The Yang-Hua theorems in several complex variables

TL;DR

This work extends the Yang–Hua theorems from one complex variable to several variables by employing Nevanlinna theory in $\mathbb{C}^m$ to study the nonlinear PDE $f^n \partial_u(f)\, g^n \partial_u(g)=1$ for non-constant meromorphic pairs $(f,g)$. It proves that all nontrivial solutions are of exponential type, namely $f=e^{\alpha}$, $g=e^{\beta}$ with $\partial_u(\alpha)=c$, $\partial_u(\beta)=-c$ and $c^2 e^{(n+1)(\alpha+\beta)}=-1$ (and analogous form in the meromorphic case for $n\ge 6$), and establishes uniqueness results under CM-sharing of nonlinear differential polynomials, showing that either $f$ and $g$ differ by an $(n+1)$-st root of unity or assume the exponential form with the prescribed directional derivatives. The paper furnishes auxiliary lemmas to support growth and value-sharing arguments, provides detailed proofs (2.2, 2.3) and extensions to higher dimensions (3.2, 3.3), and ends with an application deriving explicit entire solutions to related nonlinear PDEs, highlighting when linearity of $\beta$ along $u$ is essential. These results advance the understanding of value distribution and uniqueness in several complex variables and connect to potential physical interpretations via exponential-solution structures.

Abstract

In this paper, we investigate meromorphic solutions in $\mathbb{C}^m$ of the nonlinear differential equation \[\displaystyle f^n\partial_u(f)g^n\partial_u(g)=1,\] where $\partial_u(f)=\sum_{j=1}^mu_j\partial_j(f)$ and $\sum_{j=1}^m u_j\neq 0$. Our results extend those of Yang and Hua [{\sc C. C. Yang} and {\sc X. H. Hua}, Uniqueness and value sharing of meromorphic functions, \textit{Ann. Acad. Sci. Fenn. Math.}, \textbf{22} (1997), 395-406.] to the framework of several complex variables. Moreover, we establish new uniqueness theorems that further generalize their conclusions to higher dimensions. As an application, explicit solutions of certain nonlinear partial differential equations in several variables are derived, and their physical interpretations are summarized in tabular form.

Theorems & Definitions (33)

• Theorem 2.1
• Theorem 2.2
• Corollary 2.1
• Corollary 2.2
• Lemma 2.1
• Lemma 2.2
• proof
• proof
• Theorem 3.1
• Theorem 3.2