Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On Blow-up Profile of Ground States of Boson Stars with External
  Potential   We study minimizers of the pseudo-relativistic Hartree functional
$$\mathcal{E}_{a}(u):=\|(-\Delta+m^{2})^{1/4}u\|_{L^{2}}^{2}-\frac{a}{2}\int_{\mathbb{R}^{3}}(\left|\cdot\right|^{-1}\star
|u|^{2})(x)|u(x)|^{2}{\rm d}x+\int_{\mathbb{R}^{3}}V(x)|u(x)|^{2}{\rm d}x$$
under the mass constraint $\int_{\mathbb{R}^3}|u(x)|^2{\rm d}x=1$. Here $m>0$
is the mass of particles and $V\geq 0$ is an external potential. We prove that
minimizers exist if and only if $a$ satisfies $0\leq a<a^{*}$, and there is no
minimizer if $a\geq a^*$, where $a^*$ is called the Chandrasekhar limit}. When
$a$ approaches $a^*$ from below, the blow-up behavior of minimizers is derived
under some general external potentials $V$. Here we consider three cases of
$V$: trapping potential, i.e. $V\in L_{{\rm loc}}^{\infty}(\mathbb{R}^3)$
satisfies $\lim_{|x|\to \infty}V(x)=\infty$; periodic potential, i.e. $V\in
C(\mathbb{R}^3)$ stisfies $V(x+z)=V(x)$ for all $z\in\mathbb{Z}^3$; and
ring-shaped potential, e.g. $
  V(x)=||x|-1|^p$ for some $p>0$.

15
The categories of this paper are: nucl-ex
8
The categories of this paper are: hep-ex
12
The categories of this paper are: nucl-th
30
The categories of this paper are: q-alg
33
The categories of this paper are: plasm-ph
27
The categories of this paper are: comp-gas
25
The categories of this paper are: patt-sol
34
The categories of this paper are: chem-ph
31
The categories of this paper are: ao-sci
9
The categories of this paper are: nlin
29
The categories of this paper are: funct-an
35
The categories of this paper are: mtrl-th
24
The categories of this paper are: dg-ga
32
The categories of this paper are: atom-ph
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The categories of this paper are: q-fin
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The categories of this paper are: quant-ph