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.
Cusps and a converse to the Ambrosetti-Prodi Theorem / M. Calanchi, C. Tomei, A. Zaccur. - In: ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA. CLASSE DI SCIENZE. - ISSN 0391-173X. - 18:2(2018), pp. 483-507.
Cusps and a converse to the Ambrosetti-Prodi Theorem
M. Calanchi;
2018
Abstract
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.File | Dimensione | Formato | |
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