Existence and asymptotic behavior of non-normal conformal metrics on $\mathbb{R}^4$ with sign-changing $Q$-curvature

Abstract

We consider the following prescribed Q-curvature problem(1)We show that for every polynomial P of degree 2 such that lim |x|→+∞ P = −∞, and for every Λ ∈ (0, Λ sph ), there exists at least one solution to problem (1) which assume the form u = w + P , where w behaves logarithmically at infinity. Conversely, we prove that all solutions to (1) have the form v + P , whereand P is a polynomial of degree at most 2 bounded from above. Moreover, if u is a solution to (1), it has the following asymptotic behavior u(x) = − Λ 8π 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 2u |dx| 2 .