JournalsaihpcVol. 34, No. 2pp. 469–482

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

  • Yannick Sire

    Aix Marseille Université, CNRS, Centrale Marseille, Institut de Mathématiques de Marseille, UMR 7373, 13453 Marseille, France
  • François Hamel

    Aix Marseille Université, CNRS, Centrale Marseille, Institut de Mathématiques de Marseille, UMR 7373, 13453 Marseille, France
  • Xavier Ros-Oton

    The University of Texas at Austin, Department of Mathematics, 2515 Speedway, Austin, TX 78751, USA
  • Enrico Valdinoci

    Weierstraß Institute, Mohrenstraße 39, 10117 Berlin, Germany, Università di Milano, Dipartimento di Matematica Federigo Enriques, Via Cesare Saldini 50, 20133 Milano, Italy, The University of Melbourne, Department of Mathematics and Statistics, Parkville, VIC 3052, Australia
A one-dimensional symmetry result for a class of nonlocal semilinear equations in the plane cover
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Abstract

We consider entire solutions to Lu=f(u)\mathscr{L}u = f(u) in R2\mathbb{R}^{2}, where L\mathscr{L} is a nonlocal operator with translation invariant, even and compactly supported kernel K. Under different assumptions on the operator L\mathscr{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.

Cite this article

Yannick Sire, François Hamel, Xavier Ros-Oton, Enrico Valdinoci, A one-dimensional symmetry result for a class of nonlocal semilinear equations in the plane. Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2017), no. 2, pp. 469–482

DOI 10.1016/J.ANIHPC.2016.01.001