Entire solutions of certain nonlinear differential and delay-differential equations

Abstract

Let f(z) be an entire solution of the nonlinear differential equation f(z)n+P(z,f)=b1eλ1z+b2eλ2z, where n≥2, P(z,f) is a differential polynomial in f of degree ≤n−1 with meromorphic functions of order less than 1 as coefficients, and b1, b2, λ1, λ2 are nonzero constants and t=λ2/λ1 is real. By utilizing Nevanlinna theory, we show that t=−1 or t is a positive rational number and that in either case f(z) is expressed in terms of exponential functions. This partially answers a question in the literature. We also show that these results extend to finite-order entire solutions of the above equation when P(z,f) is replaced by a delay-differential polynomial in f with meromorphic functions of order less than 1 as coefficients.