Entire Solutions for quadratic trinomial-type partial differential-difference equations in $ \mathbb{C}^n $

TL;DR

This paper investigates the existence and form of entire solutions to quadratic trinomial-type partial differential-difference equations in complex n-dimensional space using Nevanlinna theory, extending previous results and providing explicit examples.

Contribution

It provides new existence results for entire solutions of specific quadratic trinomial-type PDEs in several complex variables, extending prior work to higher dimensions and more general coefficients.

Findings

Established conditions for existence of solutions in $ \\mathbb{C}^n $

Extended previous results from $ \\mathbb{C}^2 $ to higher dimensions

Provided explicit examples validating the theoretical results

Abstract

In this paper, utilizing Nevanlinna theory, we study existence and forms of the entire solutions $ f $ of the quadratic trinomial-type partial differential-difference equations in $ \mathbb{C}^n $ \begin{align*} a\left(\alpha\dfrac{\partial f(z)}{\partial z_i} + \beta\dfrac{\partial f(z)}{\partial z_j}\right)^2 + 2 \omega \left(\alpha\dfrac{\partial f(z)}{\partial z_i} + \beta\dfrac{\partial f(z)}{\partial z_j}\right) f(z + c) + b f(z + c)^2 = e^{g(z)} \end{align*} and \begin{align*} a\left(\alpha\dfrac{\partial f(z)}{\partial z_i} + \beta\dfrac{\partial f(z)}{\partial z_j}\right)^2 & + 2 \omega \left(\alpha\dfrac{\partial f(z)}{\partial z_i} + \beta\dfrac{\partial f(z)}{\partial z_j}\right) \Delta_cf(z) + b [\Delta_cf(z)]^2 = e^{g(z)}, \end{align*} where $ a, \omega, b\in\mathbb{C} $, $ g $ is a polynomial in $ \mathbb{C}^n $ and $ \Delta_cf(z)=f(z+c)-f(z) $. The main results of the paper improve several existence results in $ \mathbb{C}^n $ for integer $ n\geq 2 $ and $ 1\leq i<j\leq n $ and their corollaries of the paper are an extension of the results of Xu \emph{et al. } for trinomial equation with arbitrary coefficient in $ \mathbb{C}^2 $. Moreover, examples are exhibited to validate the conclusion of the main results.