On minimal non-scattering solution for focusing mass-subcritical nonlinear Schrödinger equation

Abstract

We consider time global behavior of solutions to the focusing mass-subcritical NLS equation in weighted $L^2$ space. We prove that there exists a threshold solution such that (i) it does not scatter; (ii) with respect to a certain scale-invariant quantity, this solution attains minimum value in all non-scattering solutions. In the mass-critical case, it is known that ground states are this kind of threshold solution. However, in our case, it turns out that the above threshold solution is not a standing-wave solution.

27 pages, no figure

References in corpus (1)

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