Scattering for the radial 3D cubic focusing inhomogeneous nonlinear Schr\"odinger equation

Abstract

The purpose of this work is to study the 3D focusing inhomogeneous nonlinear Schr\"odinger equation $$ i u_t +\Delta u+|x|^{-b}|u|^2 u = 0, $$ where $0<b<1/2$. Let $Q$ be the ground state solution of $-Q+\Delta Q+ |x|^{-b}|Q|^{2}Q=0$ and $s_c=(1+b)/2$. We show that if the radial initial data $u_0$ belongs to $H^1(\mathbb{R}^3)$ and satisfies $E(u_0)^{s_c}M(u_0)^{1-s_c}<E(Q)^{s_c}M(Q)^{1-s_c}$ and $\| \nabla u_0 \|_{L^2}^{s_c} \|u_0\|_{L^2}^{1-s_c}<\|\nabla Q \|_{L^2}^{s_c} \|Q\|_{L^2}^{1-s_c}$, then the corresponding solution is global and scatters in $H^1(\mathbb{R}^3)$. Our proof is based in the ideas introduced by Kenig-Merle \cite{KENIG} in their study of the energy-critical NLS and Holmer-Roudenko \cite{HOLROU} for the radial 3D cubic NLS.