Asymptotics and quantization for a mean-field equation of higher order

TL;DR

This paper investigates the asymptotic behavior and quantization phenomena of solutions to a higher-order mean field equation in bounded domains, highlighting connections with prescribing Q-curvature in differential geometry.

Contribution

It provides a detailed analysis of the limiting behavior of solutions to the higher-order mean field equation, extending understanding of quantization and asymptotics in geometric PDEs.

Findings

Describes the limiting behavior of solutions as parameters vary.

Establishes quantization properties of solutions.

Connects the problem to Q-curvature prescription.

Abstract

Given a regular bounded domain $\Omega\subset\R{2m}$, we describe the limiting behavior of sequences of solutions to the mean field equation of order $2m$, $m\geq 1$, $$(-\Delta)^m u=\rho \frac{e^{2mu}}{\int_\Omega e^{2mu}dx}\quad\text{in}\Omega,$$ under the Dirichlet boundary condition and the bound $0<\rho\leq C$. We emphasize the connection with the problem of prescribing the $Q$-curvature.