  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 $-aN^{2\alpha-1}
U(N^\alpha \cdot)$ and $bN^{4\beta-2} W(N^\beta \cdot, N^\beta \cdot))$, where
$a,b,\alpha,\beta>0$ and $\int_{\mathbb R^2}U(x) {\mathop{}\mathrm{d}} x = 1 =
\iint_{\mathbb R^{4}} W(x,y) {\mathop{}\mathrm{d}} x {\mathop{}\mathrm{d}} y$.
We derive rigorously the cubic--quintic nonlinear Schr\"odinger semiclassical
theory as the mean-field limit of the model and we investigate the behavior of
the system in the double-limit $a = a_N \to a_*$ and $b = b_N \searrow 0$.
Moreover, we also consider the homogeneous problem where the trapping potential
is removed.
