Some nonlinear problems for the superposition of fractional operators with Neumann boundary conditions

We discuss the existence theory of a nonlinear problem of nonlocal type subject to Neumann boundary conditions. Differently from the existing literature, the elliptic operator under consideration is obtained as a superposition of operators of mixed order. The setting that we introduce is very genera...

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Bibliographic Details
Published inMathematische Nachrichten Vol. 298; no. 8; pp. 2570 - 2595
Main Authors Dipierro, Serena, Proietti Lippi, Edoardo, Sportelli, Caterina, Valdinoci, Enrico
Format Journal Article
LanguageEnglish
Published Weinheim Wiley Subscription Services, Inc 01.08.2025
Subjects
Boundary conditions
Eigenvalues
Functional analysis
Neumann boundary condition
nonlocal operator
Operators (mathematics)
superposition of operators of mixed order
variational methods
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Summary:We discuss the existence theory of a nonlinear problem of nonlocal type subject to Neumann boundary conditions. Differently from the existing literature, the elliptic operator under consideration is obtained as a superposition of operators of mixed order. The setting that we introduce is very general and comprises, for instance, the sum of two fractional Laplacians, or of a fractional Laplacian and a Laplacian, as particular cases (the situation in which there are infinitely many operators, and even a continuous distribution of operators, can be considered as well). New bits of functional analysis are introduced to deal with this problem. An eigenvalue analysis divides the existence theory into two streams, one related to a mountain pass method, the other to a linking technique.
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content type line 14
ISSN:0025-584X
1522-2616
DOI:10.1002/mana.70006