Journal of the
Korean Mathematical Society JKMS

In this paper, we utilize Nevanlinna theory to study the existence and forms of solutions for quadratic trinomial complex partial differential-difference equations of the form $ aF^2+2\omega FG+bG^2=\exp(g) $, where $ ab\neq 0, \omega\in\mathbb{C} $ with $ \omega^2\neq 0, ab $ and $g$ is a polynomial in $\mathbb{C}^n$. In order to achieve a comprehensive and thorough analysis, we study the characteristics of solutions in two specific cases: one when $ \omega^2\neq 0, ab $ and the other when $ \omega=0 $. Because polynomials in several complex variables may exhibit periodic behavior, a property that differs from polynomials in single complex variables, our study of finding solutions of equations in $\mathbb{C}^n$ is significant. The main results of the paper improved several known results in $\mathbb{C}^n$ for $n\geq 2$. Additionally, the corollaries generalize results of Xu \emph{et al.} [Rocky Mountain J. Math. \textbf{52}(6) (2022), 2169--2187] for trinomial equations with arbitrary coefficients in $\mathbb{C}^n$. Finally, we provide examples that endorse the validity of the conclusions drawn from the main results and their related remarks.