On the semiclassical solutions of a two-component elliptic system in $\mathbb {R}^4$ with trapping potentials and Sobolev critical exponent: the repulsive case

Abstract

Consider the following elliptic system:

$$\begin{aligned} \left\{ \begin{array}{ll} -\varepsilon ^2\Delta u_1+V_1(x)u_1=\mu _1u_1^3+\alpha _1u_1^{p-1}+\beta u_2^2u_1&{}\quad \text {in }\mathbb {R}^4,\\ -\varepsilon ^2\Delta u_2+V_2(x)u_2=\mu _2u_2^3+\alpha _2u_2^{p-1}+\beta u_1^2u_2&{}\quad \text {in }\mathbb {R}^4,\\ u_1,u_2>0\quad \text {in }\mathbb {R}^4,\quad u_1,u_2\in H^1(\mathbb {R}^4), &{}\\ \end{array}\right. \end{aligned}$$

where $V_i(x)$ are trapping potentials, $\mu _i,\alpha _i>0(i=1,2)$ and $\beta <0$ are constants, $\varepsilon >0$ is a small parameter, and $2<p<2^*=4$. By using the variational method, we obtain a solution to this system for $\varepsilon >0$ small enough. The concentration behaviors of this solution as $\varepsilon \rightarrow 0^+$, involving the location of the spikes, are also studied by combining the uniformly elliptic estimates and local energy estimates. To the best of our knowledge, this is the first result devoted to the spikes in the Bose–Einstein condensate with trapping potentials in $\mathbb {R}^4$ for the repulsive case.