Multiplicity of positive solutions to linearly coupled Schrödinger systems involving critical exponent

We are concerned with the following linearly coupled Schrödinger system:$ \begin{equation*} \left\{ \begin{array}{ll} -\Delta u+\mu _{1}u = \left\vert u\right\vert ^{p_{1}-2}u+\beta \left( \varepsilon x\right) v, & x\in \mathbb{R}^{N}, \\ -\Delta v+\mu _{2}v = \left\vert v\right\vert ^{p_{2}-2}v+\beta \left( \varepsilon x\right) u, & x\in \mathbb{R}^{N}, \end{array} \right. \end{equation*} $where $ N\geq 3 $, $ \mu_1 $ and $ \mu_2 $ are positive constants, the exponents $ 2<p_1<2^* $ and $ 2<p_2\leq 2^{\ast } $, $ 2^* = \frac{2N}{N-2} $ is the Sobolev critical exponent, $ \beta \left( x\right) $ is a non-constant coupling coefficient, and $ \varepsilon>0 $ is a parameter. We use the parameter $ \varepsilon $ to control the shape of the graph of the non-constant coupling coefficient $ \beta \left( \varepsilon x\right) $ and study how the shape of the graph of $ \beta \left(\varepsilon x\right) $ affects the number of positive solutions of the linearly coupled system. Under some suitable assumptions on $ \beta \left(\varepsilon x\right) $, we prove that the number of positive solutions of the system is dependent on $ \beta \left( \varepsilon x\right) $ for $ \varepsilon>0 $ sufficiently small. Moreover, the asymptotic behaviors of positive solutions as $ \varepsilon \to 0 $ are studied.