Semilinear Sturm-Liouville problem with periodic nonlinearity

In this paper we study Sturm-Liouville problems (pu′)′+qu+g(u)f with periodic nonlinearities g. We are interested in the resonant case. We generalize results known by Schaaf and Schmitt and Cañada and Roca to the situations in which the corresponding nontrivial solution to the linear part change sign. Our main tools are Lyapunov-Schmidt reduction and asymptotic methods based on the stationary phase argument. These methods have been used previously by Dancer to treat the particular case g=sin(·). In this paper we essentially modify the last method to get results also in the case when g cannot be expressed as a finite sum of sines and cosines.