Nontrivial solutions for a generalized poly-Laplacian system on finite graphs
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866912006348472320 |
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| author | Qi, Wanting Zhang, Xingyong |
| author_facet | Qi, Wanting Zhang, Xingyong |
| contents | We investigate the existence and multiplicity of solutions for a class of generalized coupled system involving poly-Laplacian and a parameter $λ$ on finite graphs. By using mountain pass lemma together with cut-off technique, we obtain that system has at least a nontrivial weak solution $(u_λ,v_λ)$ for every large parameter $λ$ when the nonlinear term $F(x,u,v)$ satisfies superlinear growth conditions only in a neighborhood of origin point $(0,0)$. We also obtain a concrete form for the lower bound of parameter $λ$ and the trend of $(u_λ,v_λ)$ with the change of parameter $λ$. Moreover, by using a revised Clark's theorem together with cut-off technique, we obtain that system has a sequence of solutions tending to 0 for every $λ>0$ when the nonlinear term $F(x,u,v)$ satisfies sublinear growth conditions only in a neighborhood of origin point $(0,0)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_16364 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Nontrivial solutions for a generalized poly-Laplacian system on finite graphs Qi, Wanting Zhang, Xingyong Analysis of PDEs We investigate the existence and multiplicity of solutions for a class of generalized coupled system involving poly-Laplacian and a parameter $λ$ on finite graphs. By using mountain pass lemma together with cut-off technique, we obtain that system has at least a nontrivial weak solution $(u_λ,v_λ)$ for every large parameter $λ$ when the nonlinear term $F(x,u,v)$ satisfies superlinear growth conditions only in a neighborhood of origin point $(0,0)$. We also obtain a concrete form for the lower bound of parameter $λ$ and the trend of $(u_λ,v_λ)$ with the change of parameter $λ$. Moreover, by using a revised Clark's theorem together with cut-off technique, we obtain that system has a sequence of solutions tending to 0 for every $λ>0$ when the nonlinear term $F(x,u,v)$ satisfies sublinear growth conditions only in a neighborhood of origin point $(0,0)$. |
| title | Nontrivial solutions for a generalized poly-Laplacian system on finite graphs |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2408.16364 |