某类非线性微分方程超越亚纯解的进一步结果

本文研究非线性微分方程f~n+Q_d(z,f)=P_1(z)e~(α_1(z))+p_2(z)e~(α_2(z))超越亚纯解的存在性和形式,其中n≥4是整数,Q_d(z,f)是关于f的次数d≤n-3且系数为有理函数的微分多项式,p_1,p_2是非零的有理函数,α_1,α_2是非常数的多项式.运用Nevanlinna值分布理论,能够得到该方程存在超越亚纯解时p_1,p_2,α_1及α_2所满足的条件.特别地,还考虑了当Q_d(z,f)=a(z)ff'且n=4时方程的超越亚纯解的存在性和形式,其中a(z)是一个非零的有理函数.

Further Results on Transcendental Meromorphic Solutions of some Certain Types of Nonlinear Differential Equations

We shall deal with the existence and the form of transcendental meromorphic solutions of nonlinear differential equation of the form fⁿ+Q_d(z, f)=p₁(z)e^α₁(z)+ p₂(z)e^α₂(z), where n ≥ 4 is an integer, Q_d(z, f) is a differential polynomial in f of degree d ≤ n-3 with rational functions as its coefficients, p₁, p₂ are non-vanishing rational functions and α₁, α₂ are nonconstant polynomials. By utilizing Nevanlinna value distribution theory, we can derive conditions concerning the terms p₁, p₂, α₁ and α₂ that are necessary for the existence and the form of a transcendental meromorphic solution of the equation. In particular, we also consider the existence and the form of transcendental meromorphic solution of the equation when Q_d(z, f)=a(z)ff' with n=4, where a(z) is a non-vanishing rational function.