Classification of solutions at the mass-constrained threshold

Classify all solutions of the inhomogeneous nonlinear Schrödinger equation with inverse-square potential whose initial data satisfy E_a(u_0)=m_a(c), including the standing waves and any other threshold solutions.

Background

The paper establishes a sharp below-threshold dichotomy and shows that standing waves occur at the threshold as global, non-scattering solutions. It does not analyze or classify all solutions exactly at E_a(u_0)=m_a(c).

The authors note that the expected picture involves a one-dimensional unstable direction, with dilation perturbations leading toward scattering or blow-up. A threshold classification is linked to the unresolved nondegeneracy of ground states.

References

Classify the solutions with E_a(u_0)=m_a(c). Such a classification would require the nondegeneracy asked for in (i).

— Normalized ground states and a mass-constrained scattering threshold for the inhomogeneous NLS with an inverse-square potential  (2610.02933 - Majdoub et al., 2 Oct 2026) in Section “Complements and open problems,” subsection “Open problems,” item (vi)