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

On a smoothly bounded domain $Ω\subset\R{2m}$ we consider a sequence of positive solutions $u_k\stackrel{w}{\rightharpoondown} 0$ in $H^m(Ω)$ to the equation $(-Δ)^m u_k=λ_k u_k e^{mu_k^2}$ subject to Dirichlet boundary conditions, where $0<λ_k\to 0$. Assuming that $Λ:=\lim_{k\to\infty}\int_Ωu_k(-Δ)^m u_k dx<\infty,$ we prove that $Λ$ is an integer multiple of $Λ_1:=(2m-1)!\vol(S^{2m})$, the total $Q$-curvature of the standard $2m$-dimensional sphere.

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