Stabilization against collapse of 2D attractive Bose–Einstein condensates with repulsive, three-body interactions

We consider a trapped Bose gas of N identical bosons in two-dimensional space with both an attractive, two-body, scaled interaction and a repulsive, three-body, scaled interaction of the form \(-aN^{2\alpha -1} U(N^\alpha \cdot )\) and \(bN^{4\beta -2} W(N^\beta \cdot , N^\beta \cdot ))\), respectively, where \(a,b,\alpha ,\beta >0\) and \(\int _{\mathbb R^2}U(x) {\text {d}} x = 1 = \iint _{\mathbb R^{4}} W(x,y) {\text {d}} x {\text {d}} y\). We derive rigorously the cubic–quintic nonlinear Schrödinger semiclassical theory as the mean-field limit of the model and we investigate the behavior of the system in the double-limit \(a = a_N \rightarrow a_*\) and \(b = b_N \searrow 0\). Moreover, we also consider the homogeneous problem where the trapping potential is removed.