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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


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

Δ2u=upin N,N5,p>N+4N-4

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

Keywords: Positive Entire Solution; Radial Symmetry; Biharmonic Equation; Supercritical Exponent

MSC 2010: 31B30; 35B08


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About the article

Received: 2018-01-22

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, DOI: https://doi.org/10.1515/ans-2018-0010.

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