Spikes of the two-component elliptic system in R4 with the critical Sobolev exponent

Yuanze Wu China University of Mining and Technology Wenming Zou Tsinghua University

Analysis of PDEs mathscidoc:1901.03001

Calculus of Variations and Partial Differential Equations, 58, (1), 24, 2019.1
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 $\Omega\subset\bbr^4$ is a bounded domain, $\lambda_i,\mu_i,\alpha_i>0$ $(i=1,2)$ and $\beta\not=0$ are constants, $\ve>0$ 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 $\ve>0$. The concentration behaviors of least-energy solutions as $\ve\to0^+$ are also studied. Furthermore, by combining elliptic estimates and local energy estimates, we obtain the locations of these spikes as $\ve\to0^+$.
elliptic system; critical Sobolev exponent; spike; semi-classical solution; variational method
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@inproceedings{yuanze2019spikes,
  title={Spikes of the two-component elliptic system in R4 with the critical Sobolev exponent},
  author={Yuanze Wu, and Wenming Zou},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190109003457252089184},
  booktitle={Calculus of Variations and Partial Differential Equations},
  volume={58},
  number={1},
  pages={24},
  year={2019},
}
Yuanze Wu, and Wenming Zou. Spikes of the two-component elliptic system in R4 with the critical Sobolev exponent. 2019. Vol. 58. In Calculus of Variations and Partial Differential Equations. pp.24. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190109003457252089184.
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