  Given a smoothly bounded domain $\Omega\Subset\mathbb{R}^n$ with $n\ge 1$
odd, we study the blow-up of bounded sequences $(u_k)\subset
H^\frac{n}{2}_{00}(\Omega)$ of solutions to the non-local equation
$$(-\Delta)^\frac n2 u_k=\lambda_k u_ke^{\frac n2 u_k^2}\quad \text{in
}\Omega,$$ where $\lambda_k\to\lambda_\infty \in [0,\infty)$, and $H^{\frac
n2}_{00}(\Omega)$ denotes the Lions-Magenes spaces of functions $u\in
L^2(\mathbb{R}^n)$ which are supported in $\Omega$ and with
$(-\Delta)^\frac{n}{4}u\in L^2(\mathbb{R}^n)$. 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 subsequence
$\eta_k$ converges to the function
$\eta_0(x)=\log\left(\frac{2}{1+|x|^2}\right)$, which solves the prescribed
non-local $Q$-curvature equation $$(-\Delta)^\frac n2 \eta
=(n-1)!e^{n\eta}\quad \text{in }\mathbb{R}^n$$ recently studied by Da
Lio-Martinazzi-Rivi\`ere 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
$\Lambda:=\limsup_{k\to \infty}\|(-\Delta)^\frac n4 u_k\|_{L^2}^2\ge
\Lambda_1:= (n-1)!|S^n|$.
