Quantization for the prescribed Q-curvature equation on open domains

Details Quantization for the prescribed Q-curvature equation on open domains Quantization for the prescribed Q-curvature equation on open domains

2010-03-21

arxiv.org/abs/1003.4000v1

Contemp. Math. 13 (2011), 533-551

Mathematics

Analysis of PDEs

18 pages

Scientific paper

10.1142/S0219199711004373

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 $$\lim_{k\to \infty}\int_{\Omega_0} V_ke^{2mu_k}dx=L\Lambda_1,$$ 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.

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