Meromorphic solutions of a certain type of nonlinear differential equation

TL;DR

This paper systematically investigates the existence and forms of meromorphic solutions to a class of nonlinear differential equations involving polynomial and rational functions, improving previous criteria and explicit solution descriptions.

Contribution

It provides comprehensive existence conditions and explicit forms for solutions to a generalized nonlinear differential equation, advancing the understanding of such equations in complex analysis.

Findings

Established new existence criteria for solutions.

Derived explicit analytic forms of solutions.

Improved upon previous results by Tang, Liao, Han, and Lü.

Abstract

Our paper focuses on investigating the existence and possible forms of solutions to the nonlinear differential equation \beas f^m+\big(Rf^{(k)}\big)^n=Qe^{\alpha},\eeas where where $k$, $m$ and $n$ are three positive integers, $Q$ and $R$ are non-zero rational functions, and $\alpha$ is a polynomial. We examine this equation systematically for all positive integral values of $k$, $m$ and $n$, providing comprehensive conditions under which entire or meromorphic solutions may exist. Our results offer substantial improvements over those of Tang and Liao \cite{TL1} and Han and L\"{u} \cite{HL1}, particularly with respect to the existence criteria and the explicit analytic forms of admissible solutions. These findings contribute meaningfully to the broader study of nonlinear differential polynomials and Fermat-type functional equations in complex analysis.