On finite-energy solutions of Kazan-Warner equations on the lattice graph
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911703791304704 |
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| author | Chen, Huyuan hua, Bobo |
| author_facet | Chen, Huyuan hua, Bobo |
| contents | We investigate finite-energy solutions to Kazdan-Warner type equations in 2-dimensional integer lattice graph $$ - Δu= \varepsilon e^{κu} +βδ_0\quad {\rm in}\ \mathbb{Z}^2,$$ where $\varepsilon=\pm1$, $κ>0$ and $β\in\mathbb{R}$.
When $\varepsilon=1$, we prove the existence of a continuous family of finite-energy solutions for some parameter $κ$. This provides a partial resolution of the open problem on the existence of finite-energy solutions to the Liouville equation.
When $\varepsilon=-1$ and $β>\frac{4π}κ$, we prove that the set of finite-energy solutions exhibits a layer structure. Moreover, we derive the extremal solution in this case. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_01155 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On finite-energy solutions of Kazan-Warner equations on the lattice graph Chen, Huyuan hua, Bobo Analysis of PDEs 35J91, 05C22 F.m We investigate finite-energy solutions to Kazdan-Warner type equations in 2-dimensional integer lattice graph $$ - Δu= \varepsilon e^{κu} +βδ_0\quad {\rm in}\ \mathbb{Z}^2,$$ where $\varepsilon=\pm1$, $κ>0$ and $β\in\mathbb{R}$. When $\varepsilon=1$, we prove the existence of a continuous family of finite-energy solutions for some parameter $κ$. This provides a partial resolution of the open problem on the existence of finite-energy solutions to the Liouville equation. When $\varepsilon=-1$ and $β>\frac{4π}κ$, we prove that the set of finite-energy solutions exhibits a layer structure. Moreover, we derive the extremal solution in this case. |
| title | On finite-energy solutions of Kazan-Warner equations on the lattice graph |
| topic | Analysis of PDEs 35J91, 05C22 F.m |
| url | https://arxiv.org/abs/2509.01155 |