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

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.

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.…”