Entire and meromorphic solutions for several Fermat type partial differential difference equations in \(\mathbb{C}^2\). (English) Zbl 1510.32091

Xu, Hong Yan; Zhang, Keyu; Zheng, Xiumin

Rocky Mt. J. Math. 52, No. 6, 2169-2187 (2022).

Summary: We explore the existence and the forms of entire and meromorphic solutions for the partial differential-difference equations with more general forms of
\[ \bigg(\alpha\frac{\partial f(z_{1}, z_{2})}{\partial z_{1}}+\beta\frac{\partial f(z_{1}, z_{2})}{\partial z_{2}}\bigg)^2+f(z_1+c_1,z_2+c_2)^2=e^{g(z_{1}, z_{2})}\] and
\[ \bigg(\alpha\frac{\partial f(z_{1}, z_{2})}{\partial z_{1}}+\beta\frac{\partial f(z_{1}, z_{2})}{\partial z_{2}}\bigg)^2+[f(z_1+c_1, z_2+c_2)-f(z_1, z_2)]^2=e^{g(z_1,z_2)},\] where \(g(z_1, z_2)\) is a polynomial in \(\mathbb{C}^2\) and \(\alpha, \beta\) are constants in \(\mathbb{C}\). Some of the results about the forms of solutions for these equations that are obtained are great improvements over previous results. It is important that some of the examples show that there exist some significant differences in the forms of transcendental entire solutions of finite order of the equations between several complex variables and a single complex variable.

MSC:

32W50	Other partial differential equations of complex analysis in several variables
34M45	Ordinary differential equations on complex manifolds

Keywords:

partial differential-difference equations; entire solutions

References:

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