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Clunie lemma in several complex variables and application in PDEs

Wenjie Hao, Qingcai Zhang

Abstract

Two purposes will be shown in this paper. The first one is to extend the classic Tumura-Clunie type theorem for meromorphic functions of one complex variable to meromorphic functions of several complex variables by using Clunie lemma. The second one is to characterize entire solutions of certain partial differential equations in $\mathbb{C}^{m}$. Our results are extensions and generalizations of the previous theorems by Liao-Ye \cite{Liao-Ye} and Li \cite{Li11}.

Clunie lemma in several complex variables and application in PDEs

Abstract

Two purposes will be shown in this paper. The first one is to extend the classic Tumura-Clunie type theorem for meromorphic functions of one complex variable to meromorphic functions of several complex variables by using Clunie lemma. The second one is to characterize entire solutions of certain partial differential equations in . Our results are extensions and generalizations of the previous theorems by Liao-Ye \cite{Liao-Ye} and Li \cite{Li11}.
Paper Structure (6 sections, 14 theorems, 131 equations)

This paper contains 6 sections, 14 theorems, 131 equations.

Table of Contents

  1. Introduction and Main Results
  2. Some lemmas
  3. Proof of theorem 1.6
  4. Proof of theorem 1.8
  5. Data Availability
  6. Conflict of interests

Key Result

Theorem 1.1

(see Tumura9) Suppose that $f$ is a meromorphic function in the complex plane and has only a finite number of poles in the plane, and that $f$, $f^{(l)}$ have only a finite number of zeros for some $l\geq2$. Then where $P_{1}$, $P_{2}$, $P_{3}$, are polynomials. If further, $f$ and $f^{(l)}$ have no zeros, then $f(z)=e^{Az+B}$ or $f(z)=(Az+B)^{-n}$, where $A$, $B$ are constants such that $A\neq0$

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Example 1.1
  • Example 1.2
  • Theorem 1.7
  • Theorem 1.8
  • ...and 9 more