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

In the present note, we study the focusing NLS equation in dimension two with a point interaction in the supercritical regime, showing two results. After obtaining the (nonstandard) virial formula, we exhibit a set of initial data that shows blow-up. Moreover, we show that the standing waves e^iω tφ _ω corresponding to ground states φ _ω of the action functional are strongly unstable, at least for sufficiently high ω .