Quantization for the prescribed Q-curvature equation on open domains

We discuss compactness, blow-up and quantization phenomena for the prescribed Q-curvature equation (−Δ)muk=Vke2muk on open domains of $\R{2m}$. Under natural integral assumptions we show that when blow-up occurs, up to a subsequence
limk→∞∫Ω0Vke2mukdx=LΛ1,
where Ω0⊂⊂Ω is open and contains the blow-up points, L∈N and $\Lambda_1:=(2m-1)!\vol(S^{2m})$ is the total Q-curvature of the round sphere S2m. Moreover, under suitable assumptions, the blow-up points are isolated. We do not assume that V is positive.