Doubly nonlocal system with Hardy-Littlewood-Sobolev critical nonlinearity

This article concerns about the existence and multiplicity of weak solutions for the following nonlinear doubly nonlocal problem with critical nonlinearity in the sense of Hardy-Littlewood-Sobolev inequality{(−Δ)su=λ|u|q−2u+(∫Ω|v(y)|2μ⁎|x−y|μdy)|u|2μ⁎−2uinΩ(−Δ)sv=δ|v|q−2v+(∫Ω|u(y)|2μ⁎|x−y|μdy)|v|2μ⁎−2vinΩu=v=0inRn∖Ω, where Ω is a smooth bounded domain in Rn, n>2s, s∈(0,1), (−Δ)s is the well known fractional Laplacian, μ∈(0,n), 2μ⁎=2n−μn−2s is the upper critical exponent in the Hardy-Littlewood-Sobolev inequality, 10 are real parameters. We study the fibering maps corresponding to the functional associated with (Pλ,δ) and show that minimization over suitable subsets of Nehari manifold renders the existence of at least two non trivial solutions of (Pλ,δ) for suitable range of λ and δ.