On weak solutions to a fractional Hardy–Hénon equation: Part I: Nonexistence

Shoichi Hasegawa; Norihisa Ikoma; Tatsuki Kawakami

doi:10.3934/cpaa.2021033

Article Contents

2021, Volume 20, Issue 4: 1559-1600. Doi: 10.3934/cpaa.2021033

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On weak solutions to a fractional Hardy–Hénon equation: Part I: Nonexistence

1.
Department of Mathematics, School of Fundamental Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan
2.
Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, Kanagawa 223-8522, Japan
3.
Applied Mathematics and Informatics Course, Faculty of Advanced Science and Technology, Ryukoku University, 1-5 Yokotani, Seta Oe-cho, Otsu, Shiga 520-2194, Japan

^* Corresponding author
^* Corresponding author

Received: April 2020

Revised: January 2021

Early access: March 2021

Published: April 2021

The first author (S.H.) was supported by JSPS KAKENHI Grant Numbers JP 20J01191.The second author (N.I.) was supported by JSPS KAKENHI Grant Numbers JP 17H02851, 19H01797 and 19K03590.The third author (T.K.) was supported by JSPS KAKENHI Grant Numbers JP 19H05599 and 16K17629

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Abstract

This paper and [20] treat the existence and nonexistence of stable (resp. outside stable) weak solutions to a fractional Hardy–Hénon equation $ (-\Delta)^s u = |x|^\ell |u|^{p-1} u $ in $ \mathbb{R}^N $, where $ 0 < s < 1 $, $ \ell > -2s $, $ p>1 $, $ N \geq 1 $ and $ N > 2s $. In this paper, the nonexistence part is proved for the Joseph–Lundgren subcritical case.

Keywords:

fractional Hardy–Hénon equation,
stable (stable outside a compact set) solutions,
Liouville type theorem.

Mathematics Subject Classification: Primary: 35R11, 35J61, 35B33, 35B53; Secondary: 35D30.

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References

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