Multipeak solutions for fractional elliptic equations on conformal infinities

We consider the following problem \begin{equation*}\varepsilon^{2 s} P_{\hat{h}}^s u+u=u^p, \quad u \gt 0 \quad \text {on } \,(M, \hat{h}),\end{equation*} on an asymptotically hyperbolic manifold $(X, g^{+})$ with conformal infinity $(M,[\hat{h}])$ , where $s\in (0,1)$ , $P_{\hat{h}}^s$ is the fractional conformally invariant operators, $1 \lt p \lt \frac{n+2s}{n-2s}$ . By Lyapunov–Schmidt reduction method, we prove the existence of solutions whose peaks collapse, as $\varepsilon$ goes to zero, to a $C^1$ -stable critical point of the mean curvature $H$ for $0 \lt s \lt {1}/{2}$ or a $C^1$ -stable critical point of a function involving the scalar curvature and the second fundamental form for ${1}/{2}\le s \lt 1$ .