Nonexistence of asymptotically free solutions for a nonlinear Schrödinger equation

Let u be a nontrivial, smooth solution to iut=Δu−‖u‖ p−1u. If n=1 and 2<p≤3, then there does not exist any finite energy free solution v such that ∥u(t)−v(t)∥2→0 as t→+∞. This extends a theorem of Strauss in which the same result was proved for 1<p≤2.