Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Abundance of cusps and a converse to the Ambrosetti-Prodi theorem   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.

35
The categories of this paper are: mtrl-th
22
The categories of this paper are: chao-dyn
9
The categories of this paper are: nlin
26
The categories of this paper are: bayes-an
36
The categories of this paper are: cmp-lg
17
The categories of this paper are: q-fin
16
The categories of this paper are: stat
6
The categories of this paper are: astro-ph
1
The categories of this paper are: cs
14
The categories of this paper are: math-ph
5
The categories of this paper are: gr-qc
19
The categories of this paper are: eess
24
The categories of this paper are: dg-ga
0
The categories of this paper are: hep-ph
11
The categories of this paper are: hep-lat
4
The categories of this paper are: cond-mat