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:fn(z)+Pn−3(f)=p1eα1z+p2eα2zfn(z)+Pn−3(f)=p1eα1z+p2eα2z has no nonconstant entire solutions, where n   is an integer ⩾4, p1p1 and p2p2 are two polynomials (≢0)(≢0), α1α1, α2α2 are two nonzero constants with α1/α2≠α1/α2≠ rational number, and Pn−3(f)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−3n−3. It is conjectured that the conclusion remains to be valid when Pn−3(f)Pn−3(f) is replaced by Pn−1(f)Pn−1(f) or Pn−2(f)Pn−2(f).