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

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.