| 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 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. |
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