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A rigidity result for non-local semilinear equations

Part of: Miscellaneous topics - Partial differential equations Partial differential equations

Published online by Cambridge University Press:  31 May 2017

Alberto Farina  and
Enrico Valdinoci
Alberto Farina
Affiliation:
Department of Mathematics and Statistics, University of Melbourne, Parkville, VIC 3010, Australia and Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, 10117 Berlin, Germany and Dipartimento di Matematica, Università degli Studi di Milano, Via Cesare Saldini 50, 20133 Milano, Italy (enrico@math.utexas.edu)
Enrico Valdinoci
Affiliation:
Department of Mathematics and Statistics, University of Melbourne, Parkville, VIC 3010, Australia and Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, 10117 Berlin, Germany and Dipartimento di Matematica, Università degli Studi di Milano, Via Cesare Saldini 50, 20133 Milano, Italy (enrico@math.utexas.edu)
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We consider a possibly anisotropic integrodifferential semilinear equation, driven by a non-decreasing nonlinearity. We prove that if the solution grows less than the order of the operator at infinity, then it must be affine (possibly constant).

Keywords

non-local integrodifferential semilinear equationsLiouville-type theoremsnon-decreasing nonlinearities

MSC classification

Secondary: 35R11: Fractional partial differential equations 35B53: Liouville theorems, Phragmén-Lindelöf theorems 35R09: Integro-partial differential equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 147 , Issue 5 , October 2017 , pp. 1009 - 1018
Copyright
Copyright © Royal Society of Edinburgh 2017