A study on the existence of solutions for the class of nonlinear partial differential-difference equations in several complex variables

Abstract

We explore the non-existence and existence of transcendental entire solutions of a certain type of partial differential-difference equations in $\mathbb{C}^{n}$ of the type defined by Fermat with finite-order constraint. We also study the existence as well as the form of transcendental entire solutions for a particular class of Fermat's type partial differential-difference equations in $\mathbb{C}^{2}$ with finite-order constraint. The results provided as solutions for these problems have substantially enhanced the theorem that was previously suggested by Xu, Cao, Liu, Wang, Zhang, Zheng. In contrast to earlier articles, this one uses a different technique. By using several examples, it is shown that there are certain conditions and types of transcendental entire solutions of finite order of such equations.