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 $(-\Delta {)}^m{u}_k=V_k{e}^{2m{u}_k}$ on open domains of $\mathbb{R}2m$. Under natural integral assumptions we show that when blow-up occurs, up to a subsequence $$\underset{k\to \infty }{\lim }{\int }_{{\Omega }_0}{V}_k{e}^{2m{u}_k}dx=L{\Lambda }_1,$$ where ${\Omega }_0\subset \subset \Omega$ 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.

18 pages

References in corpus (1)

• Concentration-compactness phenomena in the higher order Liouville's equation