Minimal mass blow-up solutions for nonlinear Schr\"{o}dinger equations with a singular potential

We consider the following nonlinear Schr"{o}dinger equation with an inverse potential: $$i\frac{\mathrm{\partial }u}{\mathrm{\partial }t}+\mathrm{\Delta }u+\mathrm{\mid }u{\mathrm{\mid }}^{\frac{4}{N}}u\pm \frac{1}{\mathrm{\mid }x{\mathrm{\mid }}^{2\sigma }}\log \mathrm{\mid }x\mathrm{\mid }u=0$$ in ${\mathbb{R}}^N$. From the classical argument, the solution with subcritical mass ($\mathrm{\parallel }u{\mathrm{\parallel }}_2<\mathrm{\parallel }Q{\mathrm{\parallel }}_2$) 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 (${\parallel u_0\parallel }_2={\parallel Q\parallel }_2$).