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 respectively of the form $-a{N}^{2\alpha -1}U(N^{\alpha }\cdot )$ and $b{N}^{4\beta -2}W(N^{\beta }\cdot ,N^{\beta }\cdot ))$, where $a,b,\alpha ,\beta >0$ and ${\int }_{{\mathbb{R}}^2}U(x)\mathrm{d}x=1={\iint }_{{\mathbb{R}}^4}W(x,y)\mathrm{d}x\mathrm{d}y$. 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 $a=a_N\to a_{\ast }$ and $b=b_N\searrow 0$. Moreover, we also consider the homogeneous problem where the trapping potential is removed.