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# Advanced Nonlinear Studies

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# Radial Symmetry of Entire Solutions of a Biharmonic Equation with Supercritical Exponent

Zongming Guo
/ Long Wei
Published Online: 2018-03-20 | DOI: https://doi.org/10.1515/ans-2018-0010

## Abstract

Necessary and sufficient conditions for a regular positive entire solution u of a biharmonic equation

to be a radially symmetric solution are obtained via the exact asymptotic behavior of u at $\mathrm{\infty }$ and the moving plane method (MPM). It is known that above equation admits a unique positive radial entire solution $u\left(x\right)=u\left(|x|\right)$ for any given $u\left(0\right)>0$, and the asymptotic behavior of $u\left(|x|\right)$ at $\mathrm{\infty }$ is also known. We will see that the behavior similar to that of a radial entire solution of above equation at $\mathrm{\infty }$, in turn, determines the radial symmetry of a general positive entire solution $u\left(x\right)$ of the equation. To make the procedure of the MPM work, the precise asymptotic behavior of u at $\mathrm{\infty }$ is obtained.

MSC 2010: 31B30; 35B08

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## About the article

Accepted: 2018-03-03

Published Online: 2018-03-20

Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11171092

Award identifier / Grant number: 11571093

Research of the first author is supported by NSFC (11171092, 11571093).

Citation Information: Advanced Nonlinear Studies, ISSN (Online) 2169-0375, ISSN (Print) 1536-1365,

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