Scattering for the radial focusing INLS equation in higher dimensions

Abstract

We consider the inhomogeneous nonlinear Schr\"odinger 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.