Energy critical Schr\"odinger equation with weighted exponential nonlinearity I: Local and global well-posedness

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