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arXiv · 2608.12734

Delaunay solutions to the fractional Hartree equation with critical growth

João Henrique Andrade·Tao Feng·Paolo Piccione·Minbo Yang
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Abstract

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.

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João Henrique Andrade, Tao Feng, Paolo Piccione, Minbo Yang. 2026-08-13. Delaunay solutions to the fractional Hartree equation with critical growth. https://arxiv.org/abs/2608.12734

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