Ground state and multiple solutions for the fractional Schrödinger-Poisson system with critical Sobolev exponent

In this paper, we study the following doubly singularly perturbed fractional Schrödinger-Poissonsystem with critical Sobolev exponent ε2α(−Δ)αu+V(x)u+ϕu=|u|2α∗−2u+f(u)inRN,εθ(−Δ)s2ϕ=γsu2inRN,where α∈(12,1), N∈(2α,4α), s∈(N−2α,N), θ∈(0,s), f is a subcritical nonlinearity, ε is a small parameter, the positive potential V satisfies a local condition. By combining penalization techniques with Ljusternik-Schnirelmann theory, the number of positive solutions is estimated below by the topology of the set where the potential V attains its minimum.