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

Abstract

We consider the following prescribed [Formula: see text]-curvature problem: [Formula: see text] We show that for every polynomial [Formula: see text] of degree 2 such that [Formula: see text], and for every [Formula: see text], there exists at least one solution to problem (1) which assumes the form [Formula: see text], where [Formula: see text] behaves logarithmically at infinity. Conversely, we prove that all solutions to (1) have the form [Formula: see text], where [Formula: see text] and [Formula: see text] is a polynomial of degree at most two bounded from above. Moreover, if [Formula: see text] is a solution to (1), it has the following asymptotic behavior: [Formula: see text] As a consequence, we give a geometric characterization of solutions in terms of the scalar curvature at infinity of the associated conformal metric [Formula: see text].