In this article we investigate an asymptotically critical problem involving the fractional Laplacian operator \((-\Delta _g)^s\) in a compact Riemannian N-manifold (M, g) as follows : \({\left\{ \begin{array}{ll}(-\Delta _g)^su+hu=u^{2_s^*-1\pm \epsilon ^s} & \text {in} \ M \\ u>0 & \text {in} \ M\end{array}\right. }\) where \(2_s^*=\frac{2N}{N-2s}\) with \(0<s<1\) and \(N>2s\) with a small real parameter \(\epsilon \rightarrow 0^+\) . We take \(h:M\rightarrow \mathbb {R}\) initially a \(C^1\) -function on M. In our work, we give an asymptotic estimate of the energy functional associated to the problem via rigorous estimates of norms of the bubble function concentrated at some point \(\xi _0\in M\) . Using Lyapunov-Schmidt reduction, we then determine the condition on h such that the problem ensures a blow-up solution concentrated at \(\xi _0\) .
