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We consider the following nonlinear Schr\"{o}dinger equation with doublepower nonlinearities and an inverse power potential: \[ i\frac{\partialu}{\partial t}+\Deltau+|u|^{\frac{4}{N}}u+C_1|u|^{p-1}u+\frac{C_2}{|x|^{2\sigma}}u=0 \] 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-criticalproblem. Previous results show the existence of a minimal-mass blow-up solutionfor the equation with $C_1>0$ and $C_2=0$ or $C_1=0$ and $C_2>0$ andinvestigate the behaviour of the solution near the blow-up time. Moreover, theyhave suggested that a subcritical power nonlinearity and an inverse powerpotential 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-upsolution for the equation with $C_1<0$ and $C_2=0$ or $C_1=0$ and $C_2<0$. Inthis paper, we investigate the existence and behaviour of a minimal-massblow-up solution for the equation with $C_1>0>C_2$ or $C_1<0<C_2$, that is thesubcritical power nonlinearity and the inverse power potential cancel eachother's effects. Furthermore, we give a lower estimate of the arbitraryfinite-time blow-up solution with critical mass and show that the energies ofcritical-mass blow-up solutions are positive when $(C_1,C_2,p,\sigma)$ is undercertain conditions.

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