In the current paper, the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation including inverse-square potential is considered. First, some criteria of global existence and finite-time blow-up in the mass-critical and mass-supercritical settings with $0<c\le c^{\ast }$ are obtained. Then, by utilizing the potential well method and the sharp Sobolev constant, the sharp condition of blow-up is derived in the energy-critical case with $0<c<\frac{N^2+4N}{(N+2{)}^2}{c}^{\ast }$. Finally, we establish the mass concentration property of explosive solutions, as well as the dynamic behaviors of the minimal-mass blow-up solutions in the $L^2$-critical setting for $0<c<c^{\ast }$, by means of the variational characterization of the ground-state solution to the elliptic equation, scaling techniques and a suitable refined compactness lemma. Our results generalize and supplement the ones of some previous works.

This study is concerned with the sharp criterion of global existence and the orbital stability of standing waves for the Hartree equation in presence of a partial confinement. Using the scaling technique and constructing cross-invariant sets, we first derive the sharp threshold for global existence and blow-up of the solution in both $L^2$-critical and $L^2$-supercritical settings. Then, by taking advantage of the profile decomposition technique and concentration compact arguments together with variational methods, we explore the existence of normalized standing waves and show that these standing waves are orbitally stable in the $L^2$-subcritical, $L^2$-critical, and supercritical cases. Our conclusions complement and compensate some previous results.