A sharp scattering condition for focusing mass-subcritical nonlinear Schrödinger equation

Abstract

This article is concerned with time global behavior of solutionsto focusing mass-subcritical nonlinear Schrödinger equation of power type withdata in a critical homogeneous weighted $L^2$ space. We givea sharp sufficient condition for scattering by proving existence ofa threshold solution which does not scatter at least forone time direction and of which initial data attains minimumvalue of a norm of the weighted $L^2$ space inall initial value of non-scattering solution. Unlike in the mass-criticalor -supercritical case, ground state is not a threshold. Thisis an extension of previous author's result to the casewhere the exponent of nonlinearity is below so-called Strauss number.A main new ingredient is a stability estimate in aLorenz-modified-Bezov type spacetime norm.