Nonradial solutions for coupled elliptic system with critical exponent in exterior domain

We consider the following coupled Schr$ \ddot{o} $dinger system with critical exponent in $ \mathbb{R}^3: $$ \left \{ \begin{aligned} &-\Delta u+\lambda V(|y|)u = K_1(|y|)u^5+u^2v^3,\qquad &\text{ } \mathbb{R}^3\backslash B_\epsilon(0),\\ v+\lambda V(|y|)v K_2(|y|)v^5+v^2u^3, \qquad &u >0, v>0, \quad B_\epsilon(0), \\ & (u,v) (0,0), &\text{on \partial B_\epsilon (0), &u,v\in D^{1,2}(\mathbb{R}^3\backslash B_\epsilon(0))), \end{aligned} \right. $where V(|y|) is potential function satifying 0<V(|y|)\leq C\frac{1}{(1+|y|)^4} \lambda>0 a constant. K_i ( i 1,2 ) are smooth bounded functions satisfying some suitable assumptions. B_\epsilon(0) ball centered at origin radius \epsilon. By using Schmidt reduction arguments combine energy expansion and point theory, we prove existence of infinitely nonradial solutions for system.