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

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

2009-04-21

arxiv.org/abs/0904.3290v1

Comm. Partial Differential equations 35 (2010), 1-22

Mathematics

Analysis of PDEs

21 pages

Scientific paper

10.1080/03605300903296330

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.

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