In this paper, we prove the existence of at least one positive solution pair (u, v)∈H1(RN)×H1(RN) to the following semilinear elliptic system
{-Δu+u=f(x, v), x∈RN,
-Δv+v=g(x, u), x∈RN, (0.1)
by using a linking theorem and the concentration-compactness principle. The main conditions we imposed on the nonnegative functions f, g ∈ C0(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 semilinear elliptic boundary value problem
{-Δu+u=f(x, u), x∈Ω,
u∈H10(Ω)
where Ω（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.
