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

Part of: Partial differential equations Elliptic equations and systems

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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References

1Arioli, G., Gazzola, F. and Grunau, H. C.. Entire solutions for a semilinear fourth order elliptic problem with exponential nonlinearity. J. Differ. Equ. 230 (2006), 743–770.10.1016/j.jde.2006.05.015Google Scholar
2Berchio, E., Farina, A., Ferrero, A. and Gazzola, F.. Existence and stability of entire solutions to a semilinear fourth order elliptic problem. J. Differ. Equ. 252 (2012), 2596–2616.10.1016/j.jde.2011.09.028Google Scholar
3Chang, S. Y. A. and Chen, W. X.. A note on a class of higher order conformally covariant equations. Discrete Contin. Dyn. Syst. 7 (2001), 275–281.Google Scholar
4Chavel, I.. Eigenvalues in Riemannian geometry (New York: Academic Press, Inc., 1984).Google Scholar
5Dupaigne, L., Ghergu, M., Goubet, O. and Warnault, G.. The Gel'fand problem for the biharmonic operator. Arch. Rational Mech. Anal. 208 (2013), 725–752.10.1007/s00205-013-0613-0Google Scholar
6Guo, Z. M.. Further study of entire solutions of a biharmonic equation with exponential nonlinearity. Ann. Mat. Pura Appl. 193 (2014), 187–201.10.1007/s10231-012-0272-zGoogle Scholar
7Guo, Z. M., Huang, X. and Zhou, F.. Radial symmetry of entire solutions of a bi-harmonic equation with exponential nonlinearity. J. Funct. Anal. 268 (2015), 1972–2004.10.1016/j.jfa.2014.12.010Google Scholar
8Lin, C. S.. A classification of solutions of a conformally invariant fourth order equation in ℝn. Comment. Math. Helv. 73 (1998), 206–231.Google Scholar
9Mckenna, P. J. and Reichel, W., Radial solutions of singular nonlinear biharmonic equations and applications to conformal geometry. Electron J. Differ. Equ. 37 (2003), 13.Google Scholar
10Troy, W. C.. Symmetry properties in systems of semilinear elliptic equations. J. Differ. Equ. 42 (1981), 400–413.10.1016/0022-0396(81)90113-3Google Scholar
11Warnault, G.. Liouville theorems for stable radial solutions for the biharmonic operator. Asymptotic Anal. 69 (2010), 87–98.Google Scholar
12Wei, J. C. and Xu, X. W.. Classification of solutions of higher order conformally invariant equations. Math. Ann. 313 (1999), 207–228.10.1007/s002080050258Google Scholar
13Zou, H. H.. Symmetry of positive solutions of Δu + u p = 0 in ℝn. J. Differ. Equ. 120 (1995), 46–88.10.1006/jdeq.1995.1105Google Scholar