Bifurcation Theory for a Class of Periodic Superlinear Problems

Abstract

We analyze, mainly using bifurcation methods, an elliptic superlinear problem in one-dimension with periodic boundary conditions. One of the main novelties is that we follow for the first time a bifurcation approach, relying on a Lyapunov-Schmidt reduction and some recent global bifurcation results, that allows us to study the local and global structure of non-trivial solutions at bifurcation points where the linearized operator has a two-dimensional kernel. Indeed, at such points the classical tools in bifurcation theory, like the Crandall-Rabinowitz theorem or some generalizations of it, cannot be applied because the multiplicity of the eigenvalues is not odd, and a new approach is required. We apply this analysis to specific examples, obtaining new existence and multiplicity results for the considered periodic problems, going beyond the information variational and fixed point methods like Poincar\'e-Birkhoff theorem can provide.