Meromorphic solutions of $ f^{n}+P_{d}(f) = p_{1}e^{\alpha_{1}z}+p_{2}e^{\alpha_{2}z}+p_{3}e^{\alpha_{3}z} $

Linkui Gao; Junyang Gao; Linkui Gao; Junyang Gao

doi:10.3934/math.20221007

AIMS Mathematics

2022, Volume 7, Issue 10: 18297-18310. doi: 10.3934/math.20221007

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Research article

Meromorphic solutions of $ f^{n}+P_{d}(f) = p_{1}e^{\alpha_{1}z}+p_{2}e^{\alpha_{2}z}+p_{3}e^{\alpha_{3}z} $

Linkui Gao ,
Junyang Gao ^,

School of Science, China University of Mining and Technology, Beijing 100083, China

Received: 08 June 2022 Revised: 29 July 2022 Accepted: 05 August 2022 Published: 12 August 2022
MSC : 30D35, 34M05

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.
- Nevanlinna theory,
- complex differential equations,
- exponential sums,
- meromorphic solutions
Citation: Linkui Gao, Junyang Gao. Meromorphic solutions of $ f^{n}+P_{d}(f) = p_{1}e^{\alpha_{1}z}+p_{2}e^{\alpha_{2}z}+p_{3}e^{\alpha_{3}z} $[J]. AIMS Mathematics, 2022, 7(10): 18297-18310. doi: 10.3934/math.20221007

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Abstract

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.

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