  In this paper, we investigate the existence and uniqueness of a non-trivial
solution for a class of nonlocal equations involving the fractional
$p$-Laplacian operator defined on compact Riemannian manifold, namely,
\begin{eqnarray}\label{k1} \begin{gathered} \left\{\begin{array}{lll}
  (-\Delta_g)^s_p u(x)+ \left| u \right|^{p-2} u= f(x,u) & \text { in }&
\Omega,
  \hspace{3,4cm} u=0 & \text{in }& M\setminus\Omega,
  \end{array}\right. \end{gathered} \end{eqnarray}
  and $\Omega$ is an open bounded subset of M with a smooth boundary.
