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 R 3 . We will use an approach inspired from conformal geometry, seeing a metric on a subset of R 3 as the restriction of a metric on 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 non-positive bi-harmonic function Φ in R 4 + ), in analogy with a four-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.

Fractional operators as traces of operator-valued curves

“…The case of higher order powers of (−∆) s has been investigated firstly in [7] via conformal geometry techniques. We also cite [6,9,12,17,19], the more recent papers [5,8] and references there-in.…”