Positive solutions for some non-autonomous Schrödinger–Poisson systems

In this paper we study the Schrödinger–Poisson system(SP){−Δu+u+K(x)ϕ(x)u=a(x)|u|p−1u,x∈R3,−Δϕ=K(x)u2,x∈R3, with p∈(3,5). Assuming that a:R3→R and K:R3→R are nonnegative functions such thatlim|x|→∞a(x)=a∞>0,lim|x|→∞K(x)=0 and satisfying suitable assumptions, but not requiring any symmetry property on them, we prove the existence of positive solutions.