Delaunay solutions to the fractional Hartree equation with critical growth

We study positive solutions of the critical fractional Hartree equation with a non-removable isolated singularity at the origin. This equation is doubly nonlocal, involving both the fractional Laplacian and a Riesz convolution potential. We first prove that every positive singular solution is radially symmetric about the origin, by combining the Caffarelli–Silvestre extension with the method of moving spheres. We then establish the existence of Delaunay-type periodic singular solutions. After the Emden–Fowler transformation, the Hartree convolution survives as a genuinely nonlocal integral term, so that the resulting periodic equation cannot be reduced to an ordinary differential equation. We construct nonconstant periodic solutions for all sufficiently large periods by minimizing a Rayleigh-type quotient in a periodic fractional Sobolev space.