Asymptotic behavior of solutions to critical Hartree equations with isolated singularities

Abstract

We investigate the qualitative properties of positive classical solutions $u$ to critical Hartree equations on punctured domains, assuming that $u(x)\to +\infty$ as $x\to 0$. For the global problem set in ${\mathbb{R}}^n\setminus \{0\}$, we prove that such solutions are radial and nonincreasing in $|x|$. For the local problem in $B_R\setminus \{0\}$, we establish the upper bound $u(x)⩽C|x{|}^{-(n-2)/2}$ near the origin and the asymptotic radial symmetry estimate \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 $u$. Our approach combines integral representations, blow-up analysis, and the method of moving spheres. A main difficulty is controlling the nonlocal Hartree potential under Kelvin inversion and in the blow-up limit. We also give an alternative proof of the classification of positive entire classical solutions in ${\mathbb{R}}^n$, which is used to identify the blow-up profile.

arXiv comment35