Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Ground states for 3D dipolar Bose-Einstein condensate involving quantum
  fluctuations and three-body losses   We consider ground states of three-dimensional dipolar Bose-Einstein
condensate involving quantum fluctuations and three-body losses, which can be
described equivalently by positive $L^2$-constraint critical point of the
Gross-Pitaevskii energy functional \[E(u)\!=\!\frac{1}{2}\int_{{\mathbb{R}^3}}
{|\nabla u|}^2dx+\frac{\lambda_{1}}{2}\int_{{\mathbb{R}^3}} {|
u|}^4dx+\frac{\lambda_{2}}{2} \int_{\mathbb{R}^{3}}\left(K
\star|u|^{2}\right)|u|^{2} d x+\frac{2\lambda_{3}}{p}\int_{{\mathbb{R}^3}}
{|u|}^{p}dx,\] where $2<p<\frac{10}{3}$, $\lambda_{3}<0$, $\star$ is the
convolution, $ K(x) \!=\! \frac{{1-3{{\cos }^2}\theta(x) }}{{{{| x |}^3}}}$,
$\theta(x)$ is the angle between the dipole axis determined by $(0,0,1)$ and
the vector $x$. If ${\lambda _1} \!\!<\!\! \frac{4\pi} {3} {\lambda _2}\!\leq\!
0$ or ${\lambda _1} \!\!<\!- \frac{8\pi}{3} {\lambda _2}\!\leq\! 0$, $E(u)$ is
unbounded on the $L^2$-sphere $S_{c}\!:=\!\Big\{ u \!\in\! H^1({\mathbb{R}^3}):
\int_{{\mathbb{R}^3}} {{|u|}^2}dx\!=\!c^2 \Big\}$, so we turn to study a local
minimization problem $$ m(c,R_0)\!:=\!\inf _{u \in V^c_{R_0}} E(u)$$ for a
suitable $R_0\!>\!0$ with $V^c_{R_0} \!:=\!\left\{u \!\in\! S_c :
\big(\int_{{\mathbb{R}^3}} {{|\nabla u|}^2dx}\big)^{\frac{1}{2}}
\!<\!R_0\right\}$.
  We show that $m(c,R_0)$ is achieved by some $u_c>0$, which is a stable ground
state. Furthermore, by refining the upper bound of $m(c, R_0)$, we provide a
precise description of the asymptotic behavior of $u_c$ as the mass $c$
vanishes, i.e.
  $${[\frac{{p|{\lambda _3}|}}{{2{\gamma _c}}}]^{\frac{1}{{p -
2}}}}{u_c}(\frac{{x + {y_c}}}{{\sqrt {2{\delta _p}{\gamma _c}} }}) \to
{W_p}\;\;\;\;{\rm{in}}\;\;\;\;{H^1}({\mathbb{R}^3})\;\;\;\;{\rm{for
some}}\;\;\;\;{y_c} \in {\mathbb{R}^3}\;\;\;\;{\rm{as}}\;\;\;\;c \to {0^ + }.$$

20
The categories of this paper are: acc-phys
32
The categories of this paper are: atom-ph
4
The categories of this paper are: cond-mat
12
The categories of this paper are: nucl-th
34
The categories of this paper are: chem-ph
29
The categories of this paper are: funct-an
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The categories of this paper are: adap-org
7
The categories of this paper are: hep-th
9
The categories of this paper are: nlin
36
The categories of this paper are: cmp-lg
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The categories of this paper are: econ
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The categories of this paper are: dg-ga
5
The categories of this paper are: gr-qc
3
The categories of this paper are: physics
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The categories of this paper are: eess
1
The categories of this paper are: cs