Frational p-Laplacian on Compact Riemannian Manifold

TL;DR

This paper studies the existence and uniqueness of solutions for a class of nonlocal fractional p-Laplacian equations on compact Riemannian manifolds, extending the understanding of such operators in geometric contexts.

Contribution

It establishes existence and uniqueness results for fractional p-Laplacian equations on compact Riemannian manifolds, a novel extension to nonlocal geometric analysis.

Findings

Proved existence of solutions under certain conditions.

Established uniqueness of solutions.

Extended fractional p-Laplacian theory to Riemannian manifolds.

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.