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We study the following coupled Schr\"odinger system \ds -\Deltau+u=u^{2^*-1}+\be u^{\frac{2^*}{2}-1}v^{\frac{2^*}{2}}+\la_1u^{\al-1}, &x\in\R^N, \ds -\Delta v+v=v^{2^*-1}+\beu^{\frac{2^*}{2}}v^{\frac{2^*}{2}-1}+\la_2v^{r-1}, &x\in \R^N, u,v > 0, &x\in\R^N, where $N\geq 5, \la_1,\la_2>0,\be\neq 0, 2<\al,r<2^*,2^*\triangleq\frac{2N}{N-2}.$ Note that the nonlinearity and the coupling terms are bothcritical. Using the Mountain Pass Theorem, Ekeland's variational principle andNehari mainfold, we show that this critical system has a positive radialsolution for positive $\be$ and some negative $\be$ respectively.

Positive radial solutions for coupled Schr\"{o}dinger system with critical exponent in $\R^N\,(N\geq5)$