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 $$i\partial_t u + \Delta u - V u = \pm |x|^{-b} |u|^\alpha u, \quad (t,x) \in \mathbb{R} \times \mathbb{R}^3,$$ 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.