Periodic solutions for a fractional asymptotically linear problem

TL;DR

This paper investigates the existence and multiplicity of periodic solutions for a non-local fractional differential equation with asymptotically linear nonlinearity, using pseudo-index theory and a Caffarelli-Silvestre extension.

Contribution

It introduces a novel approach to analyze periodic solutions of fractional non-local problems with asymptotically linear behavior using a degenerate elliptic reformulation.

Findings

Established existence of periodic solutions.

Proved multiplicity results under certain conditions.

Addressed new challenges in fractional periodic nonlocal problems.

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 the pseudo-index theory developed by Bartolo, Benci and Fortunato \cite{bbf} 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 literature, some new additional difficulties and a careful analysis of the fractional spaces involved is necessary.