Conformal metrics on \(\mathbb R^{2m}\) with constant \(Q\)-curvature (Q993411)

Summary: We study the conformal metrics on \(\mathbb{R}{^2m}\) with constant \(Q\)-curvature \(Q\in\mathbb{R}\) having finite volume, particularly in the case \(Q\leq 0\). We show that when \(Q1\). Moreover we study their asymptotic behavior at infinity, in analogy with the case \(Q>0\), which we treated in a recent paper. When \(Q=0\), we show that such metrics have the form \(e^{2p}g_{\mathbb{R}^{2m}}\), where \(p\) is a polynomial such that \(2\leq \deg p\leq 2m-2\) and \(\sup_{\mathbb{R}{2m}}p<+\infty\). In dimension 4, such metrics correspond to the polynomials \(p\) of degree 2 with \(\lim_{|x|\to+\infty}p(x)=-\infty\).