Minimal mass blow-up solutions for nonlinear Schrödinger equations with a singular potential

Abstract: 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 ($\left|u_0\right|_2=\left|Q\right|_2$).