The value distribution of meromorphic solutions of some second order nonlinear difference equation

In this paper, we investigate some properties of solutions for some nonlinear difference equation, and obtain some estimates of the exponent of convergence of poles and growth of its transcendental meromorphic solutions $f(z)$ and its difference $\Delta f(z)$. Moreover, we study the existence and forms of rational solutions. We also give some examples to support our theoretical discussion.