On Solutions of Differential-Difference Equations in $\mathbb{C}^n$ (Analysis in Theory and Applications, 2025)
In this paper, we mainly explore the existence of entire solutions of thequadratic trinomial partial differential-difference equation $$af^2(z)+2ω f(z)(a_0f(z)+L^{k+s}_{1,2}(f(z)))+b(a_0f(z)+L^{k+s}_{1,2}(f(z)))^2=e^{g(z)}$$by utilizing Nevanlinna’s theory in several complex variables, where $g(z)$ is entirefunctions in $\mathbb{C}^n,$ $ω e 0$ and $a, b, ω ∈ \mathbb{C}.$ Furthermore, we get the exact froms of solutions of the above differential-difference equation when $ω = 0.$ Our results are generalizations of previous results. In addition, some examples are given to illustrate theaccuracy of the results.
Keywords: Differential-difference equations; Nevanlinna theory; finite order; entire solutions.