| Summary: | Our purpose in this article is to describe the solutions of several product-type nonlinear partial differential equations (PDEs) (a1u+b1uz1+c1uz2)(a2u+b2uz1+c2uz2)=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 (a1u+b1uz1+c1uz2)(a2u+b2uz1+c2uz2)=eg,\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 aj,bj{a}_{j},{b}_{j}, and cj(j=1,2){c}_{j}\left(j=1,2) are constants in C{\mathbb{C}}. The finite-order transcendental entire solution uu of the first equation is of the following forms: u(z1,z2)=±1a1a2+η0e1D[(a2c1−a1c2)z1+(a1b2−a2b1)z2],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(z1,z2)=12a1eQ(z1,z2)+12a2e−Q(z1,z2)+η0e1D[(a2c1−a1c2)z1+(a1b2−a2b1)z2],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=b1c2−b2c1D={b}_{1}{c}_{2}-{b}_{2}{c}_{1}, η0∈C−{0}{\eta }_{0}\in {\mathbb{C}}-\left\{0\right\}, and Q(z1,z2)=−1D[(a1c2+a2c1)z1−(a1b2+a2b1)z2]+η1,η1∈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.
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