Transcendental entire solutions for several quadratic binomial and trinomial PDEs with constant coefficients

This article is concerned with the description of the existence of entire solutions of several quadratic binomial and trinomial partial differential equations (PDEs) with constant coefficients. We established a series of theorems on the forms of finite order transcendental entire solutions for such PDEs, which are some generalization and improvement of the previous theorems given by Cao, Liu, Xu and Yang (An Stiint Univ Al I Cuza Iasi Mat (NS) 1:317-325, 2016; Mediterr J Math 15:1-14, 2018; Mediterr J Math 17:1-4, 2020). Moreover, a series of examples are given to show that the existence conditions and the forms of transcendental entire solutions with finite order of such equations are precise.

Automated summary

This article discusses the existence of entire solutions of quadratic binomial and trinomial partial differential equations with constant coefficients. The authors established a series of theorems on the forms of finite order transcendental entire solutions for these equations, which generalize and improve upon previous theorems. Examples are provided to demonstrate the precise existence conditions and forms of transcendental entire solutions with finite order for such equations.