标题： 分数阶Sobolev空间与局部有限图上的分数阶$p$-Laplace方程 显示英文标题

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

摘要： 基于图的分析具有理论和应用上的重要意义，吸引了众多研究者的关注，并在近年来取得了丰富的成果。 然而，关于分数阶问题的研究仍然有限，大多数已有的结果局限于格点图。 本文在一般的连通、局部有限且随机完备的图上构建了分数阶Sobolev空间。 在一定假设下，这些空间表现出完备性、自反性以及其他性质。 此外，我们提出了一个分数阶$p$-Laplace 算子，并研究了一些涉及该非局域算子的非线性Schrödinger型方程解的存在性。 本文的主要贡献在于建立了一套相对全面的用于研究图上分数阶问题的分析工具。 显示英文摘要

摘要： 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.