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 a scaled attractive two-body interaction of the form $-aN^{2\alpha-1} U(N^\alpha(x-y))$ and a scaled repulsive three-body interaction of the form $bN^{4\beta-2} W(N^\beta(x-y,x-z))$, where $a,b,\alpha,\beta>0$ and $\int_{\mathbb R^{2}}U(x){\rm d}x = 1 = \iint_{\mathbb R^{4}} W(x,y) {\rm d}x{\rm d}y$. The system is always stable even though $b$ is small and $a$ is large. We derive rigorously the cubic-quintic nonlinear Schr\"odinger semiclassical theory as the mean-field limit of the model. The three-body interaction is assumed stronger than the two-body interaction, in the sense $\beta>\alpha$, when the latter is too negative, i.e., $a \geq a_{*}$, where $a_{*}$ is the critical strength for the focusing cubic nonlinear Schr\"odinger. The behavior of the system is investigated in the limit $a = a_{N} \to a_{*}$ and $b = b_{N} \searrow 0$. We also consider the homogeneous problem where the trapping potential is subtracted.