On the use of Morse index and rotation numbers for multiplicity results of resonant BVPs

Motivated by a recent series of papers by K. Li and co-workers [14], [15], [16], [17] and [18], we consider the problem of the existence and multiplicity of solutions for the Neumann or periodic BVPs associated to a class of scalar equations of the form x″+f(t,x)=0x″+f(t,x)=0. The class considered is such that the behavior of its solutions near zero and near infinity may be compared with the behavior of the solutions of two suitable linear systems, one considered near zero and the other near infinity. We show how a rotation number approach, together with the Poincaré–Birkhoff theorem and a recent variant of it, allows to obtain multiplicity results in terms of the gap between the Morse indexes of the referred linear systems at zero and at infinity. These systems may also be resonant. When the gaps are sufficiently large, our multiplicity results improve the ones obtained by variational methods in the quoted papers. Also, our approach allows a description of the solutions obtained in terms of their nodal properties.