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 $e^{i\omega t} \varphi_\omega$ corresponding to ground states $\varphi_\omega$ of the action are strongly unstable, at least for sufficiently high $\omega$.