Conformally Euclidean metrics on ${\mathbb{R}}^n$ with arbitrary total $Q$-curvature

Abstract

We study the existence of solution to the problem $$(-\Delta {)}^{\frac{n}{2}}u=Q{e}^{nu}\text{in }{\mathbb{R}}^n,\kappa :={\int }_{{\mathbb{R}}^n}Q{e}^{nu}dx<\infty ,$$ where $Q\ge 0$, $\kappa \in (0,\infty )$ and $n\ge 3$. Using ODE techniques Martinazzi for $n=6$ and Huang-Ye for $n=4m+2$ proved the existence of solution to the above problem with $Q\equiv const>0$ and for every $\kappa \in (0,\infty )$. We extend these results in every dimension $n\ge 5$, 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 $4$.

References in corpus (3)

• Concentration-compactness phenomena in the higher order Liouville's equation
• Blow-up analysis of a nonlocal Liouville-type equation
• Conformal metrics in ${\mathbb{R}}^{2m}$ with constant $Q$-curvature and arbitrary volume

Cited by in corpus (3)

• The total Q-curvature, volume entropy and polynomial growth polyharmonic functions
• Existence and asymptotic behavior of non-normal conformal metrics on ${\mathbb{R}}^4$ with sign-changing $Q$-curvature
• Solutions of Liouville equations with non-trivial profile in dimensions 2 and 4