Least energy solutions of two asymptotically cubic Kirchhoff equations on locally finite graphs
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912221871734784 |
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| author | Yu, Zhangyi Zhang, Xingyong Ou, Xin |
| author_facet | Yu, Zhangyi Zhang, Xingyong Ou, Xin |
| contents | We study the existence of least energy solutions for two Kirchhoff equations with the asymptotically cubic nonlinearity $f(u)=λu+η|u|^2u$ on a locally weighted and connected finite graph $G=(V,E)$. Such nonlinearity satisfies neither $\frac{F(u)}{u^4}\to +\infty$ as $|u|\to\infty$, where $F(u)=\int_0^uf(s)ds$, nor $\frac{f(u)}{u}\to 0$ as $u\to 0$. By utilizing the constrained variational method, we prove that there exist $λ_1\ge 0$ and $η_0\ge 0$ ($λ_1^*\ge 0$ and $η_0^*\ge 0$) such that these two equations have at least a least energy solution if $|λ|<aλ_1$ ($|λ|<aλ_1^*$) and $η>η_0$ ($η>η_0^*$). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_03720 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Least energy solutions of two asymptotically cubic Kirchhoff equations on locally finite graphs Yu, Zhangyi Zhang, Xingyong Ou, Xin Analysis of PDEs We study the existence of least energy solutions for two Kirchhoff equations with the asymptotically cubic nonlinearity $f(u)=λu+η|u|^2u$ on a locally weighted and connected finite graph $G=(V,E)$. Such nonlinearity satisfies neither $\frac{F(u)}{u^4}\to +\infty$ as $|u|\to\infty$, where $F(u)=\int_0^uf(s)ds$, nor $\frac{f(u)}{u}\to 0$ as $u\to 0$. By utilizing the constrained variational method, we prove that there exist $λ_1\ge 0$ and $η_0\ge 0$ ($λ_1^*\ge 0$ and $η_0^*\ge 0$) such that these two equations have at least a least energy solution if $|λ|<aλ_1$ ($|λ|<aλ_1^*$) and $η>η_0$ ($η>η_0^*$). |
| title | Least energy solutions of two asymptotically cubic Kirchhoff equations on locally finite graphs |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2502.03720 |