Global well-posedness and scattering for defocusing energy-critical inhomogeneous NLS in dimensions $d\ge 3$

We study the defocusing energy-critical inhomogeneous nonlinear Schrödinger equation $$i{\mathrm{\partial }}_{t}u+\mathrm{\Delta }u=\mathrm{\mid }x{\mathrm{\mid }}^{-b}\mathrm{\mid }u{\mathrm{\mid }}^{\frac{4-2b}{d-2}}u,(t,x)\in \mathbb{R}\times {\mathbb{R}}^d,$$ with initial data $u_0\in {\dot{H}}_x^1({\mathbb{R}}^d)$, where $d\ge 3$ and $0<b<\min \{2,\frac{d}{2}\}$. We prove global well-posedness and scattering for arbitrary non-radial data. The main difficulties are that, when $d\ge 6$, the derivative of the critical nonlinearity is only Hölder continuous, so the short-time perturbation argument cannot be closed in ${\dot{S}}^1$, and that the singular coefficient $\mathrm{\mid }x{\mathrm{\mid }}^{-b}$ breaks translation symmetry. To handle these issues, we exploit the weak-space structure $\mathrm{\mid }x{\mathrm{\mid }}^{-b}\in L^{\frac{d}{b},\mathrm{\infty }}({\mathbb{R}}^d)$, introduce exotic Strichartz norms, and prove a long-time stability theorem for the general energy-critical inhomogeneous nonlinear Schrödinger equation. We also show that profiles escaping to spatial infinity are asymptotically linear because of the decay of $\mathrm{\mid }x{\mathrm{\mid }}^{-b}$. Consequently, almost periodic solutions are compact modulo scaling only, with neither spatial nor frequency center parameters. Combined with the concentration–compactness argument of Kenig–Merle [\emph{Invent. Math.} $\mathbf{\text{166}}$ (2006), 645–675], this yields the main theorem.