Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On existence and phase separation of solitary waves for nonlinear
  Schr\"odinger systems modelling simultaneous cooperation and competition   We study the existence of positive bound states for the nonlinear elliptic
system \[ \begin{cases} - \Delta u_i + \lambda_i u_i = \sum_{j=1}^d \beta_{ij}
u_j^2 u_i & \text{in $\Omega$} \\ u_1 =\cdots = u_d=0 & \text{on $\partial
\Omega$}, \end{cases} \] where $d \ge 2$, $\beta_{ij}= \beta_{ji}$,
$\beta_{ii},\lambda_i >0$, and $\Omega$ is either a bounded domain of
$\mathbb{R}^N$, or $\Omega=\mathbb{R}^N$, with $N \le 3$. In light of its
applicability in several physical contexts, the problem has been intensively
studied in recent years, and several results concerning existence, multiplicity
and qualitative properties of the solutions are available if either
$\beta_{ij}\le 0$ for every $i \neq j$, or $\beta_{ij}>0$ for every $i \neq j$
and some additional assumptions are satisfied. On the other hand, only very
partial results are known in the case of \emph{simultaneous cooperation and
competition}, that is, when there exist two pairs $(i_1,j_1)$ and $(i_2,j_2)$
such that $i_1 \neq j_1$, $i_2 \neq j_2$, $\beta_{i_1 j_1}>0$ and
$\beta_{i_2,j_2}<0$. In this setting, we provide sufficient conditions on the
coupling parameters $\beta_{ij}$ in order to have a positive solution. Our
first main results establishes the existence of solutions with at least $m$
positive components for every $m \le d$. Any such solution is a minimizer of
the energy functional $J$ restricted on a \emph{Nehari-type manifold}
$\mathcal{N}$. By means of level estimates on the constrained second
differential of $J$ on $\mathcal{N}$, we show that, under some additional
assumptions, any such minimizer has all nontrivial components. In order to
prove this second result, we analyse the phase separation phenomena which
involve solutions of the system in a \emph{not completely competitive}
framework.

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The categories of this paper are: acc-phys
34
The categories of this paper are: chem-ph
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The categories of this paper are: dg-ga
9
The categories of this paper are: nlin
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The categories of this paper are: cmp-lg
11
The categories of this paper are: hep-lat
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The categories of this paper are: eess
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The categories of this paper are: ao-sci
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The categories of this paper are: atom-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: cs
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The categories of this paper are: chao-dyn
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The categories of this paper are: supr-con
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The categories of this paper are: econ
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The categories of this paper are: patt-sol
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The categories of this paper are: math-ph