On entire solutions of a certain type of nonlinear differential equation (Q2777713)

Assume that a transcendental entire function of finite order \(f(z)\) is a solution of the following differential equation: NEWLINE\[NEWLINEL(f(z))-p(z)f^n(z)=h(z),NEWLINE\]NEWLINE where \(L(f)\) is a differential polynomial in \(f, h\) is an entire function, \(p\) is a polynomial and \(n\geq 3\). It is shown that either \(f\) is a unique transcendental entire solution of finite order, or any two solutions of the differential equation \(f_1\) and \(f_2\) satisfy \(L(f_1)=L(f_2)=0\).