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

In this paper, we consider the Schrödinger-Poisson system {−Δu+u+l(x)ϕu=k(x)|u|4u+μh(x)uinR3,−Δϕ=l(x)u2inR3, where μ is a positive constant and the nonlinear growth of |u|4u 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 H1(R3)×D1,2(R3) under some suitable conditions on the nonnegative functions l,k and h, but not requiring any symmetry properties on them.