A non-compactness result on the fractional Yamabe problem in large dimensions

Abstract

Let (Xn+1, g(+)) be an (n + 1)-dimensional asymptotically hyperbolic manifold with conformal infinity (M-n, [(h) over cap]). The fractional Yamabe problem addresses to solve P-gamma[g(+), (h) over cap](u) = cu(n+2 gamma/n-2 gamma), u > 0 on M where c is an element of R and P-gamma[g(+) , (h) over cap] is the fractional conformal Laplacian whose principal symbol is the Laplace-Beltrami operator (-Delta)(gamma) on M. In this paper, we construct a metric on the half space X = R-+(n+1), which is conformally equivalent to the unit ball, for which the solution set of the fractional Yamabe equation is non -compact provided that n >= 24 for gamma is an element of (0, gamma*) and n >= 25 for gamma is an element of [gamma*,1) where gamma* is an element of (0,1) is a certain transition exponent. The value of gamma* turns out to be approximately 0.940197.