Stability of standing waves for nonlinear schrödinger equations with critical power nonlinearity and potentials

We study the stability of standing waves e^iωtφ_ω(x) for a nonlinear Schrödinger equation with critical power nonlinearity |u|^4/nu and a potential V (x) in Rⁿ. Here, ω ∈ R and φ_ω(x) is a ground state of the stationary problem. Under suitable assumptions on V (x), we show that e^iωtφ_ω(x) is stable for sufficiently large ω. This result gives a different phenomenon from the case V (x) = 0 where the strong instability was proved by M. I. Weinstein [25].