On the nonexistence of entire solutions of certain type of nonlinear differential equations

Abstract

By utilizing Nevanlinna's value distribution theory of meromorphic functions, it is shown that the following type of nonlinear differential equations: f n ( z ) + P n − 3 ( f ) = p 1 e α 1 z + p 2 e α 2 z has no nonconstant entire solutions, where n is an integer ⩾4, p 1 and p 2 are two polynomials ( ≢ 0 ) , α 1 , α 2 are two nonzero constants with α 1 / α 2 ≠ rational number, and P n − 3 ( f ) denotes a differential polynomial in f and its derivatives (with polynomials in z as the coefficients) of degree no greater than n − 3 . It is conjectured that the conclusion remains to be valid when P n − 3 ( f ) is replaced by P n − 1 ( f ) or P n − 2 ( f ) .