Positive solutions for Schrödinger system with asymptotically periodic potentials

This paper is concerned with the following class of elliptic equations {−Δu+λ(x)u=μf(u)+βv2uinRN,−Δv+κ(x)v=νg(v)+βu2vinRN, where u,v∈H1(RN)u,v∈H1(RN), N≤3N≤3, μ,ν,β>0μ,ν,β>0 are coupling constants, λ(x)λ(x) and κ(x)κ(x) are asymptotically periodic functions, ff and gg are continuous functions with subcritical growth. This type of system arises, in particular, in models in Bose–Einstein condensates theory. We prove the existence of positive solution for this weakly coupled system with β>0β>0 sufficiently large. Furthermore, we obtain some sufficient conditions for the nonexistence of positive solutions.