Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Positive and sign changing solutions to a nonlinear Choquard equation   We consider the problem \[-\Delta u + W(x)u = ((1/{|x|^{\alpha}} * |u|^{p})
|u|^{p-2}u, u \in H_{0}^{1}(\Omega)\], where $\Omega$ is an exterior domain in
$\mathbb{R}^{N}$, $N\geq3,$ $\alpha \in(0,N)$, $p\in[2,(2N-\alpha)/(N-2)$, $W$
is continuous, $\inf_{\mathbb{R}^{N}}W>0,$ and $W(x)$ tends to a positive
constant as $|x|$ tends to infinity. Under symmetry assumptions on $\Omega$ and
$W$, which allow finite symmetries, and some assumptions on the decay of $W$ at
infinity, we establish the existence of a positive solution and multiple sign
changing solutions to this problem, having small energy.

35
The categories of this paper are: mtrl-th
16
The categories of this paper are: stat
12
The categories of this paper are: nucl-th
21
The categories of this paper are: adap-org
27
The categories of this paper are: comp-gas
3
The categories of this paper are: physics
34
The categories of this paper are: chem-ph
5
The categories of this paper are: gr-qc
32
The categories of this paper are: atom-ph
4
The categories of this paper are: cond-mat
18
The categories of this paper are: econ
6
The categories of this paper are: astro-ph
23
The categories of this paper are: solv-int
17
The categories of this paper are: q-fin
8
The categories of this paper are: hep-ex
10
The categories of this paper are: q-bio