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In this paper, we study the following critical system with fractional Laplacian: $$textstylebegin{cases} (-Delta)^{s}u+lambda_{1}u=mu_{1}|u|^{2^{ast}-2}u+frac{alpha gamma}{2^{ast}}|u|^{alpha-2}u|v|^{beta} u0026 text{in } Omega,  (-Delta)^{s}v+lambda_{2}v= mu_{2}|v|^{2^{ast}-2}v+frac{beta gamma}{2^{ast}}|u|^{alpha}|v|^{beta-2}v u0026 text{in } Omega,  u=v=0 u0026 text{in } mathbb{R}^{N}setminusOmega, end{cases} $$where ((-Delta)^{s}) is the fractional Laplacian, (0 0), (2^{ast}=frac{2N}{N-2s}) is a fractional critical Sobolev exponent, (Nu003e2s), (1 -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 (gammau003e0). 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).

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