  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.
