Blow-up behavior of a fractional Adams-Moser-Trudinger-type inequality in odd dimension

Given a smoothly bounded domain Ω Rn with n ≥ 1 odd, we study the blow-up of n bounded sequences (u ) ⊂ H 2 (Ω) of solutions to the non-local equation k 00 (−∆)n2 uk = λkuken2 u2k in Ω, n where λ → λ ∈ [0, ∞), and H 2 (Ω) denotes the Lions-Magenes spaces of functions u ∈ k∞ 00 L2(Rn) which are supported in Ω and with (−∆)n4 u ∈ L2(Rn). Extending previous works of Druet, Robert-Struwe and the second author, we show that if the sequence (uk) is not bounded in L∞(Ω), a suitably rescaled subsequence ηk converges to the function η0(x) = log 2 , which solves the prescribed non-local Q-curvature equation 1+|x|2 (−∆)n2 η = (n − 1)!enη in Rn recently studied by Da Lio-Martinazzi-Riviere when n = 1, Jin-Maalaoui-Martinazzi-Xiong when n = 3, and Hyder when n ≥ 5 is odd. We infer that blow-up can occur only if Λ:=limsupk→∞∥(−∆)n4 uk∥2L2 ≥Λ1 :=(n−1)!|Sn|.