Transcendental entire solutions of several complex product-type nonlinear partial differential equations in ℂ2

Abstract

Our purpose in this article is to describe the solutions of several product-type nonlinear partial differential equations (PDEs) ( a 1 u + b 1 u z 1 + c 1 u z 2 ) ( a 2 u + b 2 u z 1 + c 2 u z 2 ) = 1 , \left({a}_{1}u+{b}_{1}{u}_{{z}_{1}}+{c}_{1}{u}_{{z}_{2}})\left({a}_{2}u+{b}_{2}{u}_{{z}_{1}}+{c}_{2}{u}_{{z}_{2}})=1, and ( a 1 u + b 1 u z 1 + c 1 u z 2 ) ( a 2 u + b 2 u z 1 + c 2 u z 2 ) = e g , \left({a}_{1}u+{b}_{1}{u}_{{z}_{1}}+{c}_{1}{u}_{{z}_{2}})\left({a}_{2}u+{b}_{2}{u}_{{z}_{1}}+{c}_{2}{u}_{{z}_{2}})={e}^{g}, where g ( z ) g\left(z) is a nonconstant polynomial and a j , b j {a}_{j},{b}_{j} , and c j ( j = 1 , 2 ) {c}_{j}\left(j=1,2) are constants in C {\mathbb{C}} . The finite-order transcendental entire solution u u of the first equation is of the following forms: u ( z 1 , z 2 ) = ± 1 a 1 a 2 + η 0 e 1 D [ ( a 2 c 1 − a 1 c 2 ) z 1 + ( a 1 b 2 − a 2 b 1 ) z 2 ] , u\left({z}_{1},{z}_{2})=\pm \frac{1}{\sqrt{{a}_{1}{a}_{2}}}+{\eta }_{0}{e}^{\tfrac{1}{D}{[}\left({a}_{2}{c}_{1}-{a}_{1}{c}_{2}){z}_{1}+\left({a}_{1}{b}_{2}-{a}_{2}{b}_{1}){z}_{2}]}, or u ( z 1 , z 2 ) = 1 2 a 1 e Q ( z 1 , z 2 ) + 1 2 a 2 e − Q ( z 1 , z 2 ) + η 0 e 1 D [ ( a 2 c 1 − a 1 c 2 ) z 1 + ( a 1 b 2 − a 2 b 1 ) z 2 ] , u\left({z}_{1},{z}_{2})=\frac{1}{2{a}_{1}}{e}^{Q\left({z}_{1},{z}_{2})}+\frac{1}{2{a}_{2}}{e}^{-Q\left({z}_{1},{z}_{2})}+{\eta }_{0}{e}^{\tfrac{1}{D}{[}\left({a}_{2}{c}_{1}-{a}_{1}{c}_{2}){z}_{1}+\left({a}_{1}{b}_{2}-{a}_{2}{b}_{1}){z}_{2}]}, where D = b 1 c 2 − b 2 c 1 D={b}_{1}{c}_{2}-{b}_{2}{c}_{1} , η 0 ∈ C − { 0 } {\eta }_{0}\in {\mathbb{C}}-\left\{0\right\} , and Q ( z 1 , z 2 ) = − 1 D [ ( a 1 c 2 + a 2 c 1 ) z 1 − ( a 1 b 2 + a 2 b 1 ) z 2 ] + η 1 , η 1 ∈ C . Q\left({z}_{1},{z}_{2})=-\frac{1}{D}\left[\left({a}_{1}{c}_{2}+{a}_{2}{c}_{1}){z}_{1}-\left({a}_{1}{b}_{2}+{a}_{2}{b}_{1}){z}_{2}]+{\eta }_{1},\hspace{1em}{\eta }_{1}\in {\mathbb{C}}. The description of the forms of the solutions for these PDEs demonstrates that our results are some improvements of the previous results given by Liu, Cao, and Xu [L. Xu and T. B. Cao, Solutions of complex Fermat-type partial difference and differential-difference equations, Mediterr. J. Math. 15 (2018), 227], and [K. Liu and T. B. Cao, Entire solutions of Fermat type difference differential equations, Electron. J. Diff. Equ. 2013 (2013), No. 59, 1–10.]. Meantime, we list some examples to explain that the forms of solutions of our theorems are precise to some extent.