Entire solution of a partial differential equation
Junfeng Xu, Nabadwip Sarkar, Sujoy Majumder
TL;DR
This work investigates entire solutions of a Fermat-type partial differential equation in two complex variables by applying Nevanlinna value distribution theory. It distinguishes between the cases $n\ge 5$ and $3\le n\le 4$, showing that for large $n$ nonconstant differences $r-s$ are disallowed unless $r-s$ is constant, which forces $f$ to have an exponential form in $s$. For $3\le n\le 4$, with linear right-hand side exponents, the authors derive precise parameter relations among $(\lambda_1,\lambda_2)$ and provide explicit representations of $f$ as sums of exponentials in $z_1$ with $z_2$-dependent coefficients, subject to additional polynomial constraints. The results contribute explicit rigidity classifications and illustrate how Nevanlinna theory constrains entire solutions of Fermat-type 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 $\mathbb{C}^2$.