Fractional Laplace operator and related Schrödinger equations on locally finite graphs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909641913401344 |
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| author | Zhang, Mengjie Lin, Yong Yang, Yunyan |
| author_facet | Zhang, Mengjie Lin, Yong Yang, Yunyan |
| contents | In this paper, we first define a discrete version of the fractional Laplace operator $(-Δ)^{s}$ through the heat semigroup on a stochastically complete, connected, locally finite graph $G = (V, E, μ, w)$. Secondly, we define the fractional divergence and give another form of $(-Δ)^s$. The third point, and the foremost, is the introduction of the fractional Sobolev space $W^{s,2}(V)$, which is necessary when we study problems involving $(-Δ)^{s}$. Finally, using the mountain-pass theorem and the Nehari manifold, we obtain multiplicity solutions to a discrete fractional Schrödinger equation on $G$. We caution the readers that though these existence results are well known in the continuous case, the discrete case is quite different. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_02902 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fractional Laplace operator and related Schrödinger equations on locally finite graphs Zhang, Mengjie Lin, Yong Yang, Yunyan Analysis of PDEs In this paper, we first define a discrete version of the fractional Laplace operator $(-Δ)^{s}$ through the heat semigroup on a stochastically complete, connected, locally finite graph $G = (V, E, μ, w)$. Secondly, we define the fractional divergence and give another form of $(-Δ)^s$. The third point, and the foremost, is the introduction of the fractional Sobolev space $W^{s,2}(V)$, which is necessary when we study problems involving $(-Δ)^{s}$. Finally, using the mountain-pass theorem and the Nehari manifold, we obtain multiplicity solutions to a discrete fractional Schrödinger equation on $G$. We caution the readers that though these existence results are well known in the continuous case, the discrete case is quite different. |
| title | Fractional Laplace operator and related Schrödinger equations on locally finite graphs |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2408.02902 |