where 16305017&_mathId=si2.gif&_user=111111111&_pii=S0022247X16305017&_rdoc=1&_issn=0022247X&md5=ea3b687c18f5eb216fbba77d2574f914" title="Click to view the MathML source">(−Δ)s is fractional Laplacian, 16305017&_mathId=si3.gif&_user=111111111&_pii=S0022247X16305017&_rdoc=1&_issn=0022247X&md5=7bc0b6d5f8bdce37d5a774f8e6130790" title="Click to view the MathML source">0<s<1 and all 16305017&_mathId=si4.gif&_user=111111111&_pii=S0022247X16305017&_rdoc=1&_issn=0022247X&md5=994a1a16789a2a3a408dee7c611f94e2" title="Click to view the MathML source">λ1,λ2,μ1,μ2,γ>0, 16305017&_mathId=si13.gif&_user=111111111&_pii=S0022247X16305017&_rdoc=1&_issn=0022247X&md5=f8e8a4269f2f1444426c8ac06fdeac3f">16305017-si13.gif"> is a fractional Sobolev critical exponent, 16305017&_mathId=si6.gif&_user=111111111&_pii=S0022247X16305017&_rdoc=1&_issn=0022247X&md5=2da5dffee4e444839c5c151e8e05d6e2" title="Click to view the MathML source">N>2s, 16305017&_mathId=si7.gif&_user=111111111&_pii=S0022247X16305017&_rdoc=1&_issn=0022247X&md5=8da188a4fbb87f10989532b51f71cd60" title="Click to view the MathML source">α,β>1, 16305017&_mathId=si14.gif&_user=111111111&_pii=S0022247X16305017&_rdoc=1&_issn=0022247X&md5=98c94dc624225d21dcc252298fa714b5" title="Click to view the MathML source">α+β=2⁎, and Ω is an open bounded domain in 16305017&_mathId=si147.gif&_user=111111111&_pii=S0022247X16305017&_rdoc=1&_issn=0022247X&md5=ce265466aa1c875411996ff59f1b5152" title="Click to view the MathML source">RN with Lipschitz boundary. Under proper conditions, we establish the existence result of the ground state solution to system (0.1).