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Nonlinear equations for fractional Laplacians II: Existence, uniqueness, and qualitative properties of solutions


Authors: Xavier Cabré and Yannick Sire
Journal: Trans. Amer. Math. Soc. 367 (2015), 911-941
MSC (2010): Primary 35J05
Published electronically: October 1, 2014
MathSciNet review: 3280032
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Abstract: This paper, which is the follow-up to part I, concerns the equation $ (-\Delta )^{s} v+G'(v)=0$ in $ \mathbb{R}^{n}$, with $ s\in (0,1)$, where $ (-\Delta )^{s}$ stands for the fractional Laplacian--the infinitesimal generator of a Lévy process.

When $ n=1$, we prove that there exists a layer solution of the equation (i.e., an increasing solution with limits $ \pm 1$ at $ \pm \infty $) if and only if the potential $ G$ has only two absolute minima in $ [-1,1]$, located at $ \pm 1$ and satisfying $ G'(-1)=G'(1)=0$. Under the additional hypotheses $ G''(-1)>0$ and $ G''(1)>0$, we also establish its uniqueness and asymptotic behavior at infinity. Furthermore, we provide with a concrete, almost explicit, example of layer solution.

For $ n\geq 1$, we prove some results related to the one-dimensional symmetry of certain solutions--in the spirit of a well-known conjecture of De Giorgi for the standard Laplacian.


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Additional Information

Xavier Cabré
Affiliation: Departament de Matemàtica Aplicada I, ICREA and Universitat Politècnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain
Email: xavier.cabre@upc.edu

Yannick Sire
Affiliation: LATP, Faculté des Sciences et Techniques, Université Paul Cézanne, Case cour A, Avenue Escadrille Normandie-Niemen, F-13397 Marseille Cedex 20, France – and – CNRS, LATP, CMI, 39 rue F. Joliot-Curie, F-13453 Marseille Cedex 13, France
Address at time of publication: Institut de Mathématique de Marseille, Technopole de Chateau-Gombert, CMI, Université Aix-Marseille, 13000, Marseille, France
Email: sire@cmi.univ-mrs.fr, yannick.sire@univ-amu.fr

DOI: https://doi.org/10.1090/S0002-9947-2014-05906-0
Received by editor(s): November 3, 2011
Received by editor(s) in revised form: June 27, 2012
Published electronically: October 1, 2014
Additional Notes: The first author was supported by grants MTM2008-06349-C03-01, MTM2011-27739-C04-01 (Spain) and 2009SGR-345 (Catalunya). The second author was supported by the ANR project PREFERED
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.