Electron. J. Differential Equations, Vol. 2023 (2023), No. 43, pp. 1-18.

Solutions of complex nonlinear functional equations including second order partial differential and difference in C^2

Hong Yan Xu, Goutam Haldar

This article is devoted to exploring the existence and the form of finite order transcendental entire solutions of Fermat-type second order partial differential-difference equations $$ \Big(\frac{\partial^2 f}{\partial z_1^2}+\delta\frac{\partial^2 f}{\partial z_2^2} +\eta\frac{\partial^2 f}{\partial z_1\partial z_2}\Big)^2 +f(z_1+c_1,z_2+c_2)^2=e^{g(z_1,z_2)} $$ and $$ \Big(\frac{\partial^2 f}{\partial z_1^2}+\delta\frac{\partial^2 f}{\partial z_2^2} +\eta\frac{\partial^2 f}{\partial z_1\partial z_2}\Big)^2+(f(z_1+c_1,z_2+c_2) -f(z_1,z_2))^2=e^{g(z)}, $$ where \(\delta,\eta\in\mathbb{C}\) and \(g(z_1,z_2)\) is a polynomial in \(\mathbb{C}^2\). Our results improve the results of Liu and Dong [23] Liu et al. [24] and Liu and Yang [25] Several examples confirm that the form of transcendental entire solutions of finite order in our results are precise.

Submitted April 17, 2023. Published June 26 2023.
Math Subject Classifications: 30D35, 35M30, 32W50, 39A45.
Key Words: Functions of several complex variables; Fermat-type equations; entire solutions; Nevanlinna theory.
DOI: https://doi.org/10.58997/ejde.2023.43

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Hong Yan Xu
School of Arts and Sciences
Suqian University
Suqian, Jiangsu 223800, China
email: xhyhhh@126.com
  Goutam Haldar
Department of Mathematics
Malda College - 732101
West Bengal, India
email: goutamiit1986@gmail.com, goutamiitm@gmail.com

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