Semilinear elliptic equations involving mixed local and nonlocal operators

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...

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Bibliographic Details
Published inProceedings of the Royal Society of Edinburgh. Section A. Mathematics Vol. 151; no. 5; pp. 1611 - 1641
Main Authors Biagi, Stefano, Vecchi, Eugenio, Dipierro, Serena, Valdinoci, Enrico
Format Journal Article
LanguageEnglish
Published Edinburgh, UK Royal Society of Edinburgh Scotland Foundation 01.10.2021
Cambridge University Press
Subjects
Differential equations
Elliptic functions
Operators (mathematics)
Symmetry
qualitative properties of solutions
symmetry
35A01
existence
35B65
35R11
Operators of mixed order
moving plane
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Summary: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.
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ISSN:0308-2105
1473-7124
DOI:10.1017/prm.2020.75