Meromorphic solutions of a certain type of nonlinear differential equation

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.