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

Abstract

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

21 pages

References in corpus (2)

• Classification of solutions to the higher order Liouville's equation on R^{2m}
• Concentration-compactness phenomena in the higher order Liouville's equation

Cited by in corpus (6)

• Existence and asymptotics for solutions of a non-local Q-curvature equation in dimension three
• Improved Adams-type inequalities and their extremals in dimension 2m
• Quantization for an elliptic equation of order 2m with critical exponential non-linearity
• Uniform bounds for higher-order semilinear problems in conformal dimension
• Existence of solutions to a higher dimensional mean-field equation on manifolds
• Quantitative properties of the Hardy-type mean field equation