In this paper, we investigate the following elliptic system with Sobolev critical growth
\(\begin{cases}-\Delta u_1 = G_1(|y|) u_1^{2^*-1} + \frac{1}{2} u_1^{\frac{2^*}{2}-1} u_2^{\frac{2^*}{2}}, & y \in \mathbb{R}^N, \\-\Delta u_2 = G_2(|y|) u_2^{2^*-1} + \frac{1}{2} u_2^{\frac{2^*}{2}-1} u_1^{\frac{2^*}{2}}, & y \in \mathbb{R}^N, \\u_1, u_2 > 0, \quad u_1, u_2 \in D^{1,2}(\mathbb{R}^N),\end{cases}\)
where N ≥ 5, G1 (r) and G2(r) are positive radial potentials, \(2^*=\frac{2N}{N-2}\) . We construct an unbounded sequence of non-radial positive vector solutions of synchronized type via Lyapunov-Schmidt reduction argument. Different from the single equation, it is worth noting that when N ≥ 5, the coupling exponent \(\frac{2^*}{2}-1=\frac2{N-2}<1\) , which poses a serious obstacle to applying the perturbation argument directly. This constitutes the main difficulty of this paper and reflects the significant differences between the coupled system and a single equation. As a matter of fact, it is necessary to give an accurate point-wise estimate of the error term so that it can be absolutely controlled by less than one multiple of the approximate solution. To this end, we improve the decaying order of the error term iteratively, and it should be an effective way to deal with the ill coupled terms.
