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Semilinear elliptic equations involving mixed local and nonlocal operators

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

Published online by Cambridge University Press:  14 October 2020

Stefano Biagi ,
Eugenio Vecchi ,
Serena Dipierro  and
Enrico Valdinoci
Stefano Biagi
Affiliation:
Politecnico di Milano - Dipartimento di Matematica, Via Bonardi 9, 20133 Milano, Italy (stefano.biagi@polimi.it; eugenio.vecchi@polimi.it)
Eugenio Vecchi
Affiliation:
Politecnico di Milano - Dipartimento di Matematica, Via Bonardi 9, 20133 Milano, Italy (stefano.biagi@polimi.it; eugenio.vecchi@polimi.it)
Serena Dipierro
Affiliation:
Department of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, WA 6009, Crawley, Australia (serena.dipierro@uwa.edu.au; enrico.valdinoci@uwa.edu.au)
Enrico Valdinoci
Affiliation:
Department of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, WA 6009, Crawley, Australia (serena.dipierro@uwa.edu.au; enrico.valdinoci@uwa.edu.au)
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Abstract

In this paper, we consider an elliptic operator obtained as the superposition of a classical second-order differential operator and a nonlocal operator of fractional type. Though the methods that we develop are quite general, for concreteness we focus on the case in which the operator takes the form − Δ + ( − Δ)s, with s ∈ (0, 1). We focus here on symmetry properties of the solutions and we prove a radial symmetry result, based on the moving plane method, and a one-dimensional symmetry result, related to a classical conjecture by G.W. Gibbons.

Keywords

Operators of mixed orderexistencesymmetrymoving planequalitative properties of solutions

MSC classification

Primary: 35A01: Existence problems
Secondary: 35B65: Smoothness and regularity of solutions 35R11: Fractional partial differential equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 5 , October 2021 , pp. 1611 - 1641
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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