Calculus of Variations and Geometric Measure Theory

R. Adami - F. Boni - R. Carlone - L. Tentarelli

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

created by tentarelli on 07 Jul 2022
modified on 31 Aug 2022


Published Paper

Inserted: 7 jul 2022
Last Updated: 31 aug 2022

Journal: Calc. Var. Partial Differential Equations
Volume: 61
Number: 5
Pages: art. 195, 32 pp
Year: 2022
Doi: 10.1007/s00526-022-02310-8

ArXiv: 2109.09482 PDF


We investigate the ground states for the focusing, subcritical nonlinear Schr\"odinger 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.