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Let $(X^{n+1}, g^+)$ be an $(n+1)$-dimensional asymptotically hyperbolicmanifold with a conformal infinity $(M^n, [\hat{h}])$. The fractional Yamabeproblem addresses to solve \[P^{\gamma}[g^+,\hat{h}] (u) = cu^{n+2\gamma \overn-2\gamma}, \quad u > 0 \quad \text{on } M\] where $c \in \mathbb{R}$ and$P^{\gamma}[g^+,\hat{h}]$ is the fractional conformal Laplacian whose principalsymbol is $(-\Delta)^{\gamma}$. In this paper, we construct a metric on thehalf space $X = \mathbb{R}^{n+1}_+$, which is conformally equivalent to theunit ball, for which the solution set of the fractional Yamabe equation isnon-compact provided that $n \ge 24$ for $\gamma \in (0, \gamma^*)$ and $n \ge25$ for $\gamma \in [\gamma^*,1)$ where $\gamma^* \in (0, 1)$ is a certaintransition exponent. The value of $\gamma^*$ turns out to be approximately0.940197.

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