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Spikes of the two-component elliptic system in \({\mathbb {R}}^4\) with the critical Sobolev exponent

  • Yuanze WuEmail author
  • Wenming Zou
Article
  • 55 Downloads

Abstract

Consider the following elliptic system:
$$\begin{aligned} \left\{ \begin{aligned}&-\varepsilon ^2\Delta u_1+\lambda _1u_1=\mu _1u_1^3+\alpha _1u_1^{p-1}+\beta u_2^2u_1 \quad \text {in }\quad \Omega ,\\&-\varepsilon ^2\Delta u_2+\lambda _2u_2=\mu _2u_2^3+\alpha _2u_2^{p-1}+\beta u_1^2u_2 \quad \text {in }\quad \Omega ,\\&u_1,u_2>0\quad \text {in }\quad \Omega ,\quad u_1=u_2=0\quad \text {on }\quad \partial \Omega , \end{aligned}\right. \end{aligned}$$
where \(\Omega \subset {\mathbb {R}}^4\) is a bounded domain, \(\lambda _i,\mu _i,\alpha _i>0\) \((i=1,2)\) and \(\beta \not =0\) are constants, \(\varepsilon >0\) is a small parameter and \(2<p<2^*=4\). By using variational methods, we study the existence of ground state solutions to this system for sufficiently small \(\varepsilon >0\). The concentration behaviors of least-energy solutions as \(\varepsilon \rightarrow 0^+\) are also studied. Furthermore, by combining elliptic estimates and local energy estimates, we obtain the locations of these spikes as \(\varepsilon \rightarrow 0^+\).

Mathematics Subject Classification

35B09 35B33 35J50 

Notes

Acknowledgements

Y. Wu is supported by NSFC (11701554, 11771319), the Fundamental Research Funds for the Central Universities (2017XKQY091) and Jiangsu overseas visiting scholar program for university prominent young and middle-aged teachers and presidents. W. Zou is supported by NSFC (11771234). The authors also would like to thank the anonymous referee for very carefully reading the manuscript and wonderfully valuable comments that greatly improve this paper.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of MathematicsChina University of Mining and TechnologyXuzhouPeople’s Republic of China
  2. 2.Department of Mathematical SciencesTsinghua UniversityBeijingPeople’s Republic of China

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