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

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(α\dfrac{\partial f(z)}{\partial z_i} + β\dfrac{\partial f(z)}{\partial z_j}\right)^2 + 2 ω\left(α\dfrac{\partial f(z)}{\partial z_i} + β\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(α\dfrac{\partial f(z)}{\partial z_i} + β\dfrac{\partial f(z)}{\partial z_j}\right)^2 & + 2 ω\left(α\dfrac{\partial f(z)}{\partial z_i} + β\dfrac{\partial f(z)}{\partial z_j}\right) Δ_cf(z) + b [Δ_cf(z)]^2 = e^{g(z)}, \end{align*} where $a,\omega ,b\in \mathbb{C}$, $g$ is a polynomial in ${\mathbb{C}}^n$ and ${\Delta }_{c}f(z)=f(z+c)-f(z)$. The main results of the paper improve several existence results in ${\mathbb{C}}^n$ for integer $n\ge 2$ and $1\le i<j\le n$ 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.

20. arXiv admin note: text overlap with arXiv:2307.05549