Fractional Sobolev spaces and fractional $p$-Laplace equations on locally finite graphs
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | |
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| _version_ | 1866912419687694336 |
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| author | Zhang, Mengjie Lin, Yong Yang, Yunyan |
| author_facet | Zhang, Mengjie Lin, Yong Yang, Yunyan |
| contents | 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 Schrödinger 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_07694 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fractional Sobolev spaces and fractional $p$-Laplace equations on locally finite graphs Zhang, Mengjie Lin, Yong Yang, Yunyan Analysis of PDEs 35A15, 35R02, 35R11, 46E35 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 Schrödinger 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. |
| title | Fractional Sobolev spaces and fractional $p$-Laplace equations on locally finite graphs |
| topic | Analysis of PDEs 35A15, 35R02, 35R11, 46E35 |
| url | https://arxiv.org/abs/2506.07694 |