On symmetry and monotonicity of singular solutions to critical fractional Hartree equations

arXiv:2611.05525v1 Announce Type: new Abstract: This paper continues the study initiated in \cite{GuJevnikarYang2026b} of fractional Hartree equations with zero capacity singular sets. We are concerned here with the critical case, i.e. \[ (-\Delta)^s u=\left(\int_{\mathbb{R}^N\setminus\Gamma} \frac{u^{2_{\mu,s}^*}(y)}{|x-y|^\mu}\,d y\right)u^{2_{\mu,s}^*-1} \quad\text{in }\mathbb{R}^N\setminus\Gamma, \] where $N\ge2$, $s\in(0,1)$, $\mu\in(0,N)$ and $2_{\mu,s}^*$ is the critical exponent associated to this class of problems. We prove symmetry and monotonicity with respect to the singular set $\Gamma$. The Kelvin transforms centered at regular points provide sufficient decay at infinity for both the transformed solutions and their associated Hartree coefficients, without imposing global integrability assumptions. Combined with a moving plane argument and a comparison estimate in critical Lebesgue spaces, this allows the moving plane method to cover the full interaction range.

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