Quantization for the prescribed Q-curvature equation on open domains

TL;DR

This paper investigates the behavior of solutions to the prescribed Q-curvature equation on open domains, revealing quantization phenomena and conditions for blow-up, with implications for understanding geometric PDEs in higher dimensions.

Contribution

It establishes quantization and blow-up analysis for the prescribed Q-curvature equation without positivity assumptions on V, extending previous results to more general settings.

Findings

Blow-up solutions exhibit quantized total Q-curvature equal to integer multiples of a fundamental constant.

Under certain conditions, blow-up points are isolated.

The total Q-curvature concentrates at blow-up points, matching quantized values.

Abstract

We discuss compactness, blow-up and quantization phenomena for the prescribed $Q$-curvature equation $(-\Delta)^m u_k=V_ke^{2mu_k}$ on open domains of $\R{2m}$. Under natural integral assumptions we show that when blow-up occurs, up to a subsequence $$\lim_{k\to \infty}\int_{\Omega_0} V_ke^{2mu_k}dx=L\Lambda_1,$$ where $\Omega_0\subset\subset\Omega$ is open and contains the blow-up points, $L\in\mathbb{N}$ and $\Lambda_1:=(2m-1)!\vol(S^{2m})$ 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.