A SHARP SCATTERING CONDITION FOR FOCUSING MASS-SUBCRITICAL NONLINEAR SCHRODINGER EQUATION

This article is concerned with time global behavior of solutions to focusing mass-subcritical nonlinear Schrodinger equation of power type with data in a critical homogeneous weighted L-2 space. We give a sharp sufficient condition for scattering by proving existence of a threshold solution which does not scatter at least for one time direction and of which initial data attains minimum value of a norm of the weighted L-2 space in all initial value of non-scattering solution. Unlike in the mass-critical or -supercritical case, ground state is not a threshold. This is an extension of previous author's result to the case where the exponent of nonlinearity is below so-called Strauss number. A main new ingredient is a stability estimate in a Lorenz-modified-Bezov type spacetime norm.