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

Xia Huang (1) , Dong Ye (2)

1 ECNU - East China Normal University [Shangaï]
2 IECL - Institut Élie Cartan de Lorraine

We study the polyharmonic problem $\Delta^m u = \pm e^u$ in ${\mathbb R}^{2m}$, with $m \geq 2$. In particular, we prove that 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 any $Q_0 >0$ and arbitrary volume $V > 0$, there exist conformal metrics $g$ on ${\mathbb R}^{2m}$ with constant $Q$-curvature equal to $Q_0$ and vol$(g) =V$. This answers some open questions in Martinazzi's work

Domaines: Equations aux dérivées partielles [math.AP]

Xia Huang, Dong Ye. Conformal metrics in ${\mathbb R}^{2m}$ with constant $Q$-curvature and arbitrary volume. Calculus of Variations and Partial Differential Equations, 2015, 54 (4), pp.3373-3384. ⟨10.1007/s00526-015-0907-1⟩. ⟨hal-01279248⟩