Entire Solutions for quadratic trinomial-type partial differential-difference equations in $ \mathbb{C}^n $

In this paper, utilizing Nevanlinna theory, we study existence and forms of the entire solutions $f$ of the quadratic trinomial-type partial differential-difference equations in $\mathbb{C}^n$ \begin{align*} a\left(\alpha\dfrac{\partial f(z)}{\partial z_i} + \beta\dfrac{\partial f(z)}{\partial z_j}\right)^2 + 2 \omega \left(\alpha\dfrac{\partial f(z)}{\partial z_i} + \beta\dfrac{\partial f(z)}{\partial z_j}\right) f(z + c) + b f(z + c)^2 = e^{g(z)} \end{align*} and \begin{align*} a\left(\alpha\dfrac{\partial f(z)}{\partial z_i} + \beta\dfrac{\partial f(z)}{\partial z_j}\right)^2 & + 2 \omega \left(\alpha\dfrac{\partial f(z)}{\partial z_i} + \beta\dfrac{\partial f(z)}{\partial z_j}\right) \Delta_cf(z) + b [\Delta_cf(z)]^2 = e^{g(z)}, \end{align*} where $a, \omega, b\in\mathbb{C}$, $g$ is a polynomial in $\mathbb{C}^n$ and $\Delta_cf(z)=f(z+c)-f(z)$. The main results of the paper improve several existence results in $\mathbb{C}^n$ for integer $n\geq 2$ and $1\leq i<j\leq n$ and their corollaries of the paper are an extension of the results of Xu \emph{et al. } for trinomial equation with arbitrary coefficient in $\mathbb{C}^2$. Moreover, examples are exhibited to validate the conclusion of the main results.