All meromorphic solutions of Fermat-type functional equations

In this paper, by making use of properties of elliptic functions, we describe meromorphic solutions of Fermat-type functional equations $f(z{)}^n+f(L(z){)}^m=1$ over the complex plane $\mathbb{C}$, where $L(z)$ is a nonconstant entire function, $m$ and $n$ are two positive integers. As applications, we also consider meromorphic solutions of Fermat-type difference and $q$-difference equations.