Existence and multiplicity of solutions to discrete fractional logarithmic Kirchhoff equations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914006249242624 |
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| author | Wang, Lidan |
| author_facet | Wang, Lidan |
| contents | In this paper, we study the discrete fractional logarithmic Kirchhoff equation $$ \left(a+b \int_{\mathbb{Z}^d}|\nabla^s u|^{2} d μ\right) (-Δ)^s u+h(x) u=|u|^{p-2}u \log u^{2}, \quad x\in \mathbb{Z}^d, $$ where $a,\,b>0$ and $0<s<1$. Under suitable assumptions on $h(x)$, we first prove the existence of ground state solutions by the mountain-pass theorem for $p>4$; then we verify the existence of ground state sign-changing solutions based on the method of Nehari manifold for $p>6$. Finally, we establish the multiplicity of nontrivial weak solutions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_18840 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence and multiplicity of solutions to discrete fractional logarithmic Kirchhoff equations Wang, Lidan Analysis of PDEs 35A15, 35R02, 35R11 In this paper, we study the discrete fractional logarithmic Kirchhoff equation $$ \left(a+b \int_{\mathbb{Z}^d}|\nabla^s u|^{2} d μ\right) (-Δ)^s u+h(x) u=|u|^{p-2}u \log u^{2}, \quad x\in \mathbb{Z}^d, $$ where $a,\,b>0$ and $0<s<1$. Under suitable assumptions on $h(x)$, we first prove the existence of ground state solutions by the mountain-pass theorem for $p>4$; then we verify the existence of ground state sign-changing solutions based on the method of Nehari manifold for $p>6$. Finally, we establish the multiplicity of nontrivial weak solutions. |
| title | Existence and multiplicity of solutions to discrete fractional logarithmic Kirchhoff equations |
| topic | Analysis of PDEs 35A15, 35R02, 35R11 |
| url | https://arxiv.org/abs/2508.18840 |