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

We consider the following nonlinear Schr\"{o}dinger equation with a potential in $\mathbb{R}^N$. We studied the existence of an initial value with critical mass for which the corresponding solution blows up. A previous study demonstrated the existence of an initial value for which the corresponding solution blows up when $N=1$ or $2$. In this work, without any restrictions on the number of dimensions $N$, we construct a critical-mass initial value for which the corresponding solution blows up in finite time and derive its blow-up rate.