We investigate the ground states for the focusing, subcritical nonlinear Schrodinger 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. A refinement of a classical result on rearrangements is proved to obtain qualitative features of the ground states.

Ground states for the planar NLSE with a point defect as minimizers of the constrained energy / Adami, R; Boni, F; Carlone, R; Tentarelli, L. - In: CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS. - ISSN 0944-2669. - 61:5(2022). [10.1007/s00526-022-02310-8]

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

Adami, R;Boni, F;Tentarelli, L
2022

Abstract

We investigate the ground states for the focusing, subcritical nonlinear Schrodinger 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. A refinement of a classical result on rearrangements is proved to obtain qualitative features of the ground states.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11583/2970829