Two minimization problems on non-scattering solutions to mass-subcritical nonlinear Schr\"odinger equation

In this paper, we introduce two minimization problems on non-scattering solutions to nonlinear Schr\"odinger equation. One gives us a sharp scattering criterion, the other is concerned with minimal size of blowup profiles. We first reformulate several previous results in terms of these two minimizations. Then, the main result of the paper is existence of minimizers to the both minimization problems for mass-subcritical nonlinear Schr\"odinger equations. To consider the latter minimization, we consider the equation in a Fourier transform of generalized Morrey space. It turns out that the minimizer to the latter problem possesses a compactness property, which is so-called almost periodicity modulo symmetry.