Singular metrics of constant negative [formula omitted]-curvature in Euclidean spaces

Abstract

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 (−Δ)n/2u=−enuonRn∖{0},under a finite volume condition Λ≔∫Rnenudx. 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 Λ>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.