We study the following coupled nonlinear Schrödinger system with critical exponents, which can be seen as a coupled system of the Brezis–Nirenberg problem \(\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u=\varepsilon u+\mu _1 |u|^{2^*-2}u+\beta |v|^{\frac{2^*}{2}}|u|^{\frac{2^*}{2}-2} u\quad \text {in }\Omega ,\\ -\Delta v=\tau v+\mu _2 |v|^{2^*-2}v+\beta |u|^\frac{2^*}{2}|v|^{\frac{2^*}{2}-2} v\quad \text {in }\Omega ,\\ u=v=0 \quad \hbox {on }\partial \Omega . \end{array}\right. } \end{aligned}\) Here \(\Omega \subset \mathbb {R}^N (N\geqslant 4)\) is a smooth bounded domain and \(\varepsilon , \tau , \mu _1,\mu _2>0\) . For the attractive case \(\beta >0\) , the asymptotic behavior of least energy solutions as \((\varepsilon , \tau )\rightarrow (0,0)\) was studied by (Chen–Lin, Inter Math Res Not 2015:11045–11082, 2015), where they proved that the solutions blow up and converge to synchronized solutions of an elliptic system after scaling. In this paper, we study the repulsive case \(\beta <0\) . We show that as \((\varepsilon , \tau )\rightarrow (0,0)\) , two components of low-energy solutions both blow up but repel each other in the sense that \(\int _{\Omega }|u_{\varepsilon ,\tau }|^{{2^*}/{2}}|v_{\varepsilon ,\tau }|^{{2^*}/{2}} dx\rightarrow 0\) , and both components converge to bubble solutions of scalar equations after scaling. The pointwise estimates of both components are also given.
