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

Bull. Korean Math. Soc. Published online December 5, 2019

Peichu Hu and Manli Liu
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.