Conformal metrics in ${\mathbb{R}}^{2m}$ with constant $Q$-curvature and arbitrary volume

Abstract

We study the polyharmonic problem ${\Delta }^{m}u=\pm e^u$ in ${\mathbb{R}}^{2m}$, with $m\ge 2$. In particular, we prove that {\sl for any} $V>0$, there exist radial solutions of ${\Delta }^{m}u=-e^u$ such that $${\int }_{{\mathbb{R}}^{2m}}{e}^{u}dx=V.$$ It implies that for $m$ odd, given arbitrary volume $V>0$, there exist conformal metrics $g$ on ${\mathbb{R}}^{2m}$ with positive constant $Q$-curvature and vol$(g)=V$. This answers some open questions in Martinazzi's work.

References in corpus (1)

• Conformal metrics on ${\mathbb{R}}^{2m}$ with constant Q-curvature

Cited by in corpus (1)

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