Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2

Abstract

By utilizing the Nevanlinna theory of meromorphic functions in several complex variables, we mainly investigate the existence and the forms of entire solutions for the partial differential-difference equation of Fermat type α∂f(z1,z2)∂z1+β∂f(z1,z2)∂z2m+f(z1+c1,z2+c2)n=1{\left(\alpha \frac{\partial f\left({z}_{1},{z}_{2})}{\partial {z}_{1}}+\beta \frac{\partial f\left({z}_{1},{z}_{2})}{\partial {z}_{2}}\right)}^{m}+f{\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})}^{n}=1 and α∂f(z1,z2)∂z1+β∂f(z1,z2)∂z22+[γ1f(z1+c1,z2+c2)−γ2f(z1,z2)]2=1,{\left(\alpha \frac{\partial f\left({z}_{1},{z}_{2})}{\partial {z}_{1}}+\beta \frac{\partial f\left({z}_{1},{z}_{2})}{\partial {z}_{2}}\right)}^{2}+{\left[{\gamma }_{1}f\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})-{\gamma }_{2}f\left({z}_{1},{z}_{2})]}^{2}=1, where m,nm,n are positive integers and α,β,γ1,γ2\alpha ,\beta ,{\gamma }_{1},{\gamma }_{2} are constants in C{\mathbb{C}}. We give some results about the forms of solutions for these equations, which are great improvements of the previous theorems given by Xu and Cao et al. Moreover, it is very satisfactory that we give the corresponding examples to explain the conclusions of our theorems in each case.

Keywords

• existence
• entire solution
• partial differential-difference equation
• 35m30
• 32w50
• 39a45
• 30d35