LAUSR

Abstract Answering a question by M. Struwe [26] related to the blow-up behavior in the Nirenberg problem, we show that the prescribed Q-curvature equation Δ 2 u = ( 1 − | x | p ) e 4 u  in  R 4 , Λ : = ∫ R 4 ( 1 − | x | p ) e 4 u d x ∞ has normal solutions (namely solutions which can be written in integral form, and hence satisfy Δ u ( x ) = O ( | x | − 2 ) as | x | → ∞ ) if and only if p ∈ ( 0 , 4 ) and ( 1 + p 4 ) 8 π 2 ≤ Λ 16 π 2 . We also prove existence and non-existence results for the positive curvature case, namely for Δ 2 u = ( 1 + | x | p ) e 4 u in R 4 , and discuss some open questions.