Gluing metrics with prescribed $Q$-curvature and different asymptotic behaviour in dimension $6$

Ali Hyder, Luca Martinazzi

Published 2018-04-24Version 1

We show a new example of blow-up behaviour for the prescribed $Q$-curvature equation in dimension $6$, namely given a sequence $(V_k)\subset C^0(\mathbb{R}^6)$ suitably converging we construct a sequence $(u_k)$ of radially symmetric solutions to the equation $$(-\Delta)^3 u_k=V_k e^{6 u_k} \quad \text{in }\mathbb{R}^6,$$ with $u_k$ blowing up at the origin and on a sphere. We also prove sharp blow-up estimates.