Global dynamics for a class of inhomogeneous nonlinear Schrödinger equations with potential

We consider a class of $L^2$-supercritical inhomogeneous nonlinear Schrödinger equations with potential in three dimensions $i\partial_t u + Δu - V u = \pm |x|^{-b} |u|^αu, \quad (t,x) \in \mathbb{R} \times \mathbb{R}^3,$ where $0<b<1$ and $α>\frac{4-2b}{3}$. 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.

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