Fractional Sobolev spaces and fractional $p$-Laplace equations on locally finite graphs

TL;DR

This paper develops fractional Sobolev spaces and a fractional p-Laplace operator on general graphs, providing new analytical tools for studying fractional problems in graph analysis, with applications to nonlinear Schrödinger equations.

Contribution

It introduces fractional Sobolev spaces on general graphs and proposes a fractional p-Laplace operator, extending analysis beyond lattice graphs.

Findings

Spaces are complete and reflexive under certain conditions

Existence results for nonlinear Schrödinger equations involving the fractional p-Laplace operator

Framework enables further study of fractional problems on complex graphs

Abstract

Graph-based analysis holds both theoretical and applied significance, attracting considerable attention from researchers and yielding abundant results in recent years. However, research on fractional problems remains limited, with most of established results restricted to lattice graphs. In this paper, fractional Sobolev spaces are constructed on general graphs that are connected, locally finite and stochastically complete. Under certain assumptions, these spaces exhibit completeness, reflexivity, and other properties. Moreover, we propose a fractional $p$-Laplace operator, and study the existence of solutions to some nonlinear Schr\"odinger type equations involving this nonlocal operator. The main contribution of this paper is to establish a relatively comprehensive set of analytical tools for studying fractional problems on graphs.