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

TL;DR

This paper demonstrates that in two-dimensional Schrödinger equations with pointwise focusing nonlinearity, solutions can blow up at any power level, with a negative energy threshold and no critical power, which is unusual for local nonlinearities.

Contribution

It establishes the existence of blow-up solutions for all powers in 2D pointwise NLS and identifies a negative energy threshold coinciding with standing wave energies.

Findings

Solutions blow up at any power level in 2D pointwise NLS.

The energy threshold for blow-up is strictly negative.

No critical power exists for blow-up in this setting.

Abstract

We consider the Schr\"odinger 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 Schr\"odinger equations with local nonlinearity.