Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Ground states of nonlinear Schr\"odinger equations with sum of periodic
  and inverse-square potentials   We study the existence of solutions of the following nonlinear Schr\"odinger
equation \begin{equation*}
  -\Delta u + \Big(V(x)-\frac{\mu}{|x|^2}\Big) u = f(x,u)
  \hbox{ for } x\in\mathbb{R}^N\setminus\{0\}, \end{equation*} where
$V:\mathbb{R}^N\to\mathbb{R}$ and $f:\mathrm{R}^N\times\mathbb{R}\to\mathbb{R}$
are periodic in $x\in\mathbb{R}$. We assume that $0$ does not lie in the
spectrum of $-\Delta+V$ and $\mu<\frac{(N-2)^2}{4}$, $N\geq 3$. The superlinear
and subcritical term $f$ satisfies a weak monotonicity condition. For
sufficiently small $\mu\geq 0$ we find a ground state solution as a minimizer
of the energy functional on a natural constraint. If $\mu<0$ and $0$ lies below
the spectrum of $-\Delta+V$, then ground state solutions do not exist.

15
The categories of this paper are: nucl-ex
30
The categories of this paper are: q-alg
37
The categories of this paper are: supr-con
8
The categories of this paper are: hep-ex
19
The categories of this paper are: eess
26
The categories of this paper are: bayes-an
17
The categories of this paper are: q-fin
23
The categories of this paper are: solv-int
29
The categories of this paper are: funct-an
0
The categories of this paper are: hep-ph
11
The categories of this paper are: hep-lat
32
The categories of this paper are: atom-ph
12
The categories of this paper are: nucl-th
4
The categories of this paper are: cond-mat
5
The categories of this paper are: gr-qc
25
The categories of this paper are: patt-sol