Study on the certain type of nonlinear algebraic partial differential equation in $\mathbb{C}^m$

TL;DR

This paper investigates the existence and non-existence of entire solutions to certain nonlinear algebraic partial differential equations in complex multi-dimensional space using Nevanlinna theory, extending previous results to higher dimensions.

Contribution

It extends and improves prior results on nonlinear algebraic PDEs to higher dimensions using Nevanlinna theory, providing new existence and non-existence criteria.

Findings

Established conditions for the existence of solutions.

Proved non-existence results under specific parameter constraints.

Extended previous two-dimensional results to higher dimensions.

Abstract

In the paper, using Nevanlinna's value distribution theory of meromorphic functions in $\mathbb{C}^m$, we study for the existence of entire solutions $f$ in $\mathbb{C}^m$ of the following algebraic partial differential equation \[f^n(z)+P_d(f(z))=p(z)e^{\langle \alpha,\ol z\rangle},\] where $P_d(f)$ is an algebraic differential polynomial in $f$ of degree $d \leq n-2$, $n \geq 3$ is an integer, $p$ is a non-zero polynomial, $\alpha=(\alpha_{1},\ldots,\alpha_{m})\neq (0,\ldots,0)$ and $\langle \alpha,\ol z\rangle=\sideset{}{_{k=1}^{m}}{\sum}\alpha_{1k} z_k$. Also in the paper, we study for the non-existence of entire solutions $f$ in $\mathbb{C}^m$ of the following algebraic partial differential equation \[f^n(z)+P_d(f(z))=p_1(z)e^{\langle \alpha, \ol z\rangle}+p_2(z)e^{\langle \beta, \ol z\rangle},\] where $P_d(f)$ is an algebraic differential polynomial of degree $d \leq n-3$, $n \geq 4$ is an integer, $p_1$ and $p_2$ are two non-zero polynomials, $\alpha=(\alpha_{11},\ldots,\alpha_{1m})\neq (0,\ldots,0)$ and $\beta=(\alpha_{21},\ldots,\alpha_{2m})\neq (0,\ldots,0)$ such that $\alpha_{1i}\neq 0$, $\alpha_{2i}\neq 0$ and $\alpha_{1i}/\alpha_{2i}\not\in\mathbb{Q}$ for all $i\in\mathbb{Z}[1,m]$. Our findings extend and improve the results of Li and Yang (J. Math. Anal. Appl., 320 (2006) 827-835) and Zhang and Liao (Taiwanese J. Math., 15 (5) (2011), 2145-2157) into higher dimensions.