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

Abstract

Let $(X^{n+1}, g^+)$ be an $(n+1)$-dimensional asymptotically hyperbolic manifold with a conformal infinity $(M^n, [\hat{h}])$. The fractional Yamabe problem addresses to solve \[P^{\gamma}[g^+,\hat{h}] (u) = cu^{n+2\gamma \over n-2\gamma}, \quad u > 0 \quad \text{on } M\] where $c \in \mathbb{R}$ and $P^{\gamma}[g^+,\hat{h}]$ is the fractional conformal Laplacian whose principal symbol is $(-\Delta)^{\gamma}$. 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 $\gamma \in (0, \gamma^*)$ and $n \ge 25$ for $\gamma \in [\gamma^*,1)$ where $\gamma^* \in (0, 1)$ is a certain transition exponent. The value of $\gamma^*$ turns out to be approximately 0.940197.