Periodic fractional Ambrosetti-Prodi for one-dimensional problem with drift (Q6564658)

The authors prove in this paper some Ambrosetti-Prodi type results for periodic solutions of some one-dimensional nonlinear problems that can have a drift term whose principal operator is the fractional Laplacian of order \(s\in (0,1)\). First of all, conditions for the existence and nonexistence of solutions of those problems are stated. The proofs of the existence results are based on the sub-supersolution method combined with topological degree type arguments. A priori bounds in order to get multiplicity results are also established. Under some regularity assumptions of the nonlinearities, the solutions of the mentioned equations are classical. Finally, some Ambrosetti-Prodi type results for a problem with singular nonlinearities are obtained.