Divergence of Infinite-variance Nonradial Solutions to the 3d Nls Equation

We consider solutions u(t) to the 3d focusing NLS equation i∂tu+∆u+ |u|2u = 0 such that ‖xu(t)‖L2 = ∞ and u(t) is nonradial. Denoting by M [u] and E[u], the mass and energy, respectively, of a solution u, and byQ(x) the ground state solution to −Q+∆Q+ |Q|2Q = 0, we prove the following: if M [u]E[u] < M [Q]E[Q] and ‖u0‖L2‖∇u0‖L2 > ‖Q‖L2‖∇Q‖L2 , then either u(t) blows-up in finite positive time or u(t) exists globally for all positive time and there exists a sequence of times tn → +∞ such that ‖∇u(tn)‖L2 →∞. Similar statements hold for negative time.