Transcendental entire solutions for several quadratic binomial and trinomial PDEs with constant coefficients

The Study of Solutions of Several Systems of Nonlinear Partial Differential Difference Equations

“…Since ∂f 1 /∂z 1 is transcendental, then ∂f 2 ðz + cÞ/ ∂z 1 and ∂f 2 ðzÞ/∂z 1 are transcendental. By observing the second equation of (22), we can conclude that β 2 f 1 ðz + cÞ + β 3 f 2 is transcendental.…”

“…If ∂f 2 ðz + cÞ/∂z 1 is not transcendental, it thus leads to that ∂f 2 /∂z 1 is not transcendental. From (22), we thus get that β 2 f 1 ðz + cÞ + β 3 f 2 is not transcendental. Thus, it yields that β 2 ∂f 1 ðz + cÞ/∂z 1 + β 3 ∂f 2 /∂z 1 is not transcendental.…”

“…Proof. Assume that ðf 1 , f 2 Þ is a pair of finite-order transcendental entire functions satisfying (22). Thus, let us discuss two following cases.…”

“…Hence, if ∂f 1 /∂z 1 is transcendental, then α 2 f 2 ðz + cÞ + α 3 f 1 , β 2 f 1 ðz + cÞ + β 3 f 2 , and ∂f 2 /∂z 1 are transcendental. Hence, system (22) can be represented as…”

“…By analyzing Theorems 3-6, a natural question is as follows: What will happen about the transcendental entire solutions for the system of the PDDEs of Fermat type? Although many scholars have paid considerable attention to the complex difference equation with a single variable and the complex Fermat type difference equation in recent years, a series of important and meaningful results (including [7,[14][15][16][17][18][19][20][21][22]) were obtained, however, to our knowledge, there were not much results about the complex difference equation in several complex variables. Of course, the references involving the results of systems of complex PDDEs are even less.…”