Existence of transcendental meromorphic solutions on some types of nonlinear differential equations

Bull. Korean Math. Soc. 2020 Vol. 57, No. 4, 991-1002 https://doi.org/10.4134/BKMS.b190696 Published online December 5, 2019 Printed July 31, 2020

Peichu Hu, Manli Liu Shandong University; Shandong University

Abstract : We show that when $n>m$, the following delay differential equation \begin{equation*} f^n(z)f'(z)+p(z)(f(z+c)-f(z))^m=r(z)e^{q(z)} \end{equation*} of rational coefficients $p,r$ doesn't admit any transcendental entire solutions $f(z)$ of finite order. Furthermore, we study the conditions of $\alpha_1, \alpha_2$ that ensure existence of transcendental meromorphic solutions of the equation \begin{equation*} f^n(z) + f^{n-2}(z)f'(z)+ P_d(z,f)=p_1(z)e^{\alpha_1( z)}+p_2(z)e^{\alpha_2 (z)}. \end{equation*} These results have improved some known theorems obtained most recently by other authors.

Supported by : This work was partially supported by NSFC of Shandong (No.ZR2018MA014), PCSIRT (No.IRT1264) and the Fundamental Research Funds of Shandong University (No.2017JC019)
The authors would like to thank the referees for a careful reading of the manuscript and valuable comments and to China Scholarship Council (State Scholarship Fund No. 201906220075) for its nancial support.