Positive least energy solutions of fractional Laplacian systems with critical exponent

We study the fractional Laplacian system with critical exponent $$\displaylines{ (-\Delta)^s u+\lambda_1u =\mu_1|u|^{2_s^*-2}u+\beta|u|^{\frac{2_s^*}{2}-2}u|v|^{\frac{2_s^*}{2}} , \quad x\in \Omega , \cr (-\Delta)^s v+\lambda_2v =\mu_2|v|^{2_s^*-2}v+\beta|v|^{\frac{2_s^*}{2}-2}v |u|^{\frac{2_s^*}{2}} , \quad x\in \Omega , \cr u=v= 0\,, \quad x\in \partial \Omega , }$$ where $\Omega\subset\mathbb{R}^N$ $(N>2s)$ is a smooth bounded domain, $s\in(0,1)$, $(-\Delta)^s$ stands for the fractional Laplacian, $2_s^*:=\frac{2N}{N-2s}$ is the critical Sobolev exponent, $-\lambda_1(\Omega)0$, here $\lambda_1(\Omega)$ is the first eigenvalue of $(-\Delta)^s$ with Dirichlet boundary condition. For each fixed $\beta\geq\frac{2s}{N-2s}\max\{\mu_1,\mu_2\}$, we show that this system has a positive least energy solution.