Entire solution of a partial differential equation
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911262108024832 |
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| author | Xu, Junfeng Sarkar, Nabadwip Majumder, Sujoy |
| author_facet | Xu, Junfeng Sarkar, Nabadwip Majumder, Sujoy |
| contents | In this paper, using Nevanlinna's value distribution theory of meromorphic functions in several complex variables, we study for the existence of entire solutions $f$ in $\mathbb{C}^2$ of the following partial differential equation \[a_1\left(\frac{\partial f(z_1,z_2)}{\partial z_1}\right)^n+a_2f^n(z_1,z_2)=p_1e^{r(z_1,z_2)}+p_2e^{s(z_1,z_2)},\] where $n$ is a positive integer such that $n\geq 3$, $a_1,a_2,p_1,p_2$ are non-zero constants and $r(z_1,z_2), s(z_1,z_2)$ are arbitrary polynomials in $\mathbb{C}^2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_09579 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Entire solution of a partial differential equation Xu, Junfeng Sarkar, Nabadwip Majumder, Sujoy Complex Variables In this paper, using Nevanlinna's value distribution theory of meromorphic functions in several complex variables, we study for the existence of entire solutions $f$ in $\mathbb{C}^2$ of the following partial differential equation \[a_1\left(\frac{\partial f(z_1,z_2)}{\partial z_1}\right)^n+a_2f^n(z_1,z_2)=p_1e^{r(z_1,z_2)}+p_2e^{s(z_1,z_2)},\] where $n$ is a positive integer such that $n\geq 3$, $a_1,a_2,p_1,p_2$ are non-zero constants and $r(z_1,z_2), s(z_1,z_2)$ are arbitrary polynomials in $\mathbb{C}^2$. |
| title | Entire solution of a partial differential equation |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2511.09579 |