Threshold solutions for the $3d$ cubic INLS: the energy-subcritical case

We revisit the work [L. Campos and J. Murphy, SIAM J. Math. Anal., 55 (2023), pp. 3807–3843], which classified the dynamics of $H^1$ solutions at the ground state threshold for cubic inhomogeneous nonlinear Schrödinger equations of the form $i{\mathrm{\partial }}_{t}u+\mathrm{\Delta }u+\mathrm{\mid }x{\mathrm{\mid }}^{-b}\mathrm{\mid }u{\mathrm{\mid }}^{2}u=0$ in the range $b\in (0,\frac{1}{2})$. By modifying the modulation analysis and using Strichartz estimates in place of pointwise bounds, we extend the result to the full energy-subcritical range $b\in (0,1)$. This strategy is expected to carry over to other dispersive equations with singular potentials.