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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.02881 |
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| _version_ | 1866915658852204544 |
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| author | Zhao, Xinrong |
| author_facet | Zhao, Xinrong |
| contents | In this paper, we study the $p$-Laplacian equation
$$
-Δ_p u + V(x)|u|^{p-2}u = f(x,u)
$$
on the lattice graph $\mathbb{Z}^N$ with nonnegative potentials, where $Δ_p$ is the discrete $p$-Laplacian and $p\in(1,\infty)$. By employing the Nehari manifold method, we establish the existence of ground state solutions under suitable growth conditions on the nonlinearity $f(x,u)$, provided that the potential $V(x)$ is either periodic or bounded. Moreover, we prove that if $f$ is odd in $u$ and $p\geq2$, then the above equation admits infinitely many geometrically distinct solutions. Finally, we extend these results from $\mathbb{Z}^N$ to the more general setting of Cayley graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_02881 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ground state solutions of $p$-Laplacian equations with nonnegative potentials on Lattice graphs Zhao, Xinrong Analysis of PDEs In this paper, we study the $p$-Laplacian equation $$ -Δ_p u + V(x)|u|^{p-2}u = f(x,u) $$ on the lattice graph $\mathbb{Z}^N$ with nonnegative potentials, where $Δ_p$ is the discrete $p$-Laplacian and $p\in(1,\infty)$. By employing the Nehari manifold method, we establish the existence of ground state solutions under suitable growth conditions on the nonlinearity $f(x,u)$, provided that the potential $V(x)$ is either periodic or bounded. Moreover, we prove that if $f$ is odd in $u$ and $p\geq2$, then the above equation admits infinitely many geometrically distinct solutions. Finally, we extend these results from $\mathbb{Z}^N$ to the more general setting of Cayley graphs. |
| title | Ground state solutions of $p$-Laplacian equations with nonnegative potentials on Lattice graphs |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2512.02881 |