Three results on transcendental meromorphic solutions of certain nonlinear differential equations

TL;DR

This paper investigates the properties of transcendental meromorphic solutions to specific nonlinear differential equations, providing new results that extend previous research and partially answer open questions in the field.

Contribution

It offers new theorems on the nature of solutions to certain nonlinear differential equations involving differential polynomials and exponential functions, improving upon prior results.

Findings

Characterization of solutions when coefficients are small or rational functions

Extension of previous results to more general equations

Partial answers to open questions in the literature

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 (Complex Var. Elliptic Equ. 2015, 60(6): 748--756), and Rong-Xu (Mathematics 2019, 7, 539), etc., which partially answer a question proposed by Li (J. Math. Anal. Appl. 2011, 375: 310--319).