On the Exact Forms of Meromorphic Solutions of Certain Non-linear Delay-Differential Equations

In this paper, we consider transcendental meromorphic solutions f of finite order $$\rho $$ and few poles in the sense that $$S_{\lambda }(r,f):=O(r^{\lambda +\varepsilon })$$ , where $$\lambda <\rho $$ and $$\varepsilon \in (0,\rho -\lambda )$$ , of the delay-differential equation $$\begin{aligned} f^n+L(z,f)=p_1(z)e^{\alpha _{1}(z)}+p_2(z)e^{\alpha _{2}(z)}, \end{aligned}$$ where $$n\ge 2$$ is an integer, L(z, f) is a linear delay-differential polynomial with coefficients of growth $$S_{\lambda }(r,f)$$ . In addition, $$p_1(z)$$ , $$p_2(z)$$ are non-zero small functions of f in the sense $$S_{\lambda }(r,f)$$ and $$\alpha _{1}(z)$$ , $$\alpha _{2}(z)$$ are non-constant polynomials. In fact, we give the exact forms of all possible meromorphic solutions of the above equation and we improve some recent results.