Spikes of the two-component elliptic system in ${\text{bbr}}^4$ with Sobolev critical exponent

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 $\mathrm{\Omega }\subset {\text{bbr}}^4$ is a bounded domain, ${\lambda }_i,{\mu }_i,{\alpha }_i>0(i=1,2)$ and $\beta ≠0$ are constants, $\text{ve}>0$ is a small parameter and $2<p<2^{\ast }=4$. By using the variational method, we study the existence of the ground state solution to this system for $\text{ve}>0$ small enough. The concentration behavior of the ground state solution as $\text{ve}\to 0^{+}$ is also studied. Furthermore, by combining the elliptic estimates and local energy estimates, we also obtain the location of the spikes as $\text{ve}\to 0^{+}$. To the best of our knowledge, this is the first attempt devoted to the spikes in the Bose-Einstein condensate in ${\text{bbr}}^4$.