Positive solutions for a class of concave-convex semilinear elliptic systems with double critical exponents

In this paper, we consider the following concave-convex semilinear elliptic system with double critical exponents:~ \begin{equation} \begin{cases} -\Delta u=|u|^{2^{*}-2}u+\frac{\alpha}{2^{*}}|u|^{\alpha-2}|v|^{\beta}u+\lambda|u|^{q-2}u,~~&\mbox{in}~\Omega,\\ -\Delta v=|v|^{2^{*}-2}v+\frac{\beta}{2^{*}}|u|^{\alpha}|v|^{\beta-2}v+\mu|v|^{q-2}v,~~&\mbox{in}~\Omega,\\ u,~v>0,~~&\mbox{in}~\Omega,\\ u=v=0,~~&\mbox{on}~\partial\Omega, \end{cases} \nonumber \end{equation} where $\Omega\subset\mathbb{R}^{N}(N\geq3)$ is a bounded domain with smooth boundary,~$\lambda,~\mu>0,~11,~\beta>1,~\alpha+\beta=2^{*}=\frac{2N}{N-2}$. By the Nehari manifold method and variational method, we obtain two positive solutions which improves the recent results in the literature.