Singular metrics of constant negative $Q$-curvature in Euclidean spaces

Abstract

We study singular metrics of constant negative $Q$-curvature in the Euclidean space $\mathbb{R}^n$ for every $n \geq 1$. Precisely, we consider solutions to the problem \[ (-\Delta)^{n/2}u=-e^{nu}\quad \text{on}\quad\mathbb{R}^{n}\backslash \{0\}, \] under a finite volume condition $\Lambda:=\int_{\mathbb{R}^n}e^{nu}dx$. We classify all singular solutions of the above equation based on their behavior at infinity and zero. As a consequence of this, when $n=1,2$, we show that there is actually no singular solution. Then adapting a variational technique, we obtain that for any $n\geq 3$ and $\Lambda>0$, the equation admits solutions with prescribed asymptotic behavior. These solutions correspond to metrics of constant negative $Q$-curvature, which are either smooth or have a singularity at the origin of logarithmic or polynomial type. The present paper complements previous works on the case of positive $Q$-curvature, and also sharpens previous results in the nonsingular negative $Q$-curvature case.