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

Let $(X^{n+1},g^{+})$ be an $(n+1)$-dimensional asymptotically hyperbolic manifold with a conformal infinity $(M^n,[\widehat{h}])$. The fractional Yamabe problem addresses to solve $$P^{\gamma }[g^{+},\widehat{h}](u)=c{u}^{\frac{n+2\gamma }{n-2\gamma }},u>0\text{on }M$$ where $c\in \mathbb{R}$ and $P^{\gamma }[g^{+},\widehat{h}]$ is the fractional conformal Laplacian whose principal symbol is $(-\mathrm{\Delta }{)}^{\gamma }$. In this paper, we construct a metric on the half space $X={\mathbb{R}}_{+}^{n+1}$, which is conformally equivalent to the unit ball, for which the solution set of the fractional Yamabe equation is non-compact provided that $n\ge 24$ for $\gamma \in (0,{\gamma }^{\ast })$ and $n \ge 25$for$\gamma \in [\gamma^,1)$where$\gamma^ \in (0, 1)$ is a certain transition exponent. The value of ${\gamma }^{\ast }$ turns out to be approximately 0.940197.