We study the following coupled Schr\"odinger system \ds -\Delta u+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}+\be u^{ \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 both critical. Using the Mountain Pass Theorem, Ekeland's variational principle and Nehari mainfold, we show that this critical system has a positive radial solution for positive $\be$ and some negative $\be$ respectively.
Categories: math.AP