Meromorphic solutions of one certain type of non-linear complex differential equation

Abstract

In this paper, we are mainly concerned with one certain type of non-linear complex differential equation $f^n(z)+P_d(z,f)=p_1{e}^{{\alpha }_{1}z}+p_2{e}^{{\alpha }_{2}z},$ where $p_1,p_2,{\alpha }_1,{\alpha }_2$ are non-zero constants and $P_d(z,f)$ is a differential polynomial in f of degree d at most n−1. For this kind of equation, if it admits a meromorphic solution such that $N(r,f)=S(r,f)$, then firstly we give a positive answer to Li's question with a weaker restriction on ${\alpha }_1$ and ${\alpha }_2$ than Liu's when $\frac{{\alpha }_2}{{\alpha }_1}>0$; secondly, we show ${\alpha }_1+{\alpha }_2=0$ when $\frac{{\alpha }_2}{{\alpha }_1}<0$.

COMMUNICATED BY:

• J. Virtanen

Keywords:

• Meromorphic function
• differential equation
• Nevanlinna theory

AMS Subject Classifications:

• 30D35
• 34M10

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research work is supported by the National Natural Science Foundation of China [grant numbers 11601507, 11671191].