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We prove various results on the existence and multiplicity of harmonic and subharmonic solutions to the second order nonautonomous equation x′′+ g(x) = s+w(t, x), as s→ +∞ or s→ −∞, where g is a smooth function defined on a open interval ]a, b[⊂ R. The hypotheses we assume on the nonlinearity g(x) allow us to cover the case b = +∞ (or a = −∞) and g having superlinear growth at infinity, as well as the case b < +∞ (or a > −∞) and g having a singularity in b (respectively in a). Applications are given also to situations like g′(−∞) 6= g′(+∞) (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.

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