Periodic solutions to parameter-dependent equations with a φ-Laplacian type operator

We study the periodic boundary value problem associated with the phi-Laplacian equation of the form (phi(u'))' + f(u)' + g(t, u) = s, where s is a real parameter, f and g are continuous functions, and g is T-periodic in the variable t. The interest is in Ambrosetti-Prodi type alternatives which provide the existence of zero, one or two solutions depending on the choice of the parameter s. We investigate this problem for a broad family of nonlinearities, under non-uniform type conditions on g(t, u) as u -> +/- 8. We generalize, in a unified framework, various classical and recent results on parameter-dependent nonlinear equations.