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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2205.08216 |
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| _version_ | 1866913217582727168 |
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| author | Gao, Jia Hou, Songbo |
| author_facet | Gao, Jia Hou, Songbo |
| contents | Denote by $G=(V,E)$ a finite graph. We study a generalized Chern-Simons equation $$ Δu=λ\mathrm{e}^u(\mathrm{e}^{bu}-1)+4π\sum\limits_{j=1}^{N}δ_{p_j} $$ on $G$, where $λ$ and $b$ are positive constants; $N$ is a positive integer; $p_1, p_2, \cdot\cdot\cdot, p_N$ are distinct vertices of $V$ and $δ_{p_j}$ is the Dirac delta mass at $p_j$. We prove that there exists a critical value $λ_c$ such that the equation has a solution if $λ\geq λ_c$ and the equation has no solution if $λ<λ_c$. We also prove that if $λ>λ_c$ the equation has at least two solutions which include a local minimizer for the corresponding functional and a mountain-pass type solution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_08216 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Existence theorems for a generalized Chern-Simons equation on finite graphs Gao, Jia Hou, Songbo Analysis of PDEs Functional Analysis 35J91, 05C22 Denote by $G=(V,E)$ a finite graph. We study a generalized Chern-Simons equation $$ Δu=λ\mathrm{e}^u(\mathrm{e}^{bu}-1)+4π\sum\limits_{j=1}^{N}δ_{p_j} $$ on $G$, where $λ$ and $b$ are positive constants; $N$ is a positive integer; $p_1, p_2, \cdot\cdot\cdot, p_N$ are distinct vertices of $V$ and $δ_{p_j}$ is the Dirac delta mass at $p_j$. We prove that there exists a critical value $λ_c$ such that the equation has a solution if $λ\geq λ_c$ and the equation has no solution if $λ<λ_c$. We also prove that if $λ>λ_c$ the equation has at least two solutions which include a local minimizer for the corresponding functional and a mountain-pass type solution. |
| title | Existence theorems for a generalized Chern-Simons equation on finite graphs |
| topic | Analysis of PDEs Functional Analysis 35J91, 05C22 |
| url | https://arxiv.org/abs/2205.08216 |