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

By utilizing Nevanlinna’s value distribution theory of meromorphic functions, it is shown that the following type of nonlinear differential equations: f n(z)+Pn−3(f ) = p1eα1z +p2eα2z has no nonconstant entire solutions, where n is an integer 4, p1 and p2 are two polynomials ( ≡ 0), α1, α2 are two nonzero constants with α1/α2 = rational number, and Pn−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 Pn−3(f ) is replaced by Pn−1(f ) or Pn−2(f ). © 2005 Elsevier Inc. All rights reserved.