Periodic solutions for one-dimensional nonlinear nonlocal problem with drift including singular nonlinearities

TL;DR

This paper establishes the existence and multiplicity of periodic solutions for a class of one-dimensional nonlinear nonlocal equations involving fractional Laplacians, singular nonlinearities, and gradient terms, extending classical results to nonlocal contexts.

Contribution

It introduces new existence and multiplicity results for nonlocal equations with singular nonlinearities, using degree theory and Perron's method, expanding classical ODE results to fractional Laplacian settings.

Findings

Existence of periodic solutions under various conditions.

Multiplicity results where a priori bounds are lost.

Extension of classical ODE results to nonlocal fractional equations.

Abstract

In this paper, we prove existence results of a one-dimensional periodic solution to equations with the fractional Laplacian of order $s\in(1/2,1)$, singular nonlinearity, and gradient term under various situations, including nonlocal contra-part of classical Lienard vector equations, as well other nonlocal versions of classical results know only in the context of second-order ODE. Our proofs are based on degree theory and Perron's method, so before that we need to establish a variety of priori estimates under different assumptions on the nonlinearities appearing in the equations. Besides, we obtain also multiplicity results in a regime where a priori bounds are lost and bifurcation from infinity occurs.