Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On a doubly critical Schr\"odinger system in $\bbr^4$ with steep
  potential wells   Study the following two-component elliptic system% \begin{equation*}
\left\{\aligned&\Delta u-(\lambda a(x)+a_0)u+u^3+\beta v^2u=0\quad&\text{in
}\bbr^4,\\% &\Delta v-(\lambda b(x)+b_0)v+v^3+\beta u^2v=0\quad&\text{in
}\bbr^4,\\% &(u,v)\in\h\times\h,\endaligned\right.% \end{equation*} where
$a_0,b_0\in\bbr$ are constants; $\lambda>0$ and $\beta\in\bbr$ are parameters
and $a(x), b(x)\geq0$ are potential wells which are not necessarily to be
radial symmetric. By using the variational method, we investigate the existence
of ground state solutions and general ground state solutions (i.e., possibly
semi-trivial) to this system. Indeed, to the best of our knowledge, even the
existence of semi-trivial solutions is also unknown in the literature. We
observe some concentration behaviors of ground state solutions and general
ground state solutions. The phenomenon of phase separations is also excepted.
It seems that this is the first result definitely describing the phenomenon of
phase separation for critical system in the whole space $\bbr^4$. Note that
both the cubic nonlinearities and the coupled terms of the system are all of
critical growth with respect to the Sobolev critical exponent.

12
The categories of this paper are: nucl-th
15
The categories of this paper are: nucl-ex
28
The categories of this paper are: alg-geom
25
The categories of this paper are: patt-sol
18
The categories of this paper are: econ
33
The categories of this paper are: plasm-ph
24
The categories of this paper are: dg-ga
6
The categories of this paper are: astro-ph
1
The categories of this paper are: cs
9
The categories of this paper are: nlin
31
The categories of this paper are: ao-sci
22
The categories of this paper are: chao-dyn
36
The categories of this paper are: cmp-lg
35
The categories of this paper are: mtrl-th
7
The categories of this paper are: hep-th
37
The categories of this paper are: supr-con