Entire solutions for several second-order partial differential-difference equations of Fermat type with two complex variables

Hong Yan Xu,Da Wei Meng,Sanyang Liu,Hua Wang

doi:10.1186/s13662-020-03201-y

Abstract

This paper is concerned with description of the existence and the forms of entire solutions of several second-order partial differential-difference equations with more general forms of Fermat type. By utilizing the Nevanlinna theory of meromorphic functions in several complex variables we obtain some results on the forms of entire solutions for these equations, which are some extensions and generalizations of the previous theorems given by Xu and Cao (Mediterr. J. Math. 15:1–14, 2018; Mediterr. J. Math. 17:1–4, 2020) and Liu et al. (J. Math. Anal. Appl. 359:384–393, 2009; Electron. J. Differ. Equ. 2013:59–110, 2013; Arch. Math. 99:147–155, 2012). Moreover, by some examples we show the existence of transcendental entire solutions with finite order of such equations.

Highlights

The main purpose of this paper is investigation of the existence and the forms of transcendental entire solutions with finite order of second-order differential difference equations∂2f (z1, z2) ∂ z12+ f (z1 + c1, z2 + c2)2 = eg(z1,z2) and+ f (z1 + c1, z2 + c2) – f (z1, z2) 2 = eg(z1,z2), where g(z1, z2) is a polynomial in C2
Gross [6] discussed the existence of solutions of equation (1.1) and showed that the entire solutions are f = cos a(z), g = sin a(z) for m = n = 2, where a(z) is an entire function
2 Results and some examples In view of the above questions, this paper is concerned with description of entire solutions for several second-order partial differential-difference equations of Fermat type of more general form

Summary

Introduction

Thereom A (See [8, Theorem 1.1]) The meromorphic solutions f of the differential equation f n(z) + f n(z) = eαz+β must be entire functions, and the following statements hold: (A) Any transcendental entire solution with finite order of the partial differential-difference equation 2 Results and some examples In view of the above questions, this paper is concerned with description of entire solutions for several second-order partial differential-difference equations of Fermat type of more general form.

Objectives

Results

Conclusion