Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Existence results for double-phase problems via Morse theory   We obtain nontrivial solutions for a class of double-phase problems using
Morse theory. In the absence of a direct sum decomposition, we use a
cohomological local splitting to get an estimate of the critical groups at
zero.

10
The categories of this paper are: q-bio
33
The categories of this paper are: plasm-ph
12
The categories of this paper are: nucl-th
35
The categories of this paper are: mtrl-th
9
The categories of this paper are: nlin
28
The categories of this paper are: alg-geom
21
The categories of this paper are: adap-org
7
The categories of this paper are: hep-th
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The categories of this paper are: stat
6
The categories of this paper are: astro-ph
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The categories of this paper are: q-fin
8
The categories of this paper are: hep-ex
14
The categories of this paper are: math-ph
3
The categories of this paper are: physics
19
The categories of this paper are: eess
37
The categories of this paper are: supr-con