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Liouville type results for a nonlocal obstacle problem
Author(s) -
Brasseur Julien,
Coville Jérôme,
Hamel François,
Valdinoci Enrico
Publication year - 2019
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms.12229
Subject(s) - mathematics , bounded function , bistability , limiting , obstacle , obstacle problem , type (biology) , robustness (evolution) , nonlinear system , regular polygon , mathematical analysis , reaction–diffusion system , open set , ball (mathematics) , infinity , pure mathematics , combinatorics , geometry , physics , quantum mechanics , mechanical engineering , ecology , boundary (topology) , biochemistry , chemistry , political science , law , gene , engineering , biology
This paper is concerned with qualitative properties of solutions to nonlocal reaction–diffusion equations of the form∫ R N ∖ K J ( x − y )( u ( y ) − u ( x ) )d y + f ( u ( x ) ) = 0 , x ∈ R N ∖ K ,set in a perforated open setR N ∖ K , where K ⊂ R Nis a bounded compact ‘obstacle’ and f is a bistable nonlinearity. When K is convex, we prove some Liouville‐type results for solutions satisfying some asymptotic limiting conditions at infinity. We also establish a robustness result, assuming slightly relaxed conditions on K .
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