On the existence of entire solutions to a system of nonlinear Fermat-type partial differential-difference equations
Junfeng Xu, Sujoy Majumder, Debabrata Pramanik
TL;DR
The paper advances the theory of Fermat-type equations by classifying finite-order transcendental entire solutions to a system of nonlinear partial differential-difference equations in C^m, extending prior two-variable results. It leverages high-dimensional Nevanlinna theory and Clunie-type lemmas to establish broad nonexistence results and, in key cases, explicit solution forms. Notably, when n1=n2=m1=m2=2, solutions take sine forms with linear arguments (subject to shift conditions), while in cases with n_i>m_i the solutions adopt a quadratic-in-z1 template augmented by holomorphic g_i functions constrained by shift relations. The work thus provides a comprehensive landscape of existence/nonexistence and explicit structures across dimensions, accompanied by illustrative examples and corollaries.
Abstract
The aim of this study is to investigate the precise form of finite-order entire solutions to the following system of Fermat-type partial differential-difference equations: \beas \begin{cases} \left(\frac{\partial f_1\left(z_1, z_2, \ldots, z_m \right)}{\partial z_1}\right)^{n_1} + f_2^{m_1} \left(z_1 + c_1, z_2 + c_2, \ldots, z_m + c_m \right) = 1,\\ \left(\frac{\partial f_2\left(z_1, z_2, \ldots, z_m \right)}{\partial z_1}\right)^{n_2} + f_1^{m_2} \left(z_1 + c_1, z_2 + c_2, \ldots, z_m + c_m \right) = 1 \end{cases} \eeas for various combinations of the positive integers $n_1$, $n_2$, $m_1$ and $m_2$. Our results extend the work of Xu et al. (Entire solutions for several systems of non-linear difference and partial differential-difference equations of Fermat-type, J. Math. Anal. Appl., 483(2), 2020), generalizing the setting $\mathbb{C}^2$ to $\mathbb{C}^m$. Several examples are provided to illustrate the applicability and sharpness of the obtained results.