Global Well-posedness and Scattering for the Focusing Energy-critical Inhomogeneous Nonlinear Schr\"odinger Equation with Non-radial Data

TL;DR

This paper proves global well-posedness and scattering for the focusing energy-critical inhomogeneous nonlinear Schrödinger equation in non-radial cases across various dimensions, extending previous results with a unified approach.

Contribution

It introduces a comprehensive method that removes radial symmetry restrictions and applies to both focusing and defocusing cases for the inhomogeneous NLS.

Findings

Established global well-posedness for non-radial data

Proved scattering results in the energy-critical setting

Unified approach applicable to multiple dimensions and parameters

Abstract

We consider the focusing energy-critical inhomogeneous nonlinear Schr\"{o}dinger equation \[ iu_t + \Delta u = -|x|^{-b}|u|^{\alpha}u \] where $n \geq 3$, $0<b<\min(2, n/2)$, and $\alpha=(4-2b)/(n-2)$. We prove the global well-posedness and scattering for every $n \geq 3$, $0<b<\min(2, n/2)$, and every non-radial initial data $\phi \in \dot{H}^1(\mathbb{R}^n)$ by the concentration compactness arguments of Kenig and Merle (2006) as in the work of Guzman and Murphy (2021). Lorentz spaces are adopted during the development of the stability theory in critical spaces as in Killip and Visan (2013). The result improves multiple earlier works by providing a unified approach to the scattering problem, accepting both focusing and defocusing cases, and eliminating the radial-data assumption and additional restrictions to the range of $n$ and $b$.