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Fast and slow decay solutions for supercritical fractional elliptic problems in exterior domains

Part of: Classical differential geometry Elliptic equations and systems Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  18 January 2021

Weiwei Ao ,
Chao Liu  and
Liping Wang
Weiwei Ao
Affiliation:
School of Mathematics and Statistics, Wuhan University, Wuhan430072, China (wwao@whu.edu.cn; chaoliuwh@whu.edu.cn)
Chao Liu
Affiliation:
School of Mathematics and Statistics, Wuhan University, Wuhan430072, China (wwao@whu.edu.cn; chaoliuwh@whu.edu.cn)
Liping Wang
Affiliation:
Department of Mathematics, Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, East China Normal University, 200241, China (lpwang@math.ecnu.edu.cn)
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Abstract

We consider the fractional elliptic problem: where B1 is the unit ball in ℝN, N ⩾ 3, s ∈ (0, 1) and p > (N + 2s)/(N − 2s). We prove that this problem has infinitely many solutions with slow decay O(|x|−2s/(p−1)) at infinity. In addition, for each s ∈ (0, 1) there exists Ps > (N + 2s)/(N − 2s), for any (N + 2s)/(N − 2s) < p < Ps, the above problem has a solution with fast decay O(|x|2s−N). This result is the extension of the work by Dávila, del Pino, Musso and Wei (2008, Calc. Var. Partial Differ. Equ. 32, no. 4, 453–480) to the fractional case.

Keywords

Fractional elliptic problemssupercritical casesexistence

MSC classification

Primary: 35J61: Semilinear elliptic equations
Secondary: 35R11: Fractional partial differential equations 53A30: Conformal differential geometry
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 1 , February 2022 , pp. 28 - 53
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press

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