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.

28 pages. Keywords: standing waves, nonlinear Schrödinger, ground states, delta interaction, radially symmetric solutions, rearrangements. With respect to the previous versione, a section has been added to the Appendix and some minor revisions have been made

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• Local well-posedness and blow-up in the energy space for the 2D NLS with point interaction