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Solutions concentrating around the saddle points of the potential for critical Schrödinger equations

  • Jianjun Zhang
  • Wenming Zou
Article

Abstract

We consider the following singularly perturbed nonlinear elliptic problem
$$\begin{aligned} -\varepsilon ^2\Delta u+V(x)u=f(u),\ u\in H^1({\mathbb {R}}^N), \quad N\ge 3, \end{aligned}$$
where f is of critical growth. Using the variational techniques, we construct a solution \(u_\varepsilon \) which concentrates around the saddle points of V as \(\varepsilon \rightarrow 0\).

Mathematics Subject Classification

35J20 35B33 35J60 

Notes

Acknowledgments

The authors are deeply grateful to Professor Zhi-Qiang Wang for his kind support and fruitful discussion. They also would like to express their sincere gratitude to the anonymous referee for his/her careful reading and valuable suggestions. J. Zhang also thanks Dr. Zhijie Chen for his valuable comment on the proof of Lemma 4.7.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.College of Mathematics and StatisticsChongqing Jiaotong UniversityChongqingPeople’s Republic of China
  2. 2.Department of Mathematical SciencesTsinghua UniversityBeijingPeople’s Republic of China
  3. 3.Chern Institute of MathematicsNankai UniversityTianjinPeople’s Republic of China

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