Ground states for the planar NLSE with a point defect as minimizers of the constrained energy

Abstract

We investigate the ground states for the focusing, subcritical nonlinear Schrödinger equation with a point defect in dimension two, defined as the minimizers of the energy functional at fixed mass. We prove that ground states exist for every positive mass and show a logarithmic singularity at the defect. Moreover, up to a multiplication by a constant phase, they are positive, radially symmetric, and decreasing along the radial direction. In order to overcome the obstacles arising from the uncommon structure of the energy space, that complicates the application of standard rearrangement theory, we move to the study of the minimizers of the action functional on the Nehari manifold and then establish a connection with the original problem. An ad hoc result on rearrangements is given to prove qualitative features of the ground states.

Doubly nonlinear Schrödinger ground states on metric graphs

“…After first investigations on star graphs (see e.g. [4]), the behaviour of (4) on general non-compact graphs with half-lines has been characterized in [13,15] for L 2 -subcritical nonlinearities p ∈ (2,6) and in [16] for the L 2 -critical regime p = 6. In particular, it has been shown that whether ground states at a certain mass exist strongly depends both on topological and on metric properties of the graph.…”

“…On the other hand, fueled for instance by possible applications as the dynamic of confined charges [43,49] or the resonant tunneling [52], concentrated nonlinearities have been proposed. In Euclidean spaces, models describing the effect of a delta potential have been considered first in dimension one and three [9,10,17,18] (see also the recent papers [40,41]), whereas the analysis in dimension two is more recent [2,7,8,28,29,37]. On non-compact graphs, the presence of a single standard nonlinearity restricted to the compact core of the graph (i.e.…”

“…In this paper, we push forward this analysis by considering a model somewhat on the edge between combined and concentrated nonlinearities: the energy functional (1) involving two focusing nonlinearities, a standard one and a pointwise one. The corresponding ground states problem (2) has already been addressed both on the real line [25] and on star graphs [1]. On the real line, in the regime (3) where both the nonlinearities are L 2subcritical ground states exist for every value of the mass (see [25,Theorem 1.3] and Section 2 below).…”

“…Even though Assumption (H) is enough to prevent solutions of problem (4), this is no longer true in the doubly nonlinear case (2). The first part of our analysis provides a complete topological characterization of the problem.…”

Computing the least action ground state of the nonlinear Schrödinger equation by a normalized gradient flow

“…When Vpxq " 0, it arises in various physical contexts such as nonlinear optics and plasma physics [30]. When V is chosen as Dirac's delta function, it can be used to model a Bose-Einstein Condensation (BEC) in the presence of point defects or impurities [1,24]. Besides, recently, it has attracted a lot of attention when V is an inverse power potential (see [17,20,23] and references therein).…”