*Accepted Paper*

**Inserted:** 5 mar 2010

**Last Updated:** 17 jul 2018

**Journal:** Math. Z.

**Year:** 2010

**Abstract:**

On a smoothly bounded domain $\Omega\subset\R{2m}$ we consider a sequence of
positive solutions $u_k\stackrel{w}{\rightharpoondown} 0$ in $H^m(\Omega)$ to
the equation $(-\Delta)^m u_k=\lambda_k u_k e^{mu_k^2}$ subject to Dirichlet
boundary conditions, where $0<\lambda_k\to 0$. Assuming that
$$\Lambda:=\lim_{{k\to\infty}\int}_{\Omega} u_{k}(-\Delta)^{m} u_{k} dx<\infty,$$ we
prove that $\Lambda$ is an integer multiple of
$\Lambda_1:=(2m-1)!\vol(S^{2m})$, the total $Q$-curvature of the standard
$2m$-dimensional sphere.

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