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

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