Multiple positive solutions for nonlinear coupled fractional Laplacian system with critical exponent

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In this paper, we study the following critical system with fractional Laplacian: \t\t\t{(−Δ)su+λ1u=μ1|u|2∗−2u+αγ2∗|u|α−2u|v|βin Ω,(−Δ)sv+λ2v=μ2|v|2∗−2v+βγ2∗|u|α|v|β−2vin Ω,u=v=0in RN∖Ω,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\textstyle\\begin{cases} (-\\Delta)^{s}u+\\lambda_{1}u=\\mu_{1}|u|^{2^{\\ast}-2}u+\\frac{\\alpha \\gamma}{2^{\\ast}}|u|^{\\alpha-2}u|v|^{\\beta} & \\text{in } \\Omega, \\\\ (-\\Delta)^{s}v+\\lambda_{2}v= \\mu_{2}|v|^{2^{\\ast}-2}v+\\frac{\\beta \\gamma}{2^{\\ast}}|u|^{\\alpha}|v|^{\\beta-2}v & \\text{in } \\Omega, \\\\ u=v=0 & \\text{in } \\mathbb{R}^{N}\\setminus\\Omega, \\end{cases} $$\\end{document} where (-Delta)^{s} is the fractional Laplacian, 0< s<1, mu_{1},mu_{2}>0, 2^{ast}=frac{2N}{N-2s} is a fractional critical Sobolev exponent, N>2s, 1<alpha, beta<2, alpha+beta=2^{ast}, Ω is an open bounded set of mathbb{R}^{N} with Lipschitz boundary and lambda_{1},lambda_{2}>-lambda_{1,s}(Omega), lambda_{1,s}(Omega) is the first eigenvalue of the non-local operator (-Delta)^{s} with homogeneous Dirichlet boundary datum. By using the Nehari manifold, we prove the existence of a positive ground state solution of the system for all gamma>0. Via a perturbation argument and using the topological degree and a pseudo-gradient vector field, we show that this system has a positive higher energy solution. Then the asymptotic behaviors of the positive ground state solutions are analyzed when gammarightarrow0.