Cusps and a converse to the Ambrosetti-Prodi Theorem

By the Ambrosetti-Prodi theorem, the map F(u) = \Delta u - f (u) between appropriate functional spaces is a global fold. Among the hypotheses, the convexity of the function f is required. We show in two different ways that convexity is indeed necessary. If f is not convex, there is a point with at least four preimages under F. Even more, F generically admits cusps among its critical points. We present a larger class of nonlinearities f for which the critical set of F has cusps. The results are true for Dirichlet, Neumann and periodic boundary conditions, among others.