Uniqueness and nondegeneracy of ground states

Determine whether, for nonzero inverse-square potential coupling a and inhomogeneity parameter 0<b<2, the ground state of the stationary equation LQ+Q=|x|^{-b}|Q|^{\alpha}Q is unique up to phase, and, for a positive ground state Q, prove or disprove that the operator L+1-(\alpha+1)|x|^{-b}Q^{\alpha} has trivial kernel in H^1_a(\mathbb{R}^N).

Background

The paper establishes that normalized minimizers, sharp Gagliardo–Nirenberg optimizers on the Pohozaev manifold, and least-action ground states coincide. However, it does not establish uniqueness or nondegeneracy when both the inverse-square potential and the inhomogeneous weight are present. These properties are important for understanding the structure of the ground-state manifold and for classifying threshold dynamics.

The requested uniqueness is up to multiplication by a constant phase. Nondegeneracy refers specifically to the triviality of the kernel of the linearized operator L+1-(\alpha+1)|x|{-b}Q{\alpha} on H1_a(\mathbb{R}N).

References

When a\ne0 and b\ne0, we are not aware of any uniqueness result. By Theorem \ref{thm:level}, the ground states at a given frequency are exactly the Gagliardo--Nirenberg optimizers of the corresponding mass lying on V(c). Uniqueness of one is therefore equivalent to uniqueness of the other. The question is the following. For a\ne0 and 0<b<2, is the ground state of eq:ellipticQ unique up to phase? If Q is a positive ground state, is it nondegenerate, in the sense that the operator L+1-(\alpha+1)|x|{-b}Q{\alpha} has a trivial kernel in H1_a(RN)?

eq:ellipticQ:

LQ+Q=∣x∣−b∣Q∣αQ,L Q+Q=|x|^{-b}|Q|^{\alpha}Q ,

— 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 (i)

For a>0, are all optimizers of eq:GN radial, up to phase? Or can symmetry breaking occur for large a?

eq:GN:

$B(u)\le C\,A(u)^{\beta/2}\|u\|_{2}^{\sigma_c}, \qquad u\inH^1_a(\mathbb{R}^N), $

— 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 (ii)