Résumé en anglais

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 non-rotating case. In a second part we consider the many-particle Hamiltonian for N bosons, interacting with a potential rescaled in the mean-field manner −aNN2β−1w(Nβx),withwapositivefunctionsuchthat\int_{\mathbb{R}^2} w(x) dx = 1.Assumingthat\beta < 1/2andthata_N \to a_*sufficientlyslowly,weprovethatthemany−bodysystemisfullycondensedontheGross−PitaevskiigroundstateinthelimitN \to \infty$.