Entire solution of a partial differential equation

TL;DR

This paper investigates the existence of entire solutions to a specific nonlinear partial differential equation in two complex variables using Nevanlinna's value distribution theory, focusing on cases where the exponent n is at least 3.

Contribution

It provides new conditions for the existence of entire solutions to a class of nonlinear PDEs in several complex variables, extending previous results with the application of Nevanlinna theory.

Findings

Existence criteria for solutions when n ≥ 3

Conditions on coefficients and polynomials for solutions

Extension of Nevanlinna theory to PDEs in several complex variables

Abstract

In this paper, using Nevanlinna's value distribution theory of meromorphic functions in several complex variables, we study for the existence of entire solutions $f$ in $\mathbb{C}^2$ of the following partial differential equation \[a_1\left(\frac{\partial f(z_1,z_2)}{\partial z_1}\right)^n+a_2f^n(z_1,z_2)=p_1e^{r(z_1,z_2)}+p_2e^{s(z_1,z_2)},\] where $n$ is a positive integer such that $n\geq 3$, $a_1,a_2,p_1,p_2$ are non-zero constants and $r(z_1,z_2), s(z_1,z_2)$ are arbitrary polynomials in $\mathbb{C}^2$.