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-Pitaevksii theory. We consider a rapid rotation regime close to the transition to a giant vortex state. It was proven in [M. Correggi {\it et al}, {\it J. Math. Phys. \textbf{53}(2012)] that such a transition occurs when the angular velocity is of order $ \varepsilon ^{-4}$, with $ \varepsilon ^{-2} $ denoting the coefficient of the nonlinear term in the Gross-Pitaevskii functional and $ \varepsilon \ll 1 $ (Thomas-Fermi regime). In this paper we identify a finite value $ \Omega_{\mathrm{c}} $ such that, if $ \Omega = \Omega_0/\varepsilon ^4 $ with $ \Omega_0 > \Omega_{\mathrm{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.