199 research outputs found
Ground states for NLS on graphs: a subtle interplay of metric and topology
We review some recent results on the minimization of the energy associated to
the nonlinear Schr\"odinger Equation on non-compact graphs. Starting from
seminal results given by the author together with C. Cacciapuoti, D. Finco, and
D. Noja for the star graphs, we illustrate the achiements attained for general
graphs and the related methods, developed in collaboration with E. Serra and P.
Tilli. We emphasize ideas and examples rather than computations or proofs.Comment: 18 pages, 17 figures. A review paper for a special number of
Mathematical Modellind of Natural Phenomen
One-dimensional versions of three-dimensional system: Ground states for the NLS on the spatial grid
We investigate the existence of ground states for the focusing Nonlinear
Schr\"odinger Equation on the infinite three-dimensional cubic grid. We extend
the result found for the analogous two-dimensional grid by proving an
appropriate Sobolev inequality giving rise to a family of critical
Gagliardo-Nirenberg inequalities that hold for every nonlinearity power from
and , namely, from the -critical power for the same problem in
to the critical power for the same problem in .
Given the Gagliardo-Nirenberg inequality, the problem of the existence of
ground state can be treated as already done for the two-dimensional grid.Comment: 13 pages, 3 figure
NLS ground states on graphs
We investigate the existence of ground states for the subcritical NLS energy
on metric graphs. In particular, we find out a topological assumption that
guarantees the nonexistence of ground states, and give an example in which the
assumption is not fulfilled and ground states actually exist. In order to
obtain the result, we introduce a new rearrangement technique, adapted to the
graph where it applies. Owing to such a technique, the energy level of the
rearranged function is improved by conveniently mixing the symmetric and
monotone rearrangement procedures.Comment: 24 pages, 4 figure
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