33 research outputs found
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
Singular limit of periodic metric grids
We investigate the asymptotic behaviour of nonlinear Schr\"odinger ground
states on -dimensional periodic metric grids in the limit for the length of
the edges going to zero. We prove that suitable piecewise-affine extensions of
such states converge strongly in to the corresponding
ground states on . As an application of such convergence results,
qualitative properties of ground states and multiplicity results for fixed mass
critical points of the energy on grids are derived. Moreover, we compare
optimal constants in -dimensional Gagliardo-Nirenberg inequalities on
and on grids. For -critical and supercritical powers, we
show that the value of such constants on grids is strictly related to that on
but, contrary to , constants on grids are not
attained. The proofs of these results combine purely variational arguments with
new Gagliardo-Nirenberg inequalities on grids.Comment: 36 pages, 4 figure
Mass-constrained ground states of the stationary NLSE on periodic metric graphs
We investigate the existence of ground states with fixed mass for the
nonlinear Schr\"odinger equation with a pure power nonlinearity on periodic
metric graphs. Within a variational framework, both the -subcritical and
critical regimes are studied. In the former case, we establish the existence of
global minimizers of the NLS energy for every mass and every periodic graph. In
the critical regime, a complete topological characterization is derived,
providing conditions which allow or prevent ground states of a certain mass
from existing. Besides, a rigorous notion of periodic graph is introduced and
discussed.Comment: 25 pages, 8 figure
-limit of the cut functional on dense graph sequences
A sequence of graphs with diverging number of nodes is a dense graph sequence
if the number of edges grows approximately as for complete graphs. To each such
sequence a function, called graphon, can be associated, which contains
information about the asymptotic behavior of the sequence. Here we show that
the problem of subdividing a large graph in communities with a minimal amount
of cuts can be approached in terms of graphons and the -limit of the
cut functional, and discuss the resulting variational principles on some
examples. Since the limit cut functional is naturally defined on Young
measures, in many instances the partition problem can be expressed in terms of
the probability that a node belongs to one of the communities. Our approach can
be used to obtain insights into the bisection problem for large graphs, which
is known to be NP-complete.Comment: 25 pages, 5 figure
-critical NLS on noncompact metric graphs with localized nonlinearity: topological and metric features
Carrying on the discussion initiated in (Dovetta-Tentarelli'18), we
investigate the existence of ground states of prescribed mass for the
-critical NonLinear Schr\"odinger Equation (NLSE) on noncompact metric
graphs with localized nonlinearity. Precisely, we show that the existence (or
nonexistence) of ground states mainly depends on a parameter called reduced
critical mass, and then we discuss how the topological and metric features of
the graphs affect such a parameter, establishing some relevant differences with
respect to the case of the extended nonlinearity studied by
(Adami-Serra-Tilli'17). Our results rely on a thorough analysis of the optimal
constant of a suitable variant of the -critical Gagliardo-Nirenberg
inequality.Comment: 22 pages, 7 figures. Keywords: metric graphs, NLS, ground states,
localized nonlinearity, -critical case. Some minor revisions have been
made with respect to the previous version. Accepted for publication by Calc.
Var. Partial Differential Equation
Prescribed mass ground states for a doubly nonlinear Schr\"odinger equation in dimension one
We investigate the problem of existence and uniqueness of ground states at
fixed mass for two families of focusing nonlinear Schr\"odinger equations on
the line.
The first family consists of NLS with power nonlinearities concentrated at a
point. For such model, we prove existence and uniqueness of ground states at
every mass when the nonlinearity power is subcritical and at a threshold
value of the mass in the critical regime.
The second family is obtained by adding a standard power nonlinearity to the
previous setting. In this case, we prove existence and uniqueness at every mass
in the doubly subcritical case, namely when both the powers related to the
pointwise and the standard nonlinearity are subcritical. If only one power is
critical, then existence and uniqueness hold only at masses lower than the
critical mass associated to the critical nonlinearity. Finally, in the doubly
critical case ground states exist only at critical mass, whose value results
from a non--trivial interplay between the two nonlinearities.Comment: 17 page