Scattering for the radial focusing INLS equation in higher dimensions

Abstract

We consider the inhomogeneous nonlinear Schrödinger equation $$i{u}_t+\Delta u+|x{|}^{-b}|u{|}^{\alpha }u=0,$$ where $\frac{4-2b}{N}<\alpha <\frac{4-2b}{N-2}$ (when $N=2$, $\frac{4-2b}{N}<\alpha <\infty$) and $0<b<\min \{N/3,1\}$. For a radial initial data $u_0\in H^1({\mathbb{R}}^N)$ under a certain smallness condition we prove that the corresponding solution is global and scatters. The smallness condition is related to the ground state solution of $-Q+\Delta Q+|x{|}^{-b}|Q{|}^{\alpha }Q=0$ and the critical Sobolev index $s_c=\frac{N}{2}-\frac{2-b}{\alpha }$. This is an extension of the recent work \cite{paper2} by the same authors, where they consider the case $N=3$ and $\alpha =2$. The proof is inspired by the concentration-compactness/rigidity method developed by Kenig-Merle \cite{KENIG} to study $H^1({\mathbb{R}}^N)$-critical problem and also Holmer-Roudenko \cite{HOLROU} in the case of $H^1({\mathbb{R}}^N)$-subcritical equations.

arXiv admin note: substantial text overlap with arXiv:1610.06523

References in corpus (1)

• A sharp condition for scattering of the radial 3d cubic nonlinear Schroedinger equation

Cited by in corpus (1)

• Local and global well-posedness in $L^2({\mathbb{R}}^n)$ for the inhomogeneous nonlinear Schrödinger equation