Resonant problems with multidimensional kernel and periodic nonlinearities

Abstract

In this paper we deal with semilinear boundary value problems for systems of equations whose nonlinear part involves linear combination of periodic functions and such that the linear part has a multidimensional solution space. This kind of problems is very important in applications, specially in mechanics and electric circuits theory. By using the Liapunov-Schmidt reduction and topological degree techniques, together with a careful analysis of the oscillatory behavior of some integrals associated to the bifurcation equation, we give a qualitative and quantitative description of the range of the corresponding nonlinear operator. Also, we provide some multiplicity results.