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

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 $b$.