Calculus of Variations and Geometric Measure Theory

A. Hyder - L. Martinazzi

Conformal metrics on $R^{2m}$ with constant Q-curvature, prescribed volume and asymptotic behavior

created by martinazz on 23 Feb 2014
modified on 17 Jul 2018


Submitted Paper

Inserted: 23 feb 2014
Last Updated: 17 jul 2018

Year: 2014

ArXiv: 1401.0944 PDF


We study the solutions $u\in C^\infty(R^{2m})$ of the problem $(-\Delta)^m u= Qe^{2mu}$, where $Q=\pm (2m-1)!$, and $V :=\int_{R^{2m}}e^{2mu}dx <\infty$, particularly when $m>1$. This corresponds to finding conformal metrics $g_u:=e^{2u}
^2$ on $R^{2m}$ with constant Q-curvature $Q$ and finite volume $V$. Extending previous works of Chang-Chen, and Wei-Ye, we show that both the value $V$ and the asymptotic behavior of $u(x)$ as $
\to \infty$ can be simultaneously prescribed, under certain restrictions. When $Q=(2m-1)!$ we need to assume $V<vol(S^{2m})$, but surprisingly for $Q=-(2m-1)!$ the volume $V$ can be chosen arbitrarily.