### Abstract

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))dy+f(u(x))=0, x∈R ^{N/K} set in a perforated open set R ^{N/K} , where K ⊂ R _{N} is 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.

Original language | English |
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Pages (from-to) | 291-328 |

Journal | Proceedings of the London Mathematical Society |

Volume | 119 |

Issue number | 2 |

Early online date | 7 Feb 2019 |

DOIs | |

Publication status | E-pub ahead of print - 7 Feb 2019 |

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### Cite this

*Proceedings of the London Mathematical Society*,

*119*(2), 291-328. https://doi.org/10.1112/plms.12229

}

*Proceedings of the London Mathematical Society*, vol. 119, no. 2, pp. 291-328. https://doi.org/10.1112/plms.12229

**Liouville type results for a nonlocal obstacle problem.** / Brasseur, Julien; Coville, Jérôme; Hamel, François; Valdinoci, Enrico.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Liouville type results for a nonlocal obstacle problem

AU - Brasseur, Julien

AU - Coville, Jérôme

AU - Hamel, François

AU - Valdinoci, Enrico

PY - 2019/2/7

Y1 - 2019/2/7

N2 - 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))dy+f(u(x))=0, x∈R N/K set in a perforated open set R N/K , where K ⊂ R N is 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.

AB - 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))dy+f(u(x))=0, x∈R N/K set in a perforated open set R N/K , where K ⊂ R N is 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.

KW - 35B09

KW - 35B53

KW - 35R09

KW - 45K05

KW - 45M20 (secondary)

KW - 47G20 (primary)

UR - http://www.scopus.com/inward/record.url?scp=85061276066&partnerID=8YFLogxK

U2 - 10.1112/plms.12229

DO - 10.1112/plms.12229

M3 - Article

VL - 119

SP - 291

EP - 328

JO - Proceedings of London Mathematical Society

JF - Proceedings of London Mathematical Society

SN - 0024-6115

IS - 2

ER -