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

We consider the following nonlinear Schr\"{o}dinger equation with an inverse potential: $$i\frac{\partial u}{\partial t}+\Delta u+|u|^{\frac{4}{N}}u\pm\frac{1}{|x|^{2\sigma}}u=0$$ 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 ($\left\|u_0\right\|_2=\left\|Q\right\|_2$). Previous studies investigate the existence and behaviour of the critical-mass blow-up solution when the potential is smooth or unbounded but algebraically tractable. There exist no results when classical methods can not be used, such as the inverse power type potential. However, we construct a critical-mass initial value for which the corresponding solution blows up in finite time. Moreover, we show that the corresponding blow-up solution converges to a certain blow-up profile in virial space.