ENTIRE AND MEROMORPHIC SOLUTIONS FOR SEVERAL FERMAT TYPE PARTIAL DIFFERENTIAL DIFFERENCE EQUATIONS IN ℂ2

Abstract

We explore the existence and the forms of entire and meromorphic solutions for the partial differential-difference equations with more general forms of

$${\left(\alpha \frac{\partial f(z_1,z_2)}{\partial z_1}+\beta \frac{\partial f(z_1,z_2)}{\partial z_2}\right)}^2+f{(z_1+c_1,z_2+c_2)}^2=e^{g(z_1,z_2)}$$

and

$${\left(\alpha \frac{\partial f(z_1,z_2)}{\partial z_1}+\beta \frac{\partial f(z_1,z_2)}{\partial z_2}\right)}^2+{[f(z_1+c_1,z_2+c_2)-f(z_1,z_2)]}^2=e^{g(z_1,z_2)},$$

where $g(z_1,z_2)$ is a polynomial in ${ℂ}^2$ and $\alpha ,\beta$ are constants in $ℂ$. 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.