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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.19704 |
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| _version_ | 1866916759262461952 |
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| author | Chen, Kaizhe Zhang, Heng |
| author_facet | Chen, Kaizhe Zhang, Heng |
| contents | We derive some existence results for the solutions of the Tzitzéica equation
\begin{equation*}
-Δu + h_1(x)e^{Au} + h_2(x)e^{-Bu}=0
\end{equation*}
and the generalized Tzitzéica equation
\begin{equation*}
-Δu + h_1(x)e^{Au}(e^{Au}-1)+h_2(x)e^{-Bu}(e^{-Bu}-1)=0
\end{equation*}
on any connected finite graph \(G=(V, E)\). Here, \(h_1(x)>0\), \(h_2(x)>0\) are two given functions on \(V\), and \(A, B>0\) are two constants. Our approach involves computing the topological degree and using the connection between the degree and the critical group of an associated functional. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_19704 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence results for Tzitzéica equation via topological degree method on graphs Chen, Kaizhe Zhang, Heng Analysis of PDEs We derive some existence results for the solutions of the Tzitzéica equation \begin{equation*} -Δu + h_1(x)e^{Au} + h_2(x)e^{-Bu}=0 \end{equation*} and the generalized Tzitzéica equation \begin{equation*} -Δu + h_1(x)e^{Au}(e^{Au}-1)+h_2(x)e^{-Bu}(e^{-Bu}-1)=0 \end{equation*} on any connected finite graph \(G=(V, E)\). Here, \(h_1(x)>0\), \(h_2(x)>0\) are two given functions on \(V\), and \(A, B>0\) are two constants. Our approach involves computing the topological degree and using the connection between the degree and the critical group of an associated functional. |
| title | Existence results for Tzitzéica equation via topological degree method on graphs |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2505.19704 |