Quantization for an elliptic equation of order 2m with critical exponential non-linearity

On a smoothly bounded domain \({\Omega\subset\mathbb{R}^{2m}}\) we consider a sequence of positive solutions \({u_k\stackrel{w}{\rightharpoondown}0}\) in H m (Ω) to the equation \({(-\Delta)^m u_k=\lambda_k u_k e^{mu_k^2}}\) subject to Dirichlet boundary conditions, where 0 < λ k → 0. Assuming that
$$0 < \Lambda:=\lim_{k\to\infty}\int\limits_\Omega u_k(-\Delta)^m u_k dx < \infty,$$
we prove that Λ is an integer multiple of Λ1 := (2m − 1)! vol(S 2m ), the total Q-curvature of the standard 2m-dimensional sphere.