In the last decades, nonlinear elliptic systems have attracted researchers because of their practicality and profound mathematical depth. Also, recent development of the field allowed one to treat equations with a large class of integro-differential operators other than local elliptic operators such as the Laplacian or its square. In this talk, we consider minimal energy solutions of the fractional Lane-Emden equation, one of the simplest nonlocal Hamiltonian systems. Especially, we discuss their existence and asymptotic behavior as the exponents of the nonlinear terms approach to so called the Sobolev critical hyperbola. The close relationship between the system and the Hardy-Littlewood-Sobolev inequality is also to be explained. This work is collaborated with Woocheol Choi (KIAS, Korea).