Existence results for a fractional elliptic system with critical Sobolev‐Hardy exponents and concave‐convex nonlinearities

In this paper, we study the following nonlinear fractional Laplacian system with critical Sobolev‐Hardy exponent( − Δ ) s u − γ u | x | 2 s= λ f ( x ) | u | q − 2 u | x | α+ 2 η η + θ h ( x ) | u | η − 2 u | v | θ| x | βinΩ ,( − Δ ) s v − γ v | x | 2 s= μ g ( x ) | v | q − 2 v | x | α+ 2 θ η + θ h ( x ) | u | η | v | θ − 2 v | x | βinΩ ,u = v = 0inR N ∖ Ω ,where 0 ∈ Ω is a smooth bounded domain inR N , 0 < s < 1 , 1 ≤ q < 2 , 0 ≤ α , β < 2 s < N , 0 ≤ γ < γ H , η , θ > 1 satisfy η + θ = 2 s * ( β ) ,2 s * ( β ) = 2 ( N − β ) N − 2 sis the critical Sobolev‐Hardy exponent, λ , μ > 0 are parameters, f , g and h are nonnegative functions on Ω . Using the variational methods and analytic techniques, we prove that the critical fractional Laplacian system admits at least two positive solutions when the pair of parameters ( λ , μ ) belongs to a suitable subset ofR + 2 .