Solutions to discrete nonlinear Kirchhoff-Choquard equations
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866914759494860800 |
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| author | Wang, Lidan |
| author_facet | Wang, Lidan |
| contents | In this paper, we study the discrete Kirchhoff-Choquard equation $$ -\left(a+b \int_{\mathbb{Z}^3}|\nabla u|^{2} d μ\right) Δu+V(x) u=\left(R_α *F(u)\right)f(u),\quad x\in \mathbb{Z}^3, $$ where $a,\,b>0$ are constants, $R_α$ is the Green's function of the discrete fractional Laplacian with $α\in(0,3)$, which has no singularity but has same asymptotics as the Riesz potential. Under some suitable assumptions on $V$ and $f$, we prove the existence of nontrivial solutions and ground state solutions by variational methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_11856 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Solutions to discrete nonlinear Kirchhoff-Choquard equations Wang, Lidan Analysis of PDEs 35J20, 35R02 In this paper, we study the discrete Kirchhoff-Choquard equation $$ -\left(a+b \int_{\mathbb{Z}^3}|\nabla u|^{2} d μ\right) Δu+V(x) u=\left(R_α *F(u)\right)f(u),\quad x\in \mathbb{Z}^3, $$ where $a,\,b>0$ are constants, $R_α$ is the Green's function of the discrete fractional Laplacian with $α\in(0,3)$, which has no singularity but has same asymptotics as the Riesz potential. Under some suitable assumptions on $V$ and $f$, we prove the existence of nontrivial solutions and ground state solutions by variational methods. |
| title | Solutions to discrete nonlinear Kirchhoff-Choquard equations |
| topic | Analysis of PDEs 35J20, 35R02 |
| url | https://arxiv.org/abs/2404.11856 |