SOLUTIONS FOR QUADRATIC TRINOMIAL PARTIAL DIFFERENTIAL-DIFFERENCE EQUATIONS IN Cn
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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.
Abstract
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.
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