Entire solutions of product type nonlinear partial differential equations in $\mathbb{C}^n$

Abstract In this paper, we describe finite order entire solutions of nonlinear partial differential equation \begin{equation*} \prod _{i=1}^n\left (a_{i 1} u_{z_1}+a_{i 2} u_{z_2}+\cdots +a_{i n} u_{z_n}\right )=1 \end{equation*} with the eikonal equation as a prototype. By the elementary row transformation of a matrix, we offer an affirmative answer to a question of Xu-Liu-Xuan in Xu et al. ( J. Math. Anal. Appl. 543 (2025), ID 128885, 21 p.).