Attainment of the sharp constant at the Hardy endpoint

Determine whether the supremum C in the fractional-remainder Hardy–Gagliardo–Nirenberg inequality is attained in H^1(\mathbb{R}^N) or in the larger energy space of the critical operator -\Delta-\frac{(N-2)^2}{4}|x|^{-2}, and determine whether the limiting level from Proposition \ref{prop:hardyendpoint} is the sharp scattering and blow-up threshold for the equation at the critical Hardy coupling.

Background

The paper proves that the sharp constants and mass-constrained thresholds remain uniformly positive as the coupling approaches the critical Hardy value from above. At the critical value, however, the natural energy space is larger than H1(\mathbb{R}N), and the paper does not establish existence of optimizers or normalized solutions there.

A positive answer to the attainment question would provide normalized solutions for the critical operator. The second question asks whether the limiting variational level actually governs the dynamics of the critical-coupling equation.

References

Is the supremum C in eq:GNsharp attained, in H1(RN) or in the larger energy space of the operator -\Delta-\frac{(N-2)2}{4}|x|{-2}? A positive answer would give normalized solutions for this critical operator. Is the limiting level in Proposition \ref{prop:hardyendpoint} the sharp threshold for the equation with a=?

eq:GNsharp:

C:=sup⁡{B(u)A(u)β/2∥u∥2σc:u∈H1(RN)∖{0}}<∞.C:=\sup\Big\{\frac{B(u)}{A(u)^{\beta/2}\|u\|_{2}^{\sigma_c}}:u\in H^1(R^N)\setminus\{0\}\Big\}<\infty .

— 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 (iii)