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Periodic solutions for a fractional asymptotically linear problem

Published online by Cambridge University Press:  26 December 2018

Vincenzo Ambrosio  and
Giovanni Molica Bisci
Vincenzo Ambrosio
Affiliation:
Dipartimento di Scienze Matematiche, Informatiche e Fisiche, Università di Udine, via delle Scienze 206, 33100 Udine, Italy ([email protected])
Giovanni Molica Bisci
Affiliation:
Dipartimento PAU, Università ‘Mediterranea’ di Reggio Calabria, Salita Melissari, Feo di Vito, 89100 Reggio Calabria, Italy ([email protected])
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Abstract

We study the existence and multiplicity of periodic weak solutions for a non-local equation involving an odd subcritical nonlinearity which is asymptotically linear at infinity. We investigate such problem by applying the pseudo-index theory developed by Bartolo, Benci and Fortunato [11] after transforming the problem to a degenerate elliptic problem in a half-cylinder with a Neumann boundary condition, via a Caffarelli-Silvestre type extension in periodic setting. The periodic nonlocal case, considered here, presents, respect to the cases studied in the literature, some new additional difficulties and a careful analysis of the fractional spaces involved is necessary.

Keywords

Fractional Laplacianvariational methodsperiodic solutionsasymptotically linear problemnonresonant problemspseudo-genus35A1535S1535B10
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 3 , June 2019 , pp. 593 - 615
Copyright
Copyright © Royal Society of Edinburgh 2018 

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