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

Bull. Korean Math. Soc. Published online March 3, 2020

Jun-Fan Chen and Gui Lian
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, $\alpha_{1}(z)$, $\alpha_{2}(z)$, $\alpha_{3}(z)$ 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.