In this paper, we prove the existcnce of at least one positive solution pair (u, v) ∈ H1(RN)× H1(RN) to the following semilinear elliptic system -△u+u=f(x,v), x∈RN, (0.1)-△v+v=g(x,u), x∈RN,by using a linking theorem and the concentration-compactness principle. The main conditions we imposed on the nonnegative functions f, g ∈ CO(RN × R1) are that, f(x, t) and g(x, t) are superlinear at t = 0 as well as at t = +∞, that f and g are subcritical in t and satisfy a kind of monotonic conditions. We mention that we do not assume that f or g satisfies the Ambrosetti-Rabinowitz condition as usual.Our main result can be viewed as an extension to a recent result of Miyagaki and Souto [J. Diff. Equ. 245(2008), 3628-3638] concerning the existence of a positive solution to the senilinear elliptic boundary value problem -△u+u=f(x,u), x∈Ω,u ∈H10 (Ω)where Ω (C) RN is bounded and a result of Li and Yang [G. Li and J. Yang: Communications in P.D.E. Vol. 29(2004) Nos.5 & 6.pp.925-954, 2004] concerning (0.1) when f and g are asymptotically linear.