Mass concentration and uniqueness of ground states for mass subcritical rotational nonlinear Schrödinger equations

Abstract

This paper considers ground states of mass subcritical rotational nonlinear Schrödinger equationwhere V (x) is an external potential, Ω > 0 characterizes the rotational velocity of the trap V (x), 1 < p < 3 and ρ > 0 describes the strength of the attractive interactions. It is shown that ground states of the above equation can be described equivalently by minimizers of the L 2 − constrained variational problem. We prove that minimizers exist for any ρ ∈ (0, ∞) when 0 < Ω < Ω * , where 0 < Ω * := Ω * (V ) < ∞ denotes the critical rotational velocity of V (x). While Ω > Ω * , there admits no minimizers for any ρ ∈ (0, ∞). For fixed 0 < Ω < Ω * , by using energy estimates and blow-up analysis, we also analyze the limit behavior of minimizers as ρ → ∞. Finally, we prove that up to a constant phase, there exists a unique minimizer when ρ > 0 is large enough and Ω ∈ (0, Ω * ) is fixed.