A one-dimensional symmetry result for a class of nonlocal semilinear equations in the plane

F. Hamel, X. Ros-Oton, Y. Sire, E. Valdinoci

Research output: Contribution to journalArticle

7 Citations (Scopus)

Abstract

We consider entire solutions to Lu=f(u) in R2, where L is a nonlocal operator with translation invariant, even and compactly supported kernel K. Under different assumptions on the operator L, we show that monotone solutions are necessarily one-dimensional. The proof is based on a Liouville type approach. A variational characterization of the stability notion is also given, extending our results in some cases to stable solutions.
Original languageEnglish
Pages (from-to)469-482
Number of pages14
JournalAnnales de l'Institut Henri Poincare (C) Analyse Non Lineaire
Volume34
Issue number2
DOIs
Publication statusPublished - 2017
Externally publishedYes

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Nonlocal Equations
Semilinear Equations
Symmetry
Entire Solution
Stable Solution
Operator
Monotone
kernel
Invariant
Class

Cite this

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abstract = "We consider entire solutions to Lu=f(u) in R2, where L is a nonlocal operator with translation invariant, even and compactly supported kernel K. Under different assumptions on the operator L, we show that monotone solutions are necessarily one-dimensional. The proof is based on a Liouville type approach. A variational characterization of the stability notion is also given, extending our results in some cases to stable solutions.",
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A one-dimensional symmetry result for a class of nonlocal semilinear equations in the plane. / Hamel, F.; Ros-Oton, X.; Sire, Y.; Valdinoci, E.

In: Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire, Vol. 34, No. 2, 2017, p. 469-482.

Research output: Contribution to journalArticle

TY - JOUR

T1 - A one-dimensional symmetry result for a class of nonlocal semilinear equations in the plane

AU - Hamel, F.

AU - Ros-Oton, X.

AU - Sire, Y.

AU - Valdinoci, E.

PY - 2017

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AB - We consider entire solutions to Lu=f(u) in R2, where L is a nonlocal operator with translation invariant, even and compactly supported kernel K. Under different assumptions on the operator L, we show that monotone solutions are necessarily one-dimensional. The proof is based on a Liouville type approach. A variational characterization of the stability notion is also given, extending our results in some cases to stable solutions.

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JF - Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

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