Local asymptotics for singular solutions to critical Hartree equations
For $n\geqslant3$ and $0<α<n$, we study positive classical solutions of critical Hartree equations that blow up at the origin. For the global problem on $\mathbb R^n\setminus\{0\}$, with the Hartree potential integrated over $\mathbb R^n$, we prove that solutions in $L^p(\mathbb R^n)$, where $p=(n+α)/(n-2)$, are radial and nonincreasing in $|x|$. We establish an integral representation and apply moving spheres to this nonlocal problem. For the local problem on $B_R\setminus\{0\}$, with the Hartree potential integrated over $B_R$, we prove $u(x)\leqslant C|x|^{-(n-2)/2}$ near the origin and \begin{equation*} u(x)=\bar u(|x|)\bigl(1+\mathcal O(|x|)\bigr)\qquad\text{as }x\to0, \end{equation*} where $\bar u$ is the spherical average of $u$. A main difficulty is controlling the nonlocal Hartree potential under Kelvin inversion and in the blow-up limit. The local proof combines blow-up analysis and moving spheres with estimates of the integral tails and an auxiliary Liouville lemma. We also give an alternative proof of the classification of positive $C^2$ solutions of the critical Hartree equation in $\mathbb R^n$.