Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Multiplicity theorems involving functions with non-convex range   Here is a sample of the results proved in this paper: Let $f:{\bf R}\to {\bf
R}$ be a continuous function, let $\rho>0$ and let $\omega:[0,\rho[\to
[0,+\infty[$ be a continuous increasing function such that $\lim_{t\to
\rho^-}\omega(t)=+\infty$. Consider $C^0([0,1])\times C^0([0,1])$ endowed with
the norm $$\|(\alpha,\beta)\|=\int_0^1|\alpha(t)|dt+\int_0^1|\beta(t)|dt\ .$$
Then, the following assertions are equivalent: $(a)$ the restriction of $f$ to
$\left [-{{\sqrt{\rho}}\over {2}},{{\sqrt{\rho}}\over {2}}\right ]$ is not
constant; $(b)$ for every convex set $S\subseteq C^0([0,1])\times C^0([0,1])$
dense in $C^0([0,1])\times C^0([0,1])$, there exists $(\alpha,\beta)\in S$ such
that the problem
$$\cases{-\omega\left(\int_0^1|u'(t)|^2dt\right)u''=\beta(t)f(u)+\alpha(t) & in
$[0,1]$\cr & \cr u(0)=u(1)=0\cr & \cr \int_0^1|u'(t)|^2dt<\rho\cr}$$ has at
least two classical solutions.

9
The categories of this paper are: nlin
0
The categories of this paper are: hep-ph
4
The categories of this paper are: cond-mat
8
The categories of this paper are: hep-ex
28
The categories of this paper are: alg-geom
15
The categories of this paper are: nucl-ex
18
The categories of this paper are: econ
36
The categories of this paper are: cmp-lg
22
The categories of this paper are: chao-dyn
1
The categories of this paper are: cs
26
The categories of this paper are: bayes-an
19
The categories of this paper are: eess
7
The categories of this paper are: hep-th
34
The categories of this paper are: chem-ph
3
The categories of this paper are: physics
16
The categories of this paper are: stat