Prescribing the nodal behaviour of periodic solutions of a superlinear equation with indefinite weight

This paper deals with the existence of periodic solutions to the differential equation x'' + q(t)g(x) = 0. Here g is Lipschitz, xg(x) > 0 for all non vanishing x, g has superlinear growth at infinity and q is continuous and is allowed to change sign finitely many times. We prove that there are two periodic solutions with a precise number of zeros in each interval of positivity of q and that, moreover, for each interval of negativity, one can fix a priori whether the solution will have exactly one zero and be strictly monotone or will have no zeros and exactly one zero of the derivative. The techniques are based on the study of the Poincaré map and a careful phase plane analysis. Generalizations are discussed in order to treat more gereal Floquet-type boundary conditions.