Existence of a minimal non-scattering solution to the mass-subcritical generalized Korteweg-de Vries equation

In this article, we prove the existence of a non-scattering solution, which is minimal in some sense, to the mass-subcritical generalized Korteweg-de Vries (gKdV) equation in the scale critical Lˆr space where Lˆr={f∈S′(R)|‖f‖Lˆr=‖fˆ‖Lr′<∞}. We construct this solution by a concentration compactness argument. Then, key ingredients are a linear profile decomposition result adopted to Lˆr-framework and approximation of solutions to the gKdV equation which involves rapid linear oscillation by means of solutions to the nonlinear Schrödinger equation.