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

We consider the following nonlinear Schr\"{o}dinger equation with double power nonlinearlities and an inverse power potential: \[ i\frac{\partial u}{\partial t}+\Delta u+|u|^{\frac{4}{N}}u\pm C_1|u|^{p-1}u\pm\frac{C_2}{|x|^{2\sigma}}u=0 \] 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 $(+C_1,0)$ or $(0,+C_2)$ and investigate the behaviour of the solution near the blow-up time. Moreover, it has suggested that a subcritical power nonlinearity and an inverse power potential behave in a similar way with respect to blow-up. In this paper, we investigate the existence and behaviour of a minimal-mass blow-up solution for the equation with $(+C_1,-C_2)$ or $(-C_1,+C_2)$, that is the subcritical power nonlinearity and the inverse power potential cancel each other's effects.