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 e2pg\R2m, where p is a polynomial such that 2≤degp≤2m−2 and sup\R2mp<+∞. In dimension 4, such metrics correspond to the polynomials p of degree 2 with lim|x|→+∞p(x)=−∞.