Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Ground state and orbital stability for the NLS equation on a general
  starlike graph with potentials   We consider a nonlinear Schr\"odinger equation (NLS) posed on a graph or
network composed of a generic compact part to which a finite number of
half-lines are attached. We call this structure a starlike graph. At the
vertices of the graph interactions of $\delta$-type can be present and an
overall external potential is admitted. Under general assumptions on the
potential, we prove that the NLS is globally well-posed in the energy domain.
We are interested in minimizing the energy of the system on the manifold of
constant mass ($L^2$-norm). When existing, the minimizer is called ground state
and it is the profile of an orbitally stable standing wave for the NLS
evolution. We prove that a ground state exists for sufficiently small masses
whenever the quadratic part of the energy admits a simple isolated eigenvalue
at the bottom of the spectrum (the linear ground state). This is a wide
generalization of a result previously obtained for a star graph with a single
vertex. The main part of the proof is devoted to prove the concentration
compactness principle for starlike structures; this is non trivial due to the
lack of translation invariance of the domain. Then we show that a minimizing
bounded $H^1$ sequence for the constrained NLS energy with external linear
potentials is in fact convergent if its mass is small enough. Examples are
provided with discussion of hypotheses on the linear part.

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The categories of this paper are: atom-ph
20
The categories of this paper are: acc-phys
33
The categories of this paper are: plasm-ph
22
The categories of this paper are: chao-dyn
4
The categories of this paper are: cond-mat
35
The categories of this paper are: mtrl-th
27
The categories of this paper are: comp-gas
16
The categories of this paper are: stat
19
The categories of this paper are: eess
15
The categories of this paper are: nucl-ex
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The categories of this paper are: bayes-an
5
The categories of this paper are: gr-qc
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The categories of this paper are: econ
8
The categories of this paper are: hep-ex
31
The categories of this paper are: ao-sci
6
The categories of this paper are: astro-ph