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

Given a smoothly bounded domain Omega subset of R-n with n >= 1 odd, we study the blowup of bounded sequences (u(k)) subset of H-00(n/2)(Omega) of solutions to the nonlocal equation (-Delta)(n/2) u(k) = lambda(k)u(k)e(n/2u2/k) in Omega where lambda(k) -> lambda(infinity) is an element of[0, infinity), and H-00(n/2)(Omega) denotes the Lions-Magenes spaces of functions u is an element of L-2(R-n) which are supported in Omega and with (-Delta)(n/4) u is an element of L-2(R-n). Extending previous works of Druet, Robert-Struwe, and Martinazzi, we show that if the sequence (u(k)) is not bounded in L-infinity(Omega), a suitably rescaled subsequence n(k) converges to the function eta(0)(x) = log (2/1+vertical bar x vertical bar(2)), which solves the prescribed nonlocal Q-curvature equation (-Delta)(n/2) eta = (n = 1)(!en eta) in R-n 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 blowup can occur only if Lambda := lim sup(k ->infinity) parallel to(-Delta)(n/4) u(k)parallel to(2)(L2) >= Lambda(1) := (n = 1)!vertical bar S-n vertical bar.