Notes on the Existence of Entire Solutions for Several Partial Differential-Difference Equations

This paper is mainly devoted to discuss the existence of the finite order transcendental entire solutions for the partial differential-difference equations (∂f(z1,z2)∂z1+∂f(z1,z2)∂z2)n+f(z1+c1,z2+c2)m=1,(∂f(z1,z2)∂z1)n+f(z1+c1,z2+c2)m=1. It is confirmed that the existence of the finite order transcendental entire solutions for the above equations in the case n=2 and m=1 . This is a very powerful supplement to the previous theorems given by Xu and Cao [L. Xu and T. B. Cao, Solutions of complex Fermat-type partial difference and differential-difference equations, Mediterr. J. Math. 15 (2018), pages, 1-14.].