Sharp global existence and orbital stability of standing waves for the Schrödinger-Hartree equation with partial confinement

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.