Bulletin of the
Korean Mathematical Society
BKMS

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

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Bull. Korean Math. Soc. 2020; 57(5): 1095-1113

Online first article November 26, 2019      Printed September 30, 2020

https://doi.org/10.4134/BKMS.b190535

Copyright © The Korean Mathematical Society.

Some results on meromorphic solutions of certain nonlinear differential equations

Nan Li, Lianzhong Yang

Qilu Normal University; Shandong University

Abstract

In this paper, we investigate the transcendental meromorphic solutions for the nonlinear differential equations $f^{n}f^{(k)}+Q_{d_{*}}(z,f)=R(z)e^{\alpha(z)}$ and $f^{n}f^{(k)}+Q_{d}(z,f)=p_{1}(z)e^{\alpha_{1}(z)}+p_{2}(z)e^{\alpha_{2}(z)}$, where $Q_{d_{*}}(z,f)$ and $Q_{d}(z,f)$ are differential polynomials in $f$ with small functions as coefficients, of degree $d_{*}\, (\leq n-1)$ and $d\, (\leq n-2)$ respectively, $R,\, p_{1},\, p_{2}$ are non-vanishing small functions of $f$, and $\alpha,\, \alpha_{1},\, \alpha_{2}$ are nonconstant entire functions. In particular, we give out the conditions for ensuring the existence of these kinds of meromorphic solutions and their possible forms of the above equations.

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), and the Fund of Doctoral Program Research of University of Jinan (No. XBS1630)