Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 43, pp. 1-18.

Title: Solutions of complex nonlinear functional equations including second order partial
differential and difference in C^2
       
Authors: Hong Yan Xu (Suqian Univ., Suqian, Jiangsu, China)
         Goutam Haldar (Malda College, West Bengal, India)

Abstract: 
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.