On entire solutions for a class of product-type nonlinear PDEs in $\mathbb{C}^n$

TL;DR

This paper characterizes entire solutions to a class of nonlinear PDEs in complex space, providing explicit formulas and removing previous restrictions, thus advancing understanding of such equations.

Contribution

It introduces a novel method to explicitly solve a class of product-type nonlinear PDEs without finite order restrictions.

Findings

Explicit solutions for the PDE are derived.

The method removes the finite order restriction.

Two open questions are fully resolved.

Abstract

This paper is mainly devoted to describing the entire solutions of nonlinear partial differential equation $$ u_{z_1}u_{z_2}\cdots u_{z_n}=e^g, $$ with the eikonal equation as a prototype, where $g$ is a polynomial in $\mathbb{C}^n$. Through a novel method, we break through the restriction of finite order condition and present the explicit expressions for the entire solution of the above equation. As an application, we completely resolve two questions of Xu-Liu-Xuan in \cite{Xu}.