Blow-up and instability of standing waves for the NLS with a point interaction in dimension two

In the present note we study the NLS equation in dimension two with a point interaction and in the supercritical regime, showing two results. After obtaining the (nonstandard) virial formula, we exhibit a set of initial data that blow-up. Moreover we show the standing waves $e^{i\omega t}{\phi }_{\omega }$ corresponding to ground states ${\phi }_{\omega }$ of the action are strongly unstable, at least for sufficiently high $\omega$.