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We consider a class of $L^2$-supercritical inhomogeneous nonlinearSchr\"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 adaptingan argument of Dodson-Murphy, we first study the energy scattering below theground state for the equation with radially symmetric initial data. We thenestablish blow-up criteria for the equation whose proof is based on an argumentof Du-Wu-Zhang. In the defocusing case, we also prove the energy scattering forthe equation with radially symmetric initial data.

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