On Meromorphic Solutions of Nonlinear Complex Differential Equations

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 α_j, P_j (j = 1, 2, …, m) are nonzero constants such that ∣α₁∣ > ∣α₂∣ > … > ∣α_m∣. Also we provide the concrete forms of the solutions of the equation above, and present some examples illustrating the sharpness of our results.