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Study the following two-component elliptic system% \begin{equation*}\left\{\aligned&\Delta u-(\lambda a(x)+a_0)u+u^3+\beta v^2u=0\quad&\text{in}\bbr^4,\\% &\Delta v-(\lambda b(x)+b_0)v+v^3+\beta u^2v=0\quad&\text{in}\bbr^4,\\% &(u,v)\in\h\times\h,\endaligned\right.% \end{equation*} where$a_0,b_0\in\bbr$ are constants; $\lambda>0$ and $\beta\in\bbr$ are parametersand $a(x), b(x)\geq0$ are potential wells which are not necessarily to beradial symmetric. By using the variational method, we investigate the existenceof ground state solutions and general ground state solutions (i.e., possiblysemi-trivial) to this system. Indeed, to the best of our knowledge, even theexistence of semi-trivial solutions is also unknown in the literature. Weobserve some concentration behaviors of ground state solutions and generalground state solutions. The phenomenon of phase separations is also excepted.It seems that this is the first result definitely describing the phenomenon ofphase separation for critical system in the whole space $\bbr^4$. Note thatboth the cubic nonlinearities and the coupled terms of the system are all ofcritical growth with respect to the Sobolev critical exponent.

On a doubly critical Schr\"odinger system in $\bbr^4$ with steep potential wells