Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Axially symmetric solutions of Allen-Cahn equation with finite Morse
  index   In this paper we study axially symmetric solutions of Allen-Cahn equation
with finite Morse index. It is shown that there does not exist such a solution
in dimensions between $4$ and $10$. In dimension $3$, we prove that these
solutions have finitely many ends. Furthermore, the solution has exactly two
ends if its Morse index equals $1$.

14
The categories of this paper are: math-ph
28
The categories of this paper are: alg-geom
5
The categories of this paper are: gr-qc
21
The categories of this paper are: adap-org
26
The categories of this paper are: bayes-an
18
The categories of this paper are: econ
29
The categories of this paper are: funct-an
1
The categories of this paper are: cs
9
The categories of this paper are: nlin
22
The categories of this paper are: chao-dyn
11
The categories of this paper are: hep-lat
3
The categories of this paper are: physics
13
The categories of this paper are: quant-ph
15
The categories of this paper are: nucl-ex
6
The categories of this paper are: astro-ph
30
The categories of this paper are: q-alg