Nonlinearities and singularities in elliptic boundary value problems

This articles is concerned with the interaction between nonlinearities and the spectrum of linear differential operators in elliptic boundary value problems. The focus is on how solution multiplicity arises from spectral interaction and how singularity theory provides a framework for understanding this phenomenon. Starting with the classic Ambrosetti–Prodi theorem as a paradigm of a global fold, we progress to cusps, swallowtails, and butterfly singularities that appear when convexity assumptions are relaxed and when different types of nonlinearities are considered. The classical results in singularity theory of H. Whitney, R. Thom, V.I. Arnold, and J. Mather are presented, as well as their extension to the infinite-dimensional setting in Banach spaces. Detailed examples, including cubic nonlinearities and models with local eigenvalue crossing, illustrate the rich geometric structure of solution sets. Several open problems and recent developments are discussed.