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Radial symmetry of non-maximal entire solutions of a bi-harmonic equation with exponential nonlinearity

Part of: Elliptic equations and systems Partial differential equations

Published online by Cambridge University Press:  24 January 2019

Hongxia Guo ,
Zongming Guo  and
Fangshu Wan
Hongxia Guo
Affiliation:
School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, P.R. China (guohongxia@zzu.edu.cn)
Zongming Guo
Affiliation:
Department of Mathematics, Henan Normal University, Xinxiang 453007, P.R. China (gzm@htu.cn)
Fangshu Wan
Affiliation:
School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, P.R. China (fangshuwan@163.com)
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Abstract

We study radial symmetry of entire solutions of the equation 0.1

$$\Delta ^2u = 8(N-2)(N-4)e^u\quad {\rm in}\;R^N\;\;(N \ges 5).$$
It is known that (0.1) admits infinitely many radially symmetric entire solutions. These solutions may have either a (negative) logarithmic behaviour or a (negative) quadratic behaviour at infinity. Up to translations, we know that there is only one radial entire solution with the former behaviour, which is called ‘maximal radial entire solution’, and infinitely many radial entire solutions with the latter behaviour, which are called ‘non-maximal radial entire solutions’. The necessary and sufficient conditions for an entire solution u of (0.1) to be the maximal radial entire solution are presented in [7] recently. In this paper, we will give the necessary and sufficient conditions for an entire solution u of (0.1) to be a non-maximal radial entire solution.

Keywords

Bi-harmonic equationradial symmetryentire solutionsexponential nonlinearity

MSC classification

Primary: 35B45: A priori estimates
Secondary: 35J40: Boundary value problems for higher-order elliptic equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 6 , December 2019 , pp. 1603 - 1625
Copyright
Copyright © Royal Society of Edinburgh 2019 

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