  We consider the following nonlinear Schr\"{o}dinger equation with an inverse
potential: \[ i\frac{\partial u}{\partial t}+\Delta
u+|u|^{\frac{4}{N}}u\pm\frac{1}{|x|^{2\sigma}}\log|x|u=0 \] in $\mathbb{R}^N$.
From the classical argument, the solution with subcritical mass
($\|u\|_2<\|Q\|_2$) is global and bounded in $H^1(\mathbb{R}^N)$. Here, $Q$ is
the ground state of the mass-critical problem. Therefore, we are interested in
the existence and behaviour of blow-up solutions for the threshold
($\left\|u_0\right\|_2=\left\|Q\right\|_2$).
