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

We consider the following prescribed Q-curvature problem: {Delta(2)u = (1 - vertical bar x vertical bar(p))e(4u) on R-4, Lambda := integral(R4) (1 - vertical bar x vertical bar(p))e(4u)dx < infinity. (1) We show that for every polynomial P of degree 2 such that lim(vertical bar x vertical bar ->+infinity )P = -infinity, and for every Lambda is an element of (0, Lambda(sph)), there exists at least one solution to problem (1) which assumes the form u = w + P, where tv behaves logarithmically at. infinity. Conversely, we prove that all solutions to (1) have the form v + P, where v(x) = 1/8 pi(2) integral(R4) log (vertical bar y vertical bar/vertical bar x - y vertical bar) (1 - vertical bar y vertical bar(p))e(4u) dy 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) = - Lambda/8 pi(2) log vertical bar x vertical bar + P + o(log vertical bar x vertical bar), as vertical bar x vertical bar -> +infinity. As a consequence, we give a geometric characterization of solutions in terms of the scalar curvature at infinity of the associated conformal metric e(2u)vertical bar dx vertical bar(2).