A note on the Schrödinger equation with Coulomb potential and critical inhomogeneous source term

In this work, we study an inhomogeneous nonlinear Schrödinger equation incorporating a Coulomb-type potential. Our first main goal is to identify a sharp threshold separating global existence from finite-time blow-up for inter-critical focusing solutions below the ground state. A secondary objective is to develop a local well-posedness theory in certain Sobolev-critical regimes. The analysis is particularly challenging due to the presence of both the singular inhomogeneous term, which broke the space translation invariance and the singular Coulomb potential, which broke the scaling invariance, but also the critical Sobolev spaces. This study naturally extends and complements the results obtained in [2].