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

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. \[ (-Δ)^s u=\left(\int_{\mathbb{R}^N\setminusΓ} \frac{u^{2_{μ,s}^*}(y)}{|x-y|^μ}\,d y\right)u^{2_{μ,s}^*-1} \quad\text{in }\mathbb{R}^N\setminusΓ, \] where $N\ge2$, $s\in(0,1)$, $μ\in(0,N)$ and $2_{μ,s}^*$ is the critical exponent associated to this class of problems. We prove symmetry and monotonicity with respect to the singular set $Γ$. 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.