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On Meromorphic Solutions of Nonlinear Complex Differential Equations

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Abstract

By utilizing Nevanlinna theory of meromorphic functions, we characterize meromorphic solutions of the following nonlinear differential equation of the form

$${f^n}{f^\prime } + P(z,f,{f^\prime }, \ldots ,{f^{(t)}}) = {P_1}{e^{{\alpha _1}z}} + {P_2}{e^{{\alpha _2}z}} + \cdots + {P_m}{e^{{\alpha _m}z}},$$

where n ≥ 3, t ≥ 0 and m ≥ 1 are integers, n ≥ m, P(z, f, f′, …, f(t)) is a differential polynomial in f (z) of degree d ≤ n with small functions of f (z) as its coefficients, and αj, Pj (j = 1, 2, …, m) are nonzero constants such that ∣α1∣ > ∣α2∣ > … > ∣αm∣. Also we provide the concrete forms of the solutions of the equation above, and present some examples illustrating the sharpness of our results.

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Acknowledgement

The authors would like to thank the referee for thorough reviewing with useful suggestions and valuable comments to the paper.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Fujian Normal University, Fuzhou, 350117, P. R. China

    J.-F. Chen & Y.-Y. Feng

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  1. J.-F. Chen
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  2. Y.-Y. Feng
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Correspondence to J.-F. Chen.

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Project supported by the Natural Science Foundation of Fujian Province, China (Grant No. 2021J01651).

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Chen, JF., Feng, YY. On Meromorphic Solutions of Nonlinear Complex Differential Equations. Anal Math 49, 699–719 (2023). https://doi.org/10.1007/s10476-023-0225-3

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  • DOI: https://doi.org/10.1007/s10476-023-0225-3

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