The existence of nontrivial solutions to a semilinear elliptic system on ℝNℝN without the ambrosetti-rabinowitz condition

In this paper, we prove the existence of at least one positive solution pair (u,v)∈H1(ℝN)×H1(ℝN)(u,v)∈H1(ℝN)×H1(ℝN) to the following semilinear elliptic system equation(0.1){−Δu+u=f(x,v),x∈ℝN,−Δv+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∈C0(ℝN×ℝ1)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∈Ωu∈H01(Ω)where Ω⊂ℝNΩ⊂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.