Quantitative properties of the Hardy-type mean field equation
Lu Chen, Bohan Wang, Chunhua Wang
TL;DR
This work analyzes the Hardy-type mean field equation $-\Delta u-\frac{1}{(1-|x|^2)^2}u=\lambda e^u$ in the unit disk with zero boundary data, focusing on radial symmetry, blow-up behavior, and uniqueness as $\lambda\to 0$. The authors develop a hyperbolic-space moving-plane framework and refine heat-kernel expansions to obtain a Brezis-Merle-type control, showing that blow-up occurs only at the origin with total mass $8\pi$ and that $u_\lambda\to 8\pi G(\cdot,0)$ away from the singular point. They also establish a precise asymptotic profile near the origin via a rescaled limit $\eta_\lambda \to \eta_0=-2\log(1+|x|^2)$ and prove uniqueness for small $\lambda$ using a local Pohozaev identity and scaling arguments. Together, these results significantly advance compactness and uniqueness theory for Hardy-type mean field equations, extending classical Brezis-Merle analysis to singular Hardy potentials.
Abstract
In this paper, we consider the following Hardy-type mean field equation \[ \left\{ {\begin{array}{*{20}{c}} { - Δu-\frac{1}{(1-|x|^2)^2} u = λe^u}, & {\rm in} \ \ B_1,\\ {\ \ \ \ u = 0,} &\ {\rm on}\ \partial B_1, \end{array}} \right. \] \[\] where $λ>0$ is small and $B_1$ is the standard unit disc of $\mathbb{R}^2$. Applying the moving plane method of hyperbolic space and the accurate expansion of heat kernel on hyperbolic space, we establish the radial symmetry and Brezis-Merle lemma for solutions of Hardy-type mean field equation. Meanwhile, we also derive the quantitative results for solutions of Hardy-type mean field equation, which improves significantly the compactness results for classical mean-field equation obtained by Brezis-Merle and Li-Shafrir. Furthermore, applying the local Pohozaev identity from scaling, blow-up analysis and a contradiction argument, we prove that the solutions are unique when $λ$ is sufficiently close to 0.