Ground state solutions for the periodic fractional Schrödinger–Poisson systems with critical Sobolev exponent

Abstract

We study the fractional Schrödinger–Poisson system with critical Sobolev exponent ( − Δ ) s u + V ( x ) u + ϕ u = f ( x , u ) + K ( x ) | u | 2 s ∗ − 2 u in ℝ 3 , ( − Δ ) t ϕ = u 2 in ℝ 3 , where ( − Δ ) α denotes the fractional Laplacian of order α = s , t ∈ ( 0 , 1 ) ; V ( x ) , f ( x , u ) and K ( x ) are 1 -periodic in the x -variables; 2 s ∗ = 6 ∕ ( 3 − 2 s ) is the fractional critical Sobolev exponent in dimension 3 . Under some weaker conditions on f , we prove the existence of ground state solutions for such a system via the mountain pass theorem in combination with the concentration-compactness principle. Our results are new even for s = t = 1 .