Removal of the lower bound on the Riesz exponent

Establish whether the lower bound \(\alpha>\alpha_{\ast}\) can be removed from the existence theorem for positive Delaunay-type solutions of the critical fractional Hartree equation when \(0<s<1/2\), so that existence holds for every \(0<\alpha<n\) and every \(s\in(0,1)\).

Background

The paper proves existence of positive, nonconstant periodic Delaunay-type solutions for the critical fractional Hartree equation only when α∈(α∗,n)\alpha\in(\alpha_{\ast},n), where α∗=α∗(n,s)\alpha_{\ast}=\alpha_{\ast}(n,s) is a lower threshold that is relevant for $0L∞L^{\infty}-regularity analysis based on a De Giorgi truncation argument.

For small fractional parameters ss, the integrability requirements imposed on the periodic Hartree kernel prevent the current De Giorgi iteration from covering all 0<α<n0<\alpha<n. The unresolved issue is therefore whether a refined regularity argument can eliminate the lower restriction on α\alpha and extend the Delaunay-solution existence theorem to the full conformally invariant parameter range.

References

A natural question is whether the lower bound $\alpha > \alpha_{\ast}$ appearing in Theorem~\ref{thm:main} for $0 < s < 1/2$ can be removed entirely, so that the existence result holds for all $0 < \alpha < n$ and all $s \in (0,1)$. In this paper, we introduce a De Giorgi truncation approach to the $L\infty$ regularity estimate for Hartree-type equations. This approach is new in the Hartree setting and already enlarges the admissible parameter range beyond what the classical Moser iteration method yields. However, the integrability condition imposed on the periodic kernel in the De Giorgi iteration still requires a positive lower bound on $\alpha$ when $s$ is small. It remains open whether a refined argument can eliminate this restriction altogether.

— Delaunay solutions to the fractional Hartree equation with critical growth  (2608.12734 - Andrade et al., 13 Aug 2026) in Section 1, subsection “Open problems”