Global dynamics for a class of inhomogeneous nonlinear Schr\"odinger equations with potential

We consider a class of $L^2$-supercritical inhomogeneous nonlinear Schr"odinger equations with potential in three dimensions $$i{\mathrm{\partial }}_{t}u+\mathrm{\Delta }u-Vu=\pm \mathrm{\mid }x{\mathrm{\mid }}^{-b}\mathrm{\mid }u{\mathrm{\mid }}^{\alpha }u,(t,x)\in \mathbb{R}\times {\mathbb{R}}^3,$$ where $0<b<1$ and $\alpha >\frac{4-2b}{3}$. In the focusing case, by adapting an argument of Dodson-Murphy, we first study the energy scattering below the ground state for the equation with radially symmetric initial data. We then establish blow-up criteria for the equation whose proof is based on an argument of Du-Wu-Zhang. In the defocusing case, we also prove the energy scattering for the equation with radially symmetric initial data.