Standing waves for defocusing nonlinear Schr\"odinger equations with point interaction

We consider standing waves of the nonlinear Schr\"odinger equation $i\partial_t u = -\Delta_\alpha u + |u|^{p-1}u$ in the defocusing case in dimensions $N=2$ and $N=3$. Here, $-\Delta_\alpha$ denotes the Laplacian with a point interaction. This operator is bounded from below by a negative constant; consequently, unlike in the free case, the associated energy functional admits non-trivial minimizers. We establish existence and uniqueness of standing waves, and prove further qualitative properties, including radial symmetry, positivity, and stability. Moreover, we build an appropriate functional space for the zero-mass case and establish sharp decay estimates in this case.