Concentration phenomena for the fractional $Q$-curvature equation in dimension 3 and fractional Poisson formulas

Abstract

We study the compactness properties of metrics of prescribed fractional $Q$-curvature of order $3$ in ${\mathbb{R}}^3$. We will use an approach inspired from conformal geometry, seeing a metric on a subset of ${\mathbb{R}}^3$ as the restriction of a metric on ${\mathbb{R}}_{+}^4$ with vanishing fourth-order $Q$-curvature. We will show that a sequence of such metrics with uniformly bounded fractional $Q$-curvature can blow up on a large set (roughly, the zero set of the trace of a nonpositive biharmonic function $\Phi$ in ${\mathbb{R}}_{+}^4$), in analogy with a $4$-dimensional result of Adimurthi-Robert-Struwe, and construct examples of such behaviour. In doing so, we produce general Poisson-type representation formulas (also for higher dimension), which are of independent interest.