The Cauchy problem for the energy-critical inhomogeneous nonlinear Schr\"{o}dinger equation with inverse-square potential

In this paper, we study the Cauchy problem for the energy-critical inhomogeneous nonlinear Schr"{o}dinger equation with inverse-square potential $$i{u}_t+\mathrm{\Delta }u-c\mathrm{\mid }x{\mathrm{\mid }}^{-2}u=\lambda \mathrm{\mid }x{\mathrm{\mid }}^{-b}\mathrm{\mid }u{\mathrm{\mid }}^{\sigma }u,u(0)=u_0\in H^1,(t,x)\in \mathbb{R}\times {\mathbb{R}}^d,$$ where $d\ge 3$, $\lambda =\pm 1$, $0<b<2$, $\sigma =\frac{4-2b}{d-2}$ and $c>-c(d):=-{\left(\frac{d-2}{2}\right)}^2$. We first prove the local well-posedness as well as small data global well-posedness and scattering in $H^1$ for $c>-\frac{(d+2-2b{)}^2-4}{(d+2-2b{)}^2}c(d)$ and $0<b<\frac{4}{d}$, by using the contraction mapping principle based on the Strichartz estimates. Based on the local well-posedness result, we then establish the blowup criteria for solutions to the equation in the focusing case $\lambda =-1$. To this end, we derive the sharp Hardy-Sobolev inequality and virial estimates related to this equation.