Expressions of meromorphic solutions of a certain type of nonlinear complex differential equations

Bull. Korean Math. Soc. 2020 Vol. 57, No. 4, 1061-1073 https://doi.org/10.4134/BKMS.b190744 Published online March 3, 2020 Printed July 31, 2020

Jun-Fan Chen, Gui Lian Fujian Province University; Fujian Normal University

Abstract : In this paper, the expressions of meromorphic solutions of the following nonlinear complex differential equation of the form $$f^{n}+Q_{d}(z,f)=\sum_{i=1}^{3}p_{i}(z)e^{\alpha_{i}(z)}$$ are studied by using Nevanlinna theory, where $n\geq5$ is an integer, $Q_{d}(z,f)$ is a differential polynomial in $f$ of degree $d\leq n-4$~with rational functions as its coefficients, $p_{1}(z)$, $p_{2}(z)$, $p_{3}(z)$~are non-vanishing rational functions, and $\alpha_{1}(z)$, $\alpha_{2}(z)$, $\alpha_{3}(z)$ are nonconstant polynomials such that $\alpha_{1}'(z)$, $\alpha_{2}'(z)$, $\alpha_{3}'(z)$ are distinct each other. Moreover, examples are given to illustrate the accuracy of the condition.

Supported by : Project supported by the National Natural Science Foundation of China (Grant No. 11301076), and the Natural Science Foundation of Fujian Province, China (Grant No. 2018J01658), and Key Laboratory of Applied Mathematics of Fujian Province University (Putian University) (Grant No. SX201801)