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

TL;DR

This paper studies a nonlinear Schr"odinger equation with weighted exponential nonlinearity, establishing conditions for local and global well-posedness in subcritical and critical regimes, advancing understanding of energy-critical equations.

Contribution

It introduces new well-posedness results for a weighted exponential nonlinear Schr"odinger equation in both subcritical and critical cases.

Findings

Proved local well-posedness in subcritical regime.

Established global well-posedness in critical regime.

Analyzed energy-critical behavior of the equation.

Abstract

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