Singular metrics of constant negative Q-curvature in Euclidean spaces

We study singular metrics of constant negative Q-curvature in the Euclidean space Rn for every n >= 1. Precisely, we consider solutions to the problem (-triangle)(n/2)u = -e(nu) on R\{0}, under a finite volume condition Lambda := integral 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 >= 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.(c) 2023 Elsevier Ltd. All rights reserved.