Existence of nonconstant solutions with small periods

Determine whether nonconstant periodic solutions of the transformed critical fractional Hartree equation, equivalently nonconstant Delaunay-type singular solutions, exist for periods smaller than the threshold \(T_{0}\) supplied by the variational construction.

Background

The paper constructs positive periodic solutions by minimizing a Rayleigh-type quotient and proves that the resulting minimizers are nonconstant only for sufficiently large periods $T\geq T_{0}$. The large-period conclusion follows from comparing the diverging energy of constant profiles with the uniformly bounded energy of a localized test function.

The existence of nonconstant periodic solutions for short periods is not settled by this comparison argument. The authors emphasize that the issue is substantially different from the classical Fowler or Delaunay ODE setting, where solutions exist for every period, because the fractional Hartree equation retains a genuinely nonlocal convolution term after the Emden–Fowler transformation.