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

Seunghyeok Kim, Monica Musso, Juncheng Wei

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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.
Original languageEnglish
Pages (from-to)3759-3830
Number of pages72
JournalJournal of Functional Analysis
Issue number12
Early online date1 Aug 2017
Publication statusPublished - 15 Dec 2017


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