Blow-up for the pointwise NLS in dimension two: absence of critical power

Abstract

We consider the Schrödinger equation in dimension two with a fixed, pointwise, focusing nonlinearity and show the occurrence of a blow-up phenomenon with two peculiar features: first, the energy threshold under which all solutions blow up is strictly negative and coincides with the infimum of the energy of the standing waves. Second, there is no critical power nonlinearity, i.e. for every power there exist blow-up solutions. This last property is uncommon among the conservative Schrödinger equations with local nonlinearity.

29 pages, in press on J. Differential Equations (some changes in the structure of the paper have been made with respect to the previous version). Keywords: NLS, blow up, point interactions, standing waves