  Consider the following elliptic system: \begin{equation*}
\left\{\aligned&-\ve^2\Delta
u_1+\lambda_1u_1=\mu_1u_1^3+\alpha_1u_1^{p-1}+\beta
u_2^2u_1\quad&\text{in}\Omega,\\ &-\ve^2\Delta
u_2+\lambda_2u_2=\mu_2u_2^3+\alpha_2u_2^{p-1}+\beta
u_1^2u_2\quad&\text{in}\Omega,\\ &u_1,u_2>0\quad\text{in}\Omega,\quad
u_1=u_2=0\quad\text{on}\partial\Omega,\endaligned\right. \end{equation*} where
$\Omega\subset\bbr^4$ is a bounded domain, $\lambda_i,\mu_i,\alpha_i>0(i=1,2)$
and $\beta\not=0$ are constants, $\ve>0$ is a small parameter and $2<p<2^*=4$.
By using the variational method, we study the existence of the ground state
solution to this system for $\ve>0$ small enough. The concentration behavior of
the ground state solution as $\ve\to0^+$ is also studied. Furthermore, by
combining the elliptic estimates and local energy estimates, we also obtain the
location of the spikes as $\ve\to0^+$. To the best of our knowledge, this is
the first attempt devoted to the spikes in the Bose-Einstein condensate in
$\bbr^4$.
