Multiple positive bound states for critical Schrödinger-Poisson systems

Using variational methods we prove some results about existence and multiplicity of positive bound states of to the following Schrödinger-Poisson system: −Δu+V(x)u+K(x)ϕ(x)u=u5 −Δϕ=K(x)u2 x∈ℝ3. (SP) We remark that (SP) exhibits a “double” lack of compactness because of the unboundedness of ℝ3 and the critical growth of the nonlinear term and that in our assumptions ground state solutions of (SP) do not exist