Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2021; 58(4): 795-814

Online first article December 23, 2020      Printed July 31, 2021

https://doi.org/10.4134/BKMS.b200265

Copyright © The Korean Mathematical Society.

Three results on transcendental meromorphic solutions of certain nonlinear differential equations

Nan Li, Lianzhong Yang

Qilu Normal University; Shandong University

Abstract

In this paper, we study the transcendental meromorphic solutions for the nonlinear differential equations: $f^{n}+P(f)=R(z)e^{\alpha(z)}$ and $f^{n}+P_{*}(f)=p_{1}(z)e^{\alpha_{1}(z)}+p_{2}(z)e^{\alpha_{2}(z)}$ in the complex plane, where $P(f)$ and $P_{*}(f)$ are differential polynomials in $f$ of degree $n-1$ with coefficients being small functions and rational functions respectively, $R$ is a non-vanishing small function of $f$, $\alpha$ is a nonconstant entire function, $p_{1}, p_{2}$ are non-vanishing rational functions, and $\alpha_{1}, \alpha_{2}$ are nonconstant polynomials. Particularly, we consider the solutions of the second equation when $p_{1}, p_{2}$ are nonzero constants, and $\deg \alpha_{1}=\deg \alpha_{2}=1$. Our results are improvements and complements of Liao (\cite{LIAO2015}), and Rong-Xu (\cite{RongXu2019}), etc., which partially answer a question proposed by Li (\cite{LiP2011}).

Keywords: Meromorphic functions, nonlinear differential equations, small functions, differential polynomials

MSC numbers: Primary 34M05, 30D30, 30D35

Supported by: This work was supported by NNSF of China (No. 11801215 \& No. 11626112 \& No. 11371225), the NSF of Shandong Province, P. R. China (No. ZR2016AQ20 \& No. ZR2018MA021).