On stability and instability of standing waves for 2d-nonlinear Schrodinger equations with point interaction

We study existence and stability properties of ground-state standing waves for two-dimensional nonlinear Schrodinger equation with a point interaction and a focusing power nonlinearity. The Schrodinger operator with a point interaction (-delta alpha)(alpha is an element of R) describes a one-parameter family of self-adjoint realizations of the Laplacian with delta-like perturbation. The operator -delta alpha always has a unique simple negative eigenvalue e alpha. We prove that if the frequency of the standing wave is close to -e(alpha), it is stable. Moreover, if the frequency is sufficiently large, we have the stability in the L-2-subcritical or critical case, while the instability in the L-2-supercritical case. (C)& nbsp;2022 Elsevier Inc. All rights reserved.