Quantization for the prescribed Q-curvature equation on open domains

created by martinazz on 21 Mar 2010
modified on 17 Jul 2018

[BibTeX]

Published Paper

Inserted: 21 mar 2010
Last Updated: 17 jul 2018

Journal: Commun. Contemp. Math.
Volume: 13
Pages: 533-551
Year: 2011

ArXiv: 1003.4000 PDF

Abstract:

We discuss compactness, blow-up and quantization phenomena for the prescribed $Q$-curvature equation $(-\Delta)^m u_k=V_ke^{2mu_k}$ on open domains of $\R{2m}$. Under natural integral assumptions we show that when blow-up occurs, up to a subsequence $$\lim{k\to \infty}\int{\Omega0} Vke{2muk}dx=L\Lambda1,$$ where $\Omega_0\subset\subset\Omega$ is open and contains the blow-up points, $L\in\mathbb{N}$ and $\Lambda_1:=(2m-1)!\vol(S^{2m})$ is the total $Q$-curvature of the round sphere $S^{2m}$. Moreover, under suitable assumptions, the blow-up points are isolated. We do not assume that $V$ is positive.