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

Let (Xn+1,g+) be an (n+1)-dimensional asymptotically hyperbolic manifold with conformal infinity (Mn,[hˆ]). The fractional Yamabe problem addresses to solvePγ[g+,hˆ](u)=cun+2γn−2γ,u>0on M where c∈R and Pγ[g+,hˆ] is the fractional conformal Laplacian whose principal symbol is the Laplace-Beltrami operator (−Δ)γ 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 γ∈(0,γ⁎) and n≥25 for γ∈[γ⁎,1) where γ⁎∈(0,1) is a certain transition exponent. The value of γ⁎ turns out to be approximately 0.940197.