Global existence for the defocusing Sobolev critical Schrödinger equation under the finite variance condition of initial data

Abstract

In unbounded domain RN,N≥1, we consider the defocusing nonlinear Schrödinger equation with power-type nonlinearity containing −|u|4N−2su. For 2s∗=2N−2s, with s<N2 and s≤1, the corresponding scaling invariant space is homogeneous Sobolev Ḣxs(RN) and in this case implies that the critical regularity is sc=N2−12s∗≡s. Under the finite variance condition of initial data and the solutions of the problem satisfy the pseudo-conformal conservation law, we investigate the global existence of the solutions in Ltq−Lxp spaces for some constants p,q≥2 depending on N,s. The new results of this study encompass the existing results on mass critical (sc=0) in Dodson (2012) and energy critical (sc=1) in Killip and Visan (2010).