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

We consider the following nonlinear Schr\"{o}dinger equation with double power nonlinearities and an inverse power potential: $$i\frac{\partial u}{\partial t}+\Delta u+|u|^{\frac{4}{N}}u+C_1|u|^{p-1}u+\frac{C_2}{|x|^{2\sigma}}u=0$$ in $\mathbb{R}^N$. From the classical argument, the solution with subcritical mass ($\left\|u_0\right\|_2<\left\|Q\right\|_2$) is global and bounded in $H^1(\mathbb{R}^N)$, where $Q$ is the ground state of the mass-critical problem. Previous results show the existence of a minimal-mass blow-up solution for the equation with $C_1>0$ and $C_2=0$ or $C_1=0$ and $C_2>0$ and investigate the behaviour of the solution near the blow-up time. Moreover, they have suggested that a subcritical power nonlinearity and an inverse power potential behave in a similar way with respect to blow-up. On the other hand, the previous results also show the nonexistence of a minimal-mass blow-up solution for the equation with $C_1<0$ and $C_2=0$ or $C_1=0$ and $C_2<0$. In this paper, we investigate the existence and behaviour of a minimal-mass blow-up solution for the equation with $C_1>0>C_2$ or $C_1<0<C_2$, that is the subcritical power nonlinearity and the inverse power potential cancel each other's effects. Furthermore, we give a lower estimate of the arbitrary finite-time blow-up solution with critical mass and show that the energies of critical-mass blow-up solutions are positive when $(C_1,C_2,p,\sigma)$ is under certain conditions.