Periodic solutions for a fractional asymptotically linear problem

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 [9] 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.

Periodic fractional Ambrosetti-Prodi for one-dimensional problem with drift

“…Along with all the works, where this operator appears, we found [2,6,12] where the fractional Ambrosetti-Prodi type problem is studied, but not in the periodic setting that is the objective of the manuscript at hand. It is fair to mention that not much is known of the existence of periodic solutions in problems that involve the fractional Laplacian (see [3,4,9,11]). In the local case, s = 1, the seminal work is the one of Ambrosetti and Prodi [1], where the nonlinearity crosses the first Dirichlet eigenvalue.…”

Nonlinear equations involving the square root of the Laplacian

“…As pointed out in [11], the fractions of the Laplacian, such as the previous square root of the Laplacian A 1/2 , are the infinitesimal generators of Lévy stable diffusion processes and appear in anomalous diffusions in plasmas, flames propagation and chemical reactions in liquids, population dynamics, geophysical fluid dynamics, and American options in finance. Moreover, a lot of interest has been devoted to elliptic equations involving the fractions of the Laplacian, (see, among others, the papers [1,2,3,5,8,14,24,28,35,40] as well as [7,25,27,30,31,32,34] and the references therein). See also the papers [4,37] for related topics.…”

Nonlinear equations involving the square root of the Laplacian

“…As pointed out in [11], the fractions of the Laplacian, such as the previous square root of the Laplacian A 1/2 , are the infinitesimal generators of Lévy stable diffusion processes and appear in anomalous diffusions in plasmas, flames propagation and chemical reactions in liquids, population dynamics, geophysical fluid dynamics, and American options in finance. Moreover, a lot of interest has been devoted to elliptic equations involving the fractions of the Laplacian, (see, among others, the papers [1,2,3,5,8,14,24,28,35,40] as well as [7,25,27,30,31,32,34] and the references therein). See also the papers [4,37] for related topics.…”

“…for any t ∈ R. Assumptions (3) and (4) are quite standard in the presence of subcritical terms. Moreover, together with (5), they guarantee that the number…”