Positive and sign-changing solutions of a Schrödinger–Poisson system involving a critical nonlinearity

In this paper, we consider the Schrödinger–Poisson system { − Δ u + u + l ( x ) ϕ u = k ( x ) | u | 4 u + μ h ( x ) u in R 3 , − Δ ϕ = l ( x ) u 2 in R 3 , where μ is a positive constant and the nonlinear growth of | u | 4 u reaches the Sobolev critical exponent since 2 ∗ = 6 for three spatial dimensions. We prove the existence of at least a pair of fixed sign solutions and a pair of sign-changing solutions in H 1 ( R 3 ) × D 1 , 2 ( R 3 ) under some suitable conditions on the nonnegative functions l , k and h , but not requiring any symmetry properties on them.