Abstract
In this paper, we study the transcendental meromorphic solutions to a certain type of nonlinear complex differential equation
\(f^{n}(z)+P(z,f,f^{\prime},f^{\prime\prime},\cdots,f^{(t)})=H_{0}(z)+H_{1}(z)e^{\alpha_{1}z^{q}}+\cdots+H_{m}(z)e^{\alpha_{m}z^{q}},\)
where \(m\geq 1\) , \(n\geq m+2\) , \(t\geq 0\) , \(q\geq 1\) are integers, \(P(z,f,f^{\prime},f^{\prime\prime},\cdots,f^{(t)})\) is a differential polynomial in \(f(z)\) of degree \(d\leq n-m-1\) with small functions of \(f(z)\) as its coefficients, \(\alpha_{i}\) \((i=1,2,\cdots,m)\) are nonzero complex constants such that \(|\alpha_{1}|>|\alpha_{2}|>\cdots>|\alpha_{m}|\) , \(H_{i}(z)\) are entire functions of order less than \(q\) such \(H_{i}(z)\not\equiv 0\) for \(1\leq i\leq m\) . In fact, we give the exact forms of all possible meromorphic solutions satisfying \(N(r,f)=S(r,f)\) of the above equation. Particularly, we weaken the condition and obtain other properties of the meromorphic solutions when \(m=3\) , \(q=1\) . Some examples are given to illustrate our results.
