Blow-up behaviour of a fractional Adams-Moser-Trudinger type inequalityin odd dimension

Given a smoothly bounded domain $\Omega \Subset \mathbb{R}^n$ with $n \ge 1$ odd, we study the blow-up of bounded sequences $(u_k)\subset H_{00}^{n/2}(\Omega)$ of solutions to the non-local equation
\[
(-\Delta)^{n/2} u_k = \lambda_k u_k e^{n/2 u_k^2} in \Omega
\]
where $\lambda_k \to \lambda_\infty \in [0,\infty)$, and $H_{00}^{n/2}(\Omega)$ denotes the Lions-Magenes spaces of functions $u \in L^2(\mathbb{R}^n)$ which are supported in $\Omega$ and with $(-\Delta)^{n/4} u \in L^2(\mathbb{R}^n)$. Extending previous works of Druet, Robert-Struwe and the second author, we show that if the sequence $(u_k)$ is not bounded in $L^\infty (\Omega)$, a suitably rescaled sequence $\eta_k$ converges to the function $\eta_0(x) = \log(2/(1+|x|^2))$ which solves the prescribed non-local Q-curvature equation
\[
(-\Delta)^{n/2} \eta = (n-1)! e^{n \eta} in \mathbb{R}^n
\]
recently studied by Da Lio-Martinazzi-Rivièra when n=1, Jin-Maalaoui-Martinazzi-Xiong when n=3, and Hyder when $n\ge 5$ is odd. We infer that blow-up can occur only if $\Lambda := \limsup_{k\to\infty} \|(-\Delta)^{n/4} u_k\|_{L^2}^2 \ge \Lambda := (n-1)!|S^n|$.