Energy critical Schrödinger equation with weighted exponential nonlinearity I: Local and global well-posedness

Abstract

We investigate the initial value problem for a defocusing nonlinear Schrödinger equation with weighted exponential nonlinearity $$i{\partial }_{t}u+\Delta u=\frac{u}{|x{|}^b}(e^{\alpha |u{|}^2}-1);(t,x)\in \mathbb{R}\times {\mathbb{R}}^2,$$ where $0<b<1$ and $\alpha =2\pi (2-b)$. We establish local and global well-posedness in the subcritical and critical regimes.