Characterizations of transcendental entire solutions of trinomial partial differential-difference equations in ℂ2#

Abstract

This study is devoted to exploring the existence and the precise form of finite-order transcendental entire solutions of second-order trinomial partial differential-difference equations L(f)2+2hL(f)f(z1+c1,z2+c2)+f(z1+c1,z2+c2)2=eg(z1,z2)L{(f)}^{2}+2hL(f)f\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})+f{\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})}^{2}={e}^{g\left({z}_{1},{z}_{2})} and L˜(f)2+2hL˜(f)(f(z1+c1,z2+c2)−f(z1,z2))+(f(z1+c1,z2+c2)−f(z1,z2))2=eg(z1,z2),\tilde{L}{(f)}^{2}+2h\tilde{L}(f)(f\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})-f\left({z}_{1},{z}_{2}))+{(f\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})-f\left({z}_{1},{z}_{2}))}^{2}={e}^{g\left({z}_{1},{z}_{2})}, where L(f)L(f) and L˜(f)\tilde{L}(f) are defined in (2.1) and (2.2), respectively, and g(z)g\left(z) is a polynomial in C2{{\mathbb{C}}}^{2}. Our results are the extensions of some of the previous results of Liu et al. Also, we exhibit a series of examples to explain that the forms of transcendental entire solutions of finite-order in our results are precise.

Keywords

• functions of several complex variables
• fermat-type equations
• entire solutions
• nevanlinna theory
• 30d35
• 35m30
• 32w50
• 39a45