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Remarks on Entire Solutions for Two Fourth-Order Elliptic Problems

Part of: Elliptic equations and systems Partial differential equations

Published online by Cambridge University Press:  23 November 2015

Baishun Lai  and
Dong Ye
Baishun Lai
Affiliation:
Institute of Contemporary Mathematics, Henan University, Kaifeng 475004, People's Republic of China ([email protected])
Dong Ye
Affiliation:
IECL, UMR 7502, Département de Mathématiques, Université de Lorraine, Ile de Saulcy, 57045 Metz, France ([email protected])
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Abstract

We are interested in entire solutions for the semilinear biharmonic equation Δ2 u = f(u) in ℝN , where f(u) = eu or –u –p (p > 0). For the exponential case, we prove that for the polyharmonic problem Δ2m u = eu with positive integer m, any classical entire solution verifies Δ2m–1 u < 0; this completes the results of Dupaigne et al. (Arch. Ration. Mech. Analysis208 (2013), 725–752) and Wei and Xu (Math. Annalen313 (1999), 207–228). We also obtain a refined asymptotic expansion of the radial separatrix solution to Δ2 u = eu in ℝ3, which answers a question posed by Berchio et al. (J. Diff. Eqns252 (2012), 2569–2616). For the negative power case, we show the non-existence of the classical entire solution for any 0 < p ⩽ 1.

Keywords

biharmonic equationentire solutionasymptotic behaviournon-existence

MSC classification

Secondary: 35J91: Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian 35B08: Entire solutions 35B53: Liouville theorems, Phragmén-Lindelöf theorems 35B40: Asymptotic behavior of solutions
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 59 , Issue 3 , August 2016 , pp. 777 - 786
Copyright
Copyright © Edinburgh Mathematical Society 2015 

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