Classification of solutions to the higher order Liouville's equation on R2m

We classify the solutions to the equation (-Delta)(m)u = (2m - 1)!e(2mu) on R-2m giving rise to a metric g = e(2u)g(R2m) with finite total Q-curvature in terms of analytic and geometric properties. The analytic conditions involve the growth rate of u and the asymptotic behaviour of Delta u at infinity. As a consequence we give a geometric characterization in terms of the scalar curvature of the metric e(2u)g(R2m) at infinity, and we observe that the pull-back of this metric to S-2m via the stereographic projection can be extended to a smooth Riemannian metric if and only if it is round.