Existence and asymptotics for solutions of a non-local Q-curvature equation in dimension three

We study conformal metrics on R-3, i.e., metrics of the form g(u) = e(2u)vertical bar dx vertical bar(2), which have constant Q-curvature and finite volume. This is equivalent to studying the non-local equation (-Delta)(3/2) u = 2e(3u) in R-3, V := integral(R3) e(3u) dx < infinity, where V is the volume of g(u). Adapting a technique of A. Chang and W-X. Chen to the non-local framework, we show the existence of a large class of such metrics, particularly for V <= 2 pi(2) = vertical bar S-3 vertical bar. Inspired by previous works of C-S. Lin and L. Martinazzi, who treated the analogue cases in even dimensions, we classify such metrics based on their behavior at infinity.Swiss National Science Foundation; First Class Postdoctoral Science Foundation of China [2012M520002]SCI(E)ARTICLEtj@math.uchicago.edu; ali.maalaoui@unibas.ch; luca.martinazzi@unibas.ch; jxiong@math.pku.edu.cn3-4469-4885