一类复偏微分方程组的超越整函数解

Transcendental Entire Solutions to a Class of Complex Partial Differential Equation Systems

摘要: 本文旨在应用多变量Nevanlinna值分布理论，进一步研究一类乘积型一阶非线性复偏微分方程组$\{\begin{array}{l}{\left(a{\mu }_{z_1}+b{v}_{z_2}\right)}^{p_1}{\left(c{\mu }_{z_2}+d{v}_{z_1}\right)}^{p_2}=e^g\\ {\left(a{\mu }_{z_2}+b{v}_{z_1}\right)}^{p_3}{\left(c{\mu }_{z_1}+d{v}_{z_2}\right)}^{p_4}=e^g\end{array}$ 的超越整函数解的存在性条件与形式，其中$a,b,c,d\in ℂ-\{0\}$ ，$p_1,p_2,p_3,p_4$ 为非零实常数，$g$ 为${ℂ}^2$ 中的非常数多项式。所得结论是对徐洪焱、徐宜会等人先前结论的进一步推广和改进，同时给出例子说明求得结果是精确的。

Abstract: This article is devoted to describe the transcendental entire solutions to a class of the first order nonlinear partial differential equations (PDEs) with product type $\{\begin{array}{l}{\left(a{\mu }_{z_1}+b{v}_{z_2}\right)}^{p_1}{\left(c{\mu }_{z_2}+d{v}_{z_1}\right)}^{p_2}=e^g\\ {\left(a{\mu }_{z_2}+b{v}_{z_1}\right)}^{p_3}{\left(c{\mu }_{z_1}+d{v}_{z_2}\right)}^{p_4}=e^g\end{array}$ by using Nevanlinna theory in several complex variables, where $a,b,c,d\in ℂ-\{0\}$ , $p_1,p_2,p_3,p_4$ are non-zero real constants and $g$ is a non-constant polynomial in ${ℂ}^2$ . Our results are the generalizations and improvements of the previous theorem given by Hongyan Xu, Yihui Xu, etc. Furthermore, some examples have been exhibited to show that our results are precise.