Ground states of the inhomogeneous nonlinear fractional Schrödinger-Poisson equations

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In this paper, we are interested in the inhomogeneous nonlinear fractional Schrödinger-Poisson equations { ( − △ ) α u + V ( x ) u + A ( x ) ϕu = B ( x ) | u | p − 2 u , x ∈ R d , ( − △ ) β ϕ = A ( x ) | u | 2 , x ∈ R d , where d ≥ 3 and α , β , p are under suitable assumptions. The variable coefficients A ( x ) and B ( x ) are spherically symmetric which behave like power function with positive index. The positive potential V ( x ) is spherically symmetric. We develop the weighted Sobolev compact embedding theorem with unbounded radial potentials. The existence of the ground states of the inhomogeneous nonlinear fractional Schrödinger-Poisson equations is established by the Nehari manifold approach.