Quantitative properties of the Hardy-type mean field equation

TL;DR

This paper investigates the Hardy-type mean field equation in the unit disc, establishing symmetry, quantitative properties, and uniqueness of solutions for small parameters, thus advancing understanding of mean field equations with Hardy potentials.

Contribution

It introduces new symmetry and quantitative analysis techniques for Hardy-type mean field equations, improving upon classical results and proving solution uniqueness near zero parameter.

Findings

Solutions are radially symmetric for small bb;

Quantitative bounds on solutions are established;

Solutions are unique when bb is close to zero.

Abstract

In this paper, we consider the following Hardy-type mean field equation \[ \left\{ {\begin{array}{*{20}{c}} { - \Delta u-\frac{1}{(1-|x|^2)^2} u = \lambda e^u}, & {\rm in} \ \ B_1,\\ {\ \ \ \ u = 0,} &\ {\rm on}\ \partial B_1, \end{array}} \right. \] \[\] where $\lambda>0$ 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 $\lambda$ is sufficiently close to 0.