*Published Paper*

**Inserted:** 21 mar 2010

**Last Updated:** 17 jul 2018

**Journal:** Commun. Contemp. Math.

**Volume:** 13

**Pages:** 533-551

**Year:** 2011

**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 $$\lim_{{k\to} \infty}\int_{{\Omega}_{0}}
V_{ke}^{{2mu}_{k}dx=L\Lambda}_{1,$$} 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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