Well-posedness and scattering for a 2D inhomogeneous NLS with Aharonov-Bohm magnetic potential

Abstract

We consider the magnetic nonlinear inhomogeneous Schrödinger equation $$i{\partial }_{t}u-{\left(-i\nabla +\frac{\alpha }{|x{|}^2}(-x_2,x_1)\right)}^{2}u=\pm |x{|}^{-\varrho }|u{|}^{p-1}u,(t,x)\in \mathbb{R}\times {\mathbb{R}}^2,$$ where $\alpha \in \mathbb{R}\setminus \mathbb{Z},\varrho >0,p>1$. We prove a dichotomy of global existence and scattering versus blow-up of energy solutions under the ground state threshold in the inter-critical regime. The scattering is obtained by using the new approach of Dodson-Murphy (A new proof of scattering below the ground state for the 3D radial focusing cubic NLS, {Proc. Am. Math. Soc.} (2017)). This method is based on Tao's scattering criteria and Morawetz estimates. The novelty here is twice: we investigate the case $\varrho \alpha \ne 0$ and we consider general energy initial data (not necessarily radially symmetric).

We handle general initial data (not necessarily radially symmetric)