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

Abstract

We show a new example of blow-up behaviour for the prescribed $Q$-curvature equation in even dimension $6$ and higher, namely given a sequence $(V_k)\subset C^0({\mathbb{R}}^{2n})$ suitably converging we construct {for $n\ge 3$} a sequence $(u_k)$ of radially symmetric solutions to the equation $$(-\hat{I}”{)}^n{u}_k=V_k{e}^{2n{u}_k}\text{in }{\mathbb{R}}^{2n},$$ with $u_k$ blowing up at the origin \emph{and} on a sphere. We also prove sharp blow-up estimates. This is in sharp contrast with the $4$-dimensional case studied by F. Robert (J. Diff. Eq. 2006).