LAUSR

Abstract In this paper, we study the following doubly singularly perturbed fractional Schrodinger–Poissonsystem with critical Sobolev exponent e 2 α ( − Δ ) α u + V ( x ) u + ϕ u = | u | 2 α ∗ − 2 u + f ( u ) in R N , e θ ( − Δ ) s 2 ϕ = γ s u 2 in R N , where α ∈ ( 1 2 , 1 ) , N ∈ ( 2 α , 4 α ) , s ∈ ( N − 2 α , N ) , θ ∈ ( 0 , s ) , f is a subcritical nonlinearity, e 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.