On the third critical speed for rotating Bose-Einstein condensates

We study a two-dimensional rotating Bose-Einstein condensate confined by an anharmonic trap in the framework of the Gross-Pitaevskii theory. We consider a rapid rotation regime close to the transition to a giant vortex state. It was proven in Correggi et al. [J. Math. Phys. 53, 095203 (2012)] that such a transition occurs when the angular velocity is of order ε⁻⁴, with ε⁻² denoting the coefficient of the nonlinear term in the Gross-Pitaevskii functional and ε ≪ 1 (Thomas-Fermi regime). In this paper, we identify a finite value Ω_c such that if Ω = Ω₀/ε⁴ with Ω₀ > Ω_c, the condensate is in the giant vortex phase. Under the same condition, we prove a refined energy asymptotics and an estimate of the winding number of any Gross-Pitaevskii minimizer.