On the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation with inverse-power potential

In this paper, we study the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation with inverse-power potential $$i{u}_t+\mathrm{\Delta }u-c\mathrm{\mid }x{\mathrm{\mid }}^{-a}u=\pm \mathrm{\mid }x{\mathrm{\mid }}^{-b}\mathrm{\mid }u{\mathrm{\mid }}^{\sigma }u,(t,x)\in \mathbb{R}\times {\mathbb{R}}^d,$$ where $d\in \mathbb{N}$, $c\in \mathbb{R}$, $a,b>0$ and $\sigma >0$. First, we establish the local well-posedness in the fractional Sobolev spaces $H^s({\mathbb{R}}^d)$ with $s\ge 0$ by using contraction mapping principle based on the Strichartz estimates in Sobolev-Lorentz spaces. Next, the global existence and blow-up of $H^1$-solution are investigated. Our results extend the known results in several directions.