Nonlocal Elliptic Problems on Compact Riemannian Manifold

Abstract

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.