Conformal metrics on R-2m with constant Q-curvature

We study the conformal metrics on R2m with constant Q-curvature Q ∈ R having finite volume, particularly in the case Q ≤ 0. We show that when Q < 0 such metrics exist in R2m if and only if m > 1. Moreover we study their asymptotic behavior at infinity, in analogy with the case Q > 0, which we treated in a recent paper. When Q = 0, we show that such metrics have the form e2gR2m, where p is a polynomial such that 2 ≤ degp ≤ 2m−2 and supR2m p < +∞. In dimension 4, such metrics are exactly the polynomials p of degree 2 with lim|x|→+∞ p(x) = −∞.