Multiple solutions of asymptotically linear Schrödinger–Poisson system with radial potentials vanishing at infinity

In this paper we consider the following Schrödinger–Poisson system { − Δ u + V ( x ) u + λ ϕ u = Q ( x ) f ( u ) , x ∈ R 3 , − Δ ϕ = u 2 , x ∈ R 3 , where potentials V ∼ | x | − α , Q ∼ | x | − β ( α > 0 , β > 0 ), and f : R → R is asymptotically linear at infinity. Working in a variational setting, we prove the existence and multiplicity of solutions for the system when λ is small and α , β belong to different ranges. We also study the boundedness of the set of the solutions found, as λ → 0 + , in the spirit of Ruiz (2006) [21].