Received by editor(s) in revised form: January 12, 2013
Published electronically: August 6, 2014
Communicated by: Walter Craig
Article copyright: © Copyright 2014 American Mathematical Society

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