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

Abstract

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.