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

We study existence and stability properties of ground-state standing waves for two-dimensional nonlinear Schrödinger equation with a point interaction and a focusing power nonlinearity. The Schrödinger operator with a point interaction (−Δα)α∈R describes a one-parameter family of self-adjoint realizations of the Laplacian with delta-like perturbation. The operator −Δα always has a unique simple negative eigenvalue eα. We prove that if the frequency of the standing wave is close to −eα, it is stable. Moreover, if the frequency is sufficiently large, we have the stability in the L2-subcritical or critical case, while the instability in the L2-supercritical case.