  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.
