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Liouville-type theorems and existence results for stable solutions to weighted Lane–Emden equations

Part of: Elliptic equations and systems Partial differential equations

Published online by Cambridge University Press:  29 January 2019

Alberto Farina  and
Shoichi Hasegawa
Alberto Farina
Affiliation:
Université de Picardie Jules Verne, LAMFA, CNRS UMR 7352, 33 Rue Saint-Leu, 80039 Amiens, France (alberto.farina@u-picardie.fr)
Shoichi Hasegawa
Affiliation:
Department of Mathematics, Tokyo Institute of Technology, 2-12-1, Ookayama, Meguro-ku, Tokyo152-8551, Japan (hasegawa.s.al@m.titech.ac.jp)
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Abstract

We devote this paper to proving non-existence and existence of stable solutions to weighted Lane-Emden equations on the Euclidean space ℝN, N ⩾ 2. We first prove some new Liouville-type theorems for stable solutions which recover and considerably improve upon the known results. In particular, our approach applies to various weighted equations, which naturally appear in many applications, but that are not covered by the existing literature. A typical example is provided by the well-know Matukuma's equation. We also prove an existence result for positive, bounded and stable solutions to a large family of weighted Lane–Emden equations, which indicates that our Liouville-type theorems are somehow sharp.

Keywords

semilinear elliptic PDEstable solutionsLiouville theorems

MSC classification

Primary: 35B35: Stability
Secondary: 35B08: Entire solutions 35B53: Liouville theorems, Phragmén-Lindelöf theorems 35J61: Semilinear elliptic equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 3 , June 2020 , pp. 1567 - 1579
Copyright
Copyright © 2019 The Royal Society of Edinburgh

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