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 \[f^n(z)+P_d(f(z))=p(z)e^{\langle α,\ol z\rangle},\] where $P_d(f)$ is an algebraic differential polynomial in $f$ of degree $d\le n-2$, $n\ge 3$ is an integer, $p$ is a non-zero polynomial, $\alpha =({\alpha }_1,\dots ,{\alpha }_m)\ne (0,\dots ,0)$ and $\langle α,\ol z\rangle=\sideset{}{_{k=1}^{m}}{\sum}α_{1k} z_k$. 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 \[f^n(z)+P_d(f(z))=p_1(z)e^{\langle α, \ol z\rangle}+p_2(z)e^{\langle β, \ol z\rangle},\] where $P_d(f)$ is an algebraic differential polynomial of degree $d\le n-3$, $n\ge 4$ is an integer, $p_1$ and $p_2$ are two non-zero polynomials, $\alpha =({\alpha }_{11},\dots ,{\alpha }_{1m})\ne (0,\dots ,0)$ and $\beta =({\alpha }_{21},\dots ,{\alpha }_{2m})\ne (0,\dots ,0)$ such that ${\alpha }_{1i}\ne 0$, ${\alpha }_{2i}\ne 0$ and ${\alpha }_{1i}/{\alpha }_{2i}\notin \mathbb{Q}$ 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.

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