Positive ground state solutions of a quadratically coupled schrödinger system

In this paper, we study the following quadratically coupled Schrodinger system: \begin{document}$\begin{equation*}\left\{\begin{array}{ll}-\Delta u+\lambda_1u=\mu_1u^2+2\alpha uv+\gamma v^2, & \mbox{in }\Omega,\\-\Delta v+\lambda_2v=\mu_2v^2+2\gamma uv+\alpha u^2, & \mbox{in }\Omega,\\u=v=0, & \mbox{on }\partial\Omega,\end{array}\right.\end{equation*}$ \end{document} where $\Omega\subset\mathbb{R}^6$ is a smooth bounded domain, $-\lambda (\Omega) 0$, and $\lambda (\Omega)$ is the first eigenvalue of $-\Delta$ with the Dirichlet boundary condition. The main difficulty to investigate this kind of equations is caused by the fact that all the quadratic nonlinearities, including the coupling terms, are of critical growth. By the methods used in [Zhenyu Guo, Positive ground state solutions of a nonlinearly coupled Schrodinger system with critical exponents in [Zhenyu Guo, Positive ground state solutions of a nonlinearly coupled Schrodinger system with critical exponents in $\mathbb{R}^4$, J. Math. Anal. Appl. , 430(2):950-970, 2015], the existence of positive ground state solutions of the system is established with more ingenious hypotheses.