LAUSR

In this paper, we study the following nonlinear fractional Laplacian system with critical exponent $$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{s}u=\lambda |u|^{p-2}u+\frac{2\alpha }{\alpha +\beta }|u|^{\alpha -2}u|v|^{\beta }, &{}\quad \hbox {in} \;\ \Omega ,\\ (-\Delta )^{s}v=\mu |v|^{p-2}v+\frac{2\beta }{\alpha +\beta }|u|^{\alpha }|v|^{\beta -2}v, &{} \quad \hbox {in} \;\ \Omega ,\\ u=v=0, &{} \quad \hbox {in} \;\ {\mathbb {R}}^{N}\backslash \Omega , \end{array}\right. \end{aligned}$$(-Δ)su=λ|u|p-2u+2αα+β|u|α-2u|v|β,inΩ,(-Δ)sv=μ|v|p-2v+2βα+β|u|α|v|β-2v,inΩ,u=v=0,inRN\Ω,where $$\Omega \subset {\mathbb {R}}^{N}$$Ω⊂RN is a bounded domain with smooth boundary, $$01$$01 satisfy $$\alpha +\beta =2_{s}^{*}, 2_{s}^{*}=\frac{2N}{N-2s}$$α+β=2s∗,2s∗=2NN-2s is the critical Sobolev exponent, and $$N>4s, \lambda , \mu >0$$N>4s,λ,μ>0 are parameters. Using the $${\mathcal {N}}$$N ehari manifold, fibering maps and the Lusternik–Schnirelmann category, we prove that the problem has at least $$\hbox {cat}(\Omega )+1$$cat(Ω)+1 distinct positive solutions, where $$\hbox {cat} (\Omega )$$cat(Ω) denotes the Lusternik–Schnirelmann category of $$\Omega $$Ω in itself.