Abstract
Abstract We study the metrics of constant $Q$-curvature in the Euclidean space with a prescribed singularity at the origin, namely solutions to the equation $$\begin{equation*} (-\Delta)^{\frac{n}{2}}w=e^{nw}-c\delta_{0} \ \textrm{on}\ {\mathbb{R}}^n, \end{equation*}$$under a finite volume condition. We analyze the asymptotic behavior at infinity and the existence of solutions for every $n\ge 3$ also in a supercritical regime. Finally, we state some open problems.
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