In this paper, we study the Cauchy problem for the focusing inhomogeneous nonlinear Schr\"{ o}dinger equation with inverse-square potential \[iu_{ t} +\Delta u-c|x|^{ -2}u+|x|^{ -b} |u|^{ \sigma } u=0,\; u(0)=u_{ 0} \in H_{ c}^{ 1},\;(t,x)\in \mathbb R\times\mathbb R^{ d},\] where $d\ge3$, $0<b<2$, $\frac{ 4-2b}{ d}<\sigma<\frac{ 4-2b}{ d-2}$ and $c>-c(d):=-\left(\frac{ d-2}{ 2}\right)^{ 2}$. We first establish the criteria for global existence and blow-up of general (not necessarily radial or finite variance) solutions to the equation. Using these criteria, we study the global existence and blow-up of solutions to the equation with general data lying below, at, and above the ground state threshold. Our results extend the global existence and blow-up results of Campos-Guzm\'{ a}n (Z. Angew. Math. Phys., 2021) and Dinh-Keraani (SIAM J. Math. Anal., 2021).
Categories: math.AP