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

Let (Xn+1,g+)bean(n+ 1)dimensional asymptotically hyperbolic manifold with conformal infinity (Mn,[ˆh]). The fractional Yamabe problem addresses to solve Pγ[g+,ˆh(u)=cun+2γn−2γ,u>0 on M where c ∈ R and Pγ[g+,ˆh] is the fractional conformal Laplacian whose principal symbol is the Laplace-Beltramioperator (−Δ)γ on M. In this paper, we construct a metric on the half space X=Rn+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 o fγ∗ turns out to be approximately 0.940197.