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

Abstract

Consider the following elliptic system: \begin{equation*} \left\{\aligned&-\ve^2Δu_1+λ_1u_1=μ_1u_1^3+α_1u_1^{p-1}+βu_2^2u_1\quad&\text{in}Ω,\\ &-\ve^2Δu_2+λ_2u_2=μ_2u_2^3+α_2u_2^{p-1}+βu_1^2u_2\quad&\text{in}Ω,\\ &u_1,u_2>0\quad\text{in}Ω,\quad u_1=u_2=0\quad\text{on}\partialΩ,\endaligned\right. \end{equation*} where $Ω\subset\bbr^4$ is a bounded domain, ${\lambda }_i,{\mu }_i,{\alpha }_i>0(i=1,2)$ and $\beta \ne 0$ are constants, $\ve>0$ 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 $\ve>0$ 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 $\bbr^4$.

39 pages