Positive solutions to fractional $p$-Laplacian Choquard equation on lattice graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911083746295808 |
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| author | Wang, Lidan |
| author_facet | Wang, Lidan |
| contents | In this paper, we study the fractional $p$-Laplacian Choquard equation $$ (-Δ)_{p}^{s} u+h(x)|u|^{p-2} u=\left(R_α *F(u)\right)f(u) $$ on lattice graphs $\mathbb{Z}^d$, where $s\in(0,1)$, $ p\geq 2$, $α\in(0, d)$ and $R_α$ represents the Green's function of the discrete fractional Laplacian that behaves as the Riesz potential. Under suitable assumptions on the potential function $h$, we first prove the existence of a strictly positive solution by the mountain-pass theorem for the nonlinearity $f$ satisfying some growth conditions. Moreover, if we add some monotonicity condition, we establish the existence of a positive ground state solution by the method of Nehari manifold. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_22552 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Positive solutions to fractional $p$-Laplacian Choquard equation on lattice graphs Wang, Lidan Analysis of PDEs 35A15, 35R02, 35R11 In this paper, we study the fractional $p$-Laplacian Choquard equation $$ (-Δ)_{p}^{s} u+h(x)|u|^{p-2} u=\left(R_α *F(u)\right)f(u) $$ on lattice graphs $\mathbb{Z}^d$, where $s\in(0,1)$, $ p\geq 2$, $α\in(0, d)$ and $R_α$ represents the Green's function of the discrete fractional Laplacian that behaves as the Riesz potential. Under suitable assumptions on the potential function $h$, we first prove the existence of a strictly positive solution by the mountain-pass theorem for the nonlinearity $f$ satisfying some growth conditions. Moreover, if we add some monotonicity condition, we establish the existence of a positive ground state solution by the method of Nehari manifold. |
| title | Positive solutions to fractional $p$-Laplacian Choquard equation on lattice graphs |
| topic | Analysis of PDEs 35A15, 35R02, 35R11 |
| url | https://arxiv.org/abs/2507.22552 |