Conformally Euclidean metrics on ℝn with arbitrary total Q-curvature

Abstract

We study the existence of solution to the problemwhere Q ≥ 0, κ ∈ (0, ∞) and n ≥ 3. Using ODE techniques Martinazzi for n = 6 and Huang-Ye for n = 4m + 2 proved the existence of solution to the above problem with Q ≡ const > 0 and for every κ ∈ (0, ∞). We extend these results in every dimension n ≥ 5, thus completely answering the problem opened by Martinazzi. Our approach also extends to the case in which Q is non-constant, and under some decay assumptions on Q we can also treat the cases n = 3 and 4.

Gluing metrics with prescribed $Q$-curvature and different asymptotic behaviour in high dimension

“…It follows from i) that re u k is monotone decreasing on (c p r k , t k ). Using that r k = o(t k ) and (11) we obtain for any L > c p and for k large…”

“…We set c p = 1 + p 2−p . For any L > c p and for r ∈ (c p r k , Lr k ), using (11), which follows from Lemma 2.5, we get (r p e u k (r) ) (r) = pr p−1 + r p u k (r) e u k (r)…”

“…Such solutions exist for every Λ > 0, thanks to [5,10,11,22]. Take u k (x) := u( x k ) − log k, which is also a solution to (77).…”

Large blow-up sets for the prescribed Q-curvature equation in the Euclidean space

“…Arguing as in [13] one can show that the operator T k has a fixed point, say v k . We set u k = v k + c v k .…”

Higher order conformally invariant equations in <inline-formula><tex-math id="M1">\begin{document}$ {\mathbb R}^3 $\end{document}</tex-math></inline-formula> with prescribed volume

“…However, if n ≥ 5 then for every V ∈ (0, ∞) there exists a radial solution to (7). See [3,10,11,15,16,19] and the references therein.…”