Abundance of cusps and a converse to the Ambrosetti-Prodi theorem

TL;DR

This paper demonstrates that convexity of the nonlinearity is necessary for the Ambrosetti-Prodi theorem's global fold property, showing that non-convex functions lead to multiple preimages and cusps in the critical set.

Contribution

It proves that convexity is essential for the theorem and characterizes the occurrence of cusps and multiple preimages for non-convex nonlinearities.

Findings

Non-convex $f$ leads to points with at least four preimages.

Critical points of $F$ can generically be cusps.

Results hold for various boundary conditions.

Abstract

According to 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, under mild conditions, convexity is indeed necessary. If $f$ is not convex, there is a point with at least four preimages under $F$. 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 a class of boundary conditions.