Blow-up profile of rotating 2D focusing bose gases

We consider the Gross–Pitaevskii equation describing an attractive Bose gas trapped to a quasi 2D layer by means of a purely harmonic potential, and which rotates at a fixed speed of rotation Ω. First, we study the behavior of the ground state when the coupling constant approaches a*, the critical strength of the cubic nonlinearity for the focusing nonlinear Schrödinger equation. We prove that blow-up always happens at the center of the trap, with the blow-up profile given by the Gagliardo–Nirenberg solution. In particular, the blow-up scenario is independent of Ω, to leading order. This generalizes results obtained by Guo and Seiringer (Lett. Math. Phys., 2014, vol. 104, p. 141–156) in the nonrotating case. In a second part, we consider the many-particle Hamiltonian for N bosons, interacting with a potential rescaled in the mean-field manner (Formula Presented), a positive function such that (Formula Presented). Assuming that β>1/2 and that a_N→a* sufficiently slowly, we prove that the many-body system is fully condensed on the Gross–Pitaevskii ground state in the limit N→∞.