Multiplicity results for periodic solutions of second order ODEs with asymmetric nonlinearities

We prove various results on the existence and multiplicity of harmonic and subharmonic solutions to the second order nonautonomous equation , as or where is a smooth function defined on a open interval The hypotheses we assume on the nonlinearity allow us to cover the case (or ) and having superlinear growth at infinity, as well as the case (or ) and having a singularity in (respectively in ). Applications are given also to situations like (including the so-called ``jumping nonlinearities'). Our results are connected to the periodic Ambrosetti - Prodi problem and related problems arising from the Lazer - McKenna suspension bridges model.