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

  • Yuanze Wu


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.


Elliptic system Sobolev critical exponent Spike Trapping potential Variational method 

Mathematics Subject Classification

35B09 35B33 35J50 



The author was supported by the Fundamental Research Funds for the Central Universities (2017XKQY091). The author also would like to thank the anonymous referee for very carefully reading the manuscript and wonderful valuable comments that greatly improve this paper.


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Authors and Affiliations

  1. 1.School of MathematicsChina University of Mining and TechnologyXuzhouPeople’s Republic of China

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