Meromorphic solutions of f n + P d ( f ) = p 1 e α 1 z + p 2 e α 2 z + p 3 e α 3 z " role="presentation"> f n + P d ( f ) = p 1 e α 1 z + p 2 e α 2 z + p 3 e α 3 z f n + P d ( f ) = p 1 e α 1 z + p 2 e α 2 z + p 3 e α 3 z f^{n}+P_{d}(f) = p_{1}e^{\alpha_{1}z}+p_{2}e^{\alpha_{2}z}+p_{3}e^{\alpha_{3}z}

AIMS Mathematics

article

Linkui Gao¹  Junyang Gao¹ 

[1] School of Science, China University of Mining and Technology

关键词: Nevanlinna theory;    complex differential equations;    exponential sums;    meromorphic solutions;   

DOI  :  10.3934/math.20221007

学科分类：地球科学(综合)

来源: AIMS Press

【 摘 要 】

By using Nevanlinna of the value distribution of meromorphic functions, we investigate the transcendental meromorphic solutions of the non-linear differential equation$ \begin{equation*} f^{n}+P_{d}(f) = p_{1}e^{\alpha_{1}z}+p_{2}e^{\alpha_{2}z}+p_{3}e^{\alpha_{3}z}, \end{equation*} $where $ P_{d}(f) $ is a differential polynomial in $ f $ of degree $ d(0\leq d\leq n-3) $ with small meromorphic coefficients and $ p_{i}, \alpha_{i}(i = 1, 2, 3) $ are nonzero constants. We show that the solutions of this type equation are exponential sums and they are in $ \Gamma_{0}\cup\Gamma_{1}\cup\Gamma_{3} $ which will be given in Section $ 1 $. Moreover, we give some examples to illustrate our results.

【 授权许可】

CC BY