Existence and asymptotic behavior of non-normal conformal metrics on ℝ4 with sign-changing Q-curvature.

We consider the following prescribed Q -curvature problem: Δ 2 u = (1 − | x | p) e 4 u on  ℝ 4 , Λ : = ∫ ℝ 4 (1 − | x | p) e 4 u d x  <  ∞. ⁽¹⁾ We show that for every polynomial P of degree 2 such that lim | x | → + ∞ P = − ∞ , and for every Λ ∈ (0 , Λ s p h) , there exists at least one solution to problem  (1) which assumes the form u = w + P , where w behaves logarithmically at infinity. Conversely, we prove that all solutions to  (1) have the form v + P , where v (x) = 1 8 π 2 ∫ ℝ 4 log | y | | x − y | (1 − | y | p) e 4 u d y and P is a polynomial of degree at most two bounded from above. Moreover, if u is a solution to  (1), it has the following asymptotic behavior: u (x) = − Λ 8 π 2 log | x | + P + o (log | x |) , as  | x | → + ∞. As a consequence, we give a geometric characterization of solutions in terms of the scalar curvature at infinity of the associated conformal metric e 2 u | d x | 2 . [ABSTRACT FROM AUTHOR]