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

In this paper we consider the following Schrödinger–Poisson system{−Δu+V(x)u+λϕu=Q(x)f(u),x∈R3,−Δϕ=u2,x∈R3, where potentials V∼|x|−αV∼|x|−α, Q∼|x|−βQ∼|x|−β (α>0α>0, β>0β>0), and f:R→Rf: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,βα,β belong to different ranges. We also study the boundedness of the set of the solutions found, as λ→0+λ→0+, in the spirit of Ruiz (2006) [21].