Scattering and blow-up criteria for 3D cubic focusing nonlinear inhomogeneous NLS with a potential

In this paper, we consider the 3d cubic focusing inhomogeneous nonlinear Schr"{o}dinger equation with a potential $$i{u}_t+\mathrm{\Delta }u-Vu+\mathrm{\mid }x{\mathrm{\mid }}^{-b}\mathrm{\mid }u{\mathrm{\mid }}^{2}u=0,(t,x)\in \mathbf{R}\times {\mathbf{R}}^3,$$ where $0<b<1$. We first establish global well-posedness and scattering for the radial initial data $u_0$ in $H^1({\mathbf{R}}^3)$ satisfying $M(u_0{)}^{1-s_c}E(u_0{)}^{s_c}<\mathcal{E}$ and $\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{L^2}^{2(1-s_c)}\mathrm{\parallel }{H}^{\frac{1}{2}}{u}_0{\mathrm{\parallel }}_{L^2}^{2s_c}<\mathcal{K}$ provided that $V$ is repulsive, where $\mathcal{E}$ and $\mathcal{K}$ are the mass-energy and mass-kinetic of the ground states, respectively. Our result extends the results of Hong \cite{H} and Farah-Guzm$\acute{\mathrm{a}}$n \cite{FG1} with $b\in (0,\frac{1}{2})$ to the case $0<b<1$. We then obtain a blow-up result for initial data $u_0$ in $H^1({\mathbf{R}}^3)$ satisfying $M(u_0{)}^{1-s_c}E(u_0{)}^{s_c}<\mathcal{E}$ and $\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{L^2}^{2(1-s_c)}\mathrm{\parallel }{H}^{\frac{1}{2}}{u}_0{\mathrm{\parallel }}_{L^2}^{2s_c}>\mathcal{K}$ if $V$ satisfies some additional assumptions.0}|{L^{2}}^{2(1-s{c})}|H^{\frac{1}{2}}u_{0}|{L^{2}}^{2s{c}}>$\mathcal{K}$$if$V$ satisfies some additional assumptions.