标题： 关于$\mathbb{C}^m$中某种非线性代数偏微分方程的研究 显示英文标题

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

摘要： 在论文中，利用亚纯函数的Nevanlinna值分布理论在$\mathbb{C}^m$中，我们研究了以下代数偏微分方程在$\mathbb{C}^m$中整解$f$的存在性，其中\[f^n(z)+P_d(f(z))=p(z)e^{\langle \alpha,\ol z\rangle},\]，这里$P_d(f)$是在$f$中的次数为$d \leq n-2$的代数微分多项式，$n \geq 3$是一个整数，$p$是一个非零多项式，$\alpha=(\alpha_{1},\ldots,\alpha_{m})\neq (0,\ldots,0)$和$\langle \alpha,\ol z\rangle=\sideset{}{_{k=1}^{m}}{\sum}\alpha_{1k} z_k$。 同时在论文中，我们研究了以下代数偏微分方程在$\mathbb{C}^m$中整体解$f$的不存在性\[f^n(z)+P_d(f(z))=p_1(z)e^{\langle \alpha, \ol z\rangle}+p_2(z)e^{\langle \beta, \ol z\rangle},\]，其中$P_d(f)$是次数为$d \leq n-3$的代数微分多项式，$n \geq 4$是一个整数，$p_1$和$p_2$是两个非零多项式，$\alpha=(\alpha_{11},\ldots,\alpha_{1m})\neq (0,\ldots,0)$和$\beta=(\alpha_{21},\ldots,\alpha_{2m})\neq (0,\ldots,0)$满足$\alpha_{1i}\neq 0$，$\alpha_{2i}\neq 0$和$\alpha_{1i}/\alpha_{2i}\not\in\mathbb{Q}$对所有$i\in\mathbb{Z}[1,m]$成立。 我们的研究结果将Li和Yang（J. Math. Anal. Appl., 320 (2006) 827-835）以及Zhang和Liao（Taiwanese J. Math., 15 (5) (2011), 2145-2157）的结果扩展并改进到更高维空间。 显示英文摘要

摘要： 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.