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

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.