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Semiclassical limits of ground state solutions to Schrödinger systems

  • Yanheng Ding
  • Cheng Lee
  • Fukun Zhao
Article

Abstract

This paper is concerned with the existence and concentration properties of the ground state solutions to the following coupled Schrödinger systems
$$\begin{aligned} \left\{ \begin{array}{l} -\varepsilon ^2\varDelta u+u+V(x)v=W(x)G_{v}(z)~\hbox { in }\ {\mathbb {R}}^N,\\ -\varepsilon ^2\varDelta v+v+V(x)u=W(x)G_{u}(z)~\hbox {in } \ {\mathbb {R}}^N,\\ u(x)\rightarrow 0\ \hbox {and }v(x)\rightarrow 0\ \hbox {as } \ |x|\rightarrow \infty , \end{array} \right. \end{aligned}$$
and
$$\begin{aligned} \left\{ \begin{array}{l} -\varepsilon ^2\varDelta u+u+V(x)v=W(x)(G_{v}(z)+|z|^{2^*-2}v)~\hbox {in } \ {\mathbb {R}}^N,\\ -\varepsilon ^2\varDelta v+v+V(x)u=W(x)(G_{u}(z)+|z|^{2^*-2}u)~\hbox {in } \ {\mathbb {R}}^N,\\ u(x)\rightarrow 0\ \hbox {and }v(x)\rightarrow 0\ \hbox {as } \ |x|\rightarrow \infty , \end{array} \right. \end{aligned}$$
where \(z=(u,v)\in {\mathbb {R}}^2\), \(G\) is a power type nonlinearity, having superquadratic growth at both \(0\) and infinity but subcritical, \(V\) can be sign-changing and \(\inf W>0\). We prove the existence, exponential decay, \(H^2\)-convergence and concentration phenomena of the ground state solutions for small \(\varepsilon >0\).

Mathematics Subject Classification

35J50 58E05 

Notes

Acknowledgments

The authors would express their thanks to unknown referee for his/her careful reading and suggestions which improve the work. This work was completed during a visit by Y. Ding and F. Zhao to the Department of Mathematics of NCUE in Taiwan. They would like to thank NCUE for its hospitality and support. Also, the first author was supported by NSFC(11331010 and 10421001), China. The second author was supported by NSC(101-2115-M018-001). The corresponding author was supported by NSFC (11061040 and 11361078), Key Project of Chinese Ministry of Education (No212162) and NSFY of Yunnan Province (2011CI020), China.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Institute of Mathematics, AMSS, CASBeijingPeople’s Republic of China
  2. 2.Department of MathematicsNational Changhua University of EducationChanghuaTaiwan
  3. 3.Department of MathematicsYunnan Normal UniversityKunmingPeople’s Republic of China

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