Solutions of Fermat-type partial differential-difference equations in ${\mathbb{C}}^n$

Abstract

For two meromorphic functions $f$ and $g$, the equation $f^m+g^m=1$ can be regarded as Fermat-type equations. Using Nevanlinna theory for meromorphic functions in several complex variables, the main purpose of this paper is to investigate the properties of the transcendental entire solutions of Fermat-type difference and partial differential-difference equations in ${\mathbb{C}}^n$. In addition, we find the precise form of the transcendental entire solutions in ${\mathbb{C}}^2$ with finite order of the Fermat-type partial differential-difference equation $${\left(\frac{\partial f(z_1,z_2)}{\partial z_1}\right)}^2+(f(z_1+c_1,z_2+c_2)-f(z_1,z_2){)}^2=1$$ and $$f^2(z_1,z_2)+P^2(z_1,z_2){\left(\frac{\partial f(z_1+c_1,z_2+c_2)}{\partial z_1}-\frac{\partial f(z_1,z_2)}{\partial z_1}\right)}^2=1,$$ where $P(z_1,z_2)$ is a polynomial in ${\mathbb{C}}^2$. Moreover, one of the main results of the paper significantly improved the result of Xu and Cao [Mediterr. J. Math. (2018) 15:227 , 1-14 and Mediterr. J. Math. (2020) 17:8, 1-4].