  In this paper, we consider the following equation: \[ i\frac{\partial
u}{\partial t}+\Delta u+g(x)|u|^{\frac{4}{N}}u-Wu=0. \] We construct a
critical-mass solution that blows up at a finite time and describe the
behaviour of the solution in the neighbourhood of the blow-up time.
Banica-Carles-Duyckaertz (2011) has shown the existence of a critical-mass
blow-up solution under the assumptions that $N\leq 2$, that $g$ and $W$ are
sufficiently smooth and that each derivative of these is bounded. In this
paper, we show the existence of a critical-mass blow-up solution under weaker
assumptions regarding smoothness and boundedness of $g$ and $W$. In particular,
it includes the cases where $W$ is growth at spatial infinity or not Lipschitz
continuous.
