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

Abstract

We study the defocusing energy-critical inhomogeneous nonlinear Schr\"odinger equation \[ i\partial_tu+\Delta u=|x|^{-b}|u|^{\frac{4-2b}{d-2}}u, \qquad (t,x)\in\R\times\R^d, \] with initial data $u_0\in\dot H_x^1(\R^d)$, where $d\ge 3$ and $0<b<\min\{2,\frac d2\}$. 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 H\"older continuous, so the short-time perturbation argument cannot be closed in $\dot S^1$, and that the singular coefficient $|x|^{-b}$ breaks translation symmetry. To handle these issues, we exploit the weak-space structure $|x|^{-b}\in L^{\frac{d}{b},\infty}(\R^d)$, introduce exotic Strichartz norms, and prove a long-time stability theorem for the general energy-critical inhomogeneous nonlinear Schr\"odinger equation. We also show that profiles escaping to spatial infinity are asymptotically linear because of the decay of $|x|^{-b}$. 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.} \textbf{166} (2006), 645--675], this yields the main theorem.