Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On mass - critical NLS with local and non-local nonlinearities   We consider the following nonlinear Schr\"{o}dinger equation with the double
$L^2$-critical nonlinearities \begin{align*} iu_t+\Delta
u+|u|^\frac{4}{3}u+\mu\left(|x|^{-2}*|u|^2\right)u=0\ \ \ \text{in
$\mathbb{R}^3$,} \end{align*} where $\mu>0$ is small enough. Our first goal is
to prove the existence and the non-degeneracy of the ground state $Q_{\mu}$. In
particular, we develop an appropriate perturbation approach to prove the radial
non-degeneracy property and then obtain the general non-degeneracy of the
ground state $Q_{\mu}$. We then show the existence of finite time blowup
solution with minimal mass $\|u_0\|_{L^2}=\|Q_{\mu}\|_{L^2}$. More precisely,
we construct the minimal mass blowup solutions that are parametrized by the
energy $E_{\mu}(u_0)>0$ and the momentum $P_{\mu}(u_0)$. In addition, the
non-degeneracy property plays crucial role in this construction.

28
The categories of this paper are: alg-geom
17
The categories of this paper are: q-fin
30
The categories of this paper are: q-alg
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
34
The categories of this paper are: chem-ph
4
The categories of this paper are: cond-mat
14
The categories of this paper are: math-ph
27
The categories of this paper are: comp-gas
25
The categories of this paper are: patt-sol
33
The categories of this paper are: plasm-ph
5
The categories of this paper are: gr-qc
10
The categories of this paper are: q-bio
13
The categories of this paper are: quant-ph
15
The categories of this paper are: nucl-ex