Entire solutions of Fermat-Type Partial Differential-Difference equations in higher dimensions

Abstract

The purpose of this paper is to investigate the finite-order entire solutions of the following Fermat-type partial differential-difference equations: \begin{align*} (P_1(z)f(z+c))^{m_1}+\left(P_2(z)f(z)+\sideset{}{_{j=1}^m}{\sum} Q_j(z)\partial_{z_j}(f(z))\right)^{m_2}=P(z), \end{align*} \begin{align*} (P_1(z)f(z))^{m_1}+\left(P_2(z)f(z+c)+\sideset{}{_{j=1}^m}{\sum} Q_j(z)\partial_{z_j}(f(z))\right)^{m_2}=P(z) \end{align*} and \begin{align*} \left(\sideset{}{_{j=1}^m}{\sum} Q_j(z)\partial_{z_j}(f(z))\right)^{m_1}+\left(P_1(z)f(z+c)+P_2(z)f(z)\right)^{m_2}=P(z), \end{align*} in $\mathbb{C}^m$, where $m_1,m_2\in\mathbb{N}$ such that $m_1+m_2>2$, $c=(c_1,c_2,\ldots,c_m)\in\mathbb{C}^m$, $P(\not\equiv 0)$, $P_1$, $P_2$ and $Q_i\;(i=1,2,\ldots,m)$ are polynomials in $\mathbb{C}^m$ such that $P(z)$ is irreducible in $\mathbb{C}^m$. The results established in this paper generalize those of Xu et al. \cite{Xu-Luo-Cao-2025} from $\mathbb{C}$ to $\mathbb{C}^m$.