Positive solutions of an asymptotically periodic Schrödinger–Poisson system with critical exponent

Existence of one positive solution of the generalized Schrödinger–Poisson system {−Δu+V(x)u−K(x)ϕ|u|3u=f(x,u)inR3,−Δϕ=K(x)|u|5inR3, where V,K,f are asymptotically periodic functions of xx, is proved by the mountain pass theorem and the concentration-compactness principle. The system with subcritical nonlocal term has been studied extensively in the last twenty years, while the system with critical nonlocal term has seldom been studied. It turns out that new techniques are needed in dealing with the case of critical nonlocal term.