Entire solution of a partial differential equation

Junfeng Xu; Nabadwip Sarkar; Sujoy Majumder

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arXiv · 2511.09579

Entire solution of a partial differential equation

Junfeng Xu·Nabadwip Sarkar·Sujoy Majumder

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Abstract

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 $\mathbb{C}^2$.

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Junfeng Xu, Nabadwip Sarkar, Sujoy Majumder. 2025-11-12. Entire solution of a partial differential equation. https://arxiv.org/abs/2511.09579

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