Instability of Vortex Solitons for 2d Focusing Nls

We study instability of a vortex soliton eφω,m(r) to iut +∆u+ |u| u = 0, for x ∈ R, t > 0, where n = 2, m ∈ N and (r, θ) are polar coordinates in R. Grillakis [11] proved that every radially standing wave solutions are unstable if p > 1+4/n. However, we do not have any examples of unstable standing wave solutions in the subcritical case (p < 1 + n/4). Suppose φω,m is nonnegative. We investigate a limiting profile of φω,m as m → ∞ and prove that for every p > 1, there exists an m∗ ∈ N such that for m ≥ m∗, a vortex soliton e φω,m(r) becomes unstable to the perturbations of the form ev(r) with 1 ≪ j ≪ m.