On the Cauchy problem for the inhomogeneous nonlinear Schr\"odinger equation with inverse-power potential

In this paper, we study the Cauchy problem for the inhomogeneous nonlinear Schr\"{o}dinger equation with inverse-power potential \[iuₜ +Δ u-c|x|⁻ᵃu=± |x|⁻ᵇ |u|σ u,\;\;(t,x)∈ R× Rᵈ,\] where d∈ N, c∈ R, $a,b>0$ and σ>0. First, we establish the local well-posedness in the fractional Sobolev spaces Hˢ( Rᵈ) with s≥ 0 by using contraction mapping principle based on the Strichartz estimates in Sobolev-Lorentz spaces. Next, the global existence and blow-up of H¹-solution are investigated. Our results extend the known results in several directions.