Asymptotically linear Schrödinger–Poisson systems with potentials vanishing at infinity

In this paper, we are concerned with the following nonlinear Schrödinger–Poisson equations( P ) { − Δ u + V ( x ) u + λ ϕ ( x ) u = K ( x ) f ( u ) , x ∈ R 3 , − Δ ϕ = u 2 , lim | x | → ∞ ϕ ( x ) = 0 , where λ > 0 is a parameter, the potential V ( x ) may be vanishing at infinity, f ( s ) is asymptotically linear at infinity, that is f ( s ) ∼ O ( s ) as s → ∞ . For this kind of potential, it seems difficult to find solutions in H 1 ( R 3 ) . Under some assumptions on V ( x ) , K ( x ) and f ( s ) , we prove that problem (aaaa) has a positive solution for λ small and has no any nontrivial solution for λ large.