Quantization for the prescribed Q-curvature equation on open domains

Abstract

We discuss compactness, blow-up and quantization phenomena for the prescribed $Q$-curvature equation $(-Δ)^m u_k=V_ke^{2mu_k}$ on open domains of $\R{2m}$. Under natural integral assumptions we show that when blow-up occurs, up to a subsequence $$\lim_{k\to \infty}\int_{Ω_0} V_ke^{2mu_k}dx=LΛ_1,$$ where $Ω_0\subset\subsetΩ$ is open and contains the blow-up points, $L\in\mathbb{N}$ and $Λ_1:=(2m-1)!\vol(S^{2m})$ is the total $Q$-curvature of the round sphere $S^{2m}$. Moreover, under suitable assumptions, the blow-up points are isolated. We do not assume that $V$ is positive.