Entire solutions of several quadratic binomial and trinomial partial differential-difference equations in $$ {\mathbb {C}}^2 $$

Abstract

In this paper, utilizing Nevanlinna theory, we study existence and forms of the entire solutions f of the quadratic trinomial-type partial differentialdifference equations in C n

Solutions of Fermat-Type Partial Differential–Difference Equations in $$\pmb {{\mathbb {C}}^n}$$

“…In 1995, Khavinson pointed out that in C 2 , the entire solution of the Fermat type partial differential equations f 2 z 1 + f 2 z 2 = 1 must necessarily be linear. After the development of the difference Nevanlinna theory in several complex variables, specially the difference version of logarithmic derivative lemma (see [1,3,17]), many mathematicians have started to study the existence and the precise form of entire and meromorphic solutions of different variants of Fermat type difference and partial differential difference equations, and obtained very remarkable and interesting results (see [12,13,[38][39][40][41]43]).…”

Entire solutions to Fermat-type difference and partial differential-difference equations in C^n

“…where L(z) = α 1 z 1 + α 2 z 2 , α 1 , α 2 , A ∈ C, Φ(t) is a polynomial in t := c 2 z 1 − c 1 z 2 in C, and L(z) satisfies As far as our knowledge is concerned, although there are some remarkable results about the existence and forms of transcendental entire solutions of Fermat-type difference and partial differential-difference equations in several complex variables (see [17,49,50,51,55,11,12,14,45,13,47,48,46]), the number of results about the solutions of the system of Fermat-type equations in the literature (see [52]) are scanty. We would like to discuss some of these results which are relevant to the content of this article.…”

Solutions of complex nonlinear functional equations including second order partial differential and difference in C^2

“…Hereinafter, we denote by z + w = (z 1 + w 1 , z 2 + w 2 ) for any z = (z 1 , z 2 ), w = (w 1 , w 2 ) ∈ C 2 . The study of several characteristics of the solutions to partial differential equations in several complex variables is an important topic; see [1,2,3,8,9,12,14,18,26,34,35,36,37,38]). It was Saleebly, who in 1999, first studied the existence and form of entire and meromorphic solutions of Fermat-type partial differential equations (see [30,31,32]).…”