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

TL;DR

This paper investigates the existence, nonexistence, and multiplicity of periodic solutions for one-dimensional fractional Laplacian problems with drift, using topological and a priori methods, including cases with singular nonlinearities.

Contribution

It extends Ambrosetti-Prodi results to fractional Laplacian problems with drift, providing new existence, nonexistence, and regularity results, including for singular nonlinearities.

Findings

Established conditions for solution existence and nonexistence.

Proved solutions are classical under certain regularity assumptions.

Extended results to problems with singular nonlinearities.

Abstract

We establish Ambrosetti -Prodi type results for periodic solutions of one -dimensional nonlinear problems with drift term and drift -less whose principal operator is the fractional Laplacian of order $s\in(0,1)$. We establish conditions for the existence and nonexistence of solutions. The proofs of the existence results are based on the sub-supersolution method combined with topological degree type arguments. We also establish a priori bounds in order to get multiplicity results. We also prove that the solutions are $C^{1,\alpha}$ under some regularity assumptions in the nonlinearities, that is, the solutions of equations are classical. We finish the work obtaining existence results for problems with the fractional Laplacian with singular nonlinearity. In particular, we establish an Ambrosetti-Prodi type problem with singular nonlinearities.