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Multi-bump solutions for Choquard equation with deepening potential well

  • Claudianor O. Alves
  • Alânnio B. Nóbrega
  • Minbo Yang
Article

Abstract

In this paper we study the existence of multi-bump solutions for the following Choquard equation
$$\begin{aligned} \begin{array}{ll} -\Delta u + (\lambda a(x)+1)u=\displaystyle \big (\frac{1}{|x|^{\mu }}*|u|^p\big )|u|^{p-2}u \text{ in } \,\,\, \mathbb {R}^3, \end{array} \end{aligned}$$
where \(\mu \in (0,3), p\in (2, 6-\mu )\), \(\lambda \) is a positive parameter and the nonnegative continuous function a(x) has a potential well \( \Omega :=int (a^{-1}(0))\) which possesses k disjoint bounded components \( \Omega :=\cup _{j=1}^{k}\Omega _j\). We prove that if the parameter \(\lambda \) is large enough, then the equation has at least \(2^{k}-1\) multi-bump solutions.

Mathematics Subject Classification

35J20 35J65 

Notes

Acknowledgments

The authors would like to thank the anonymous referee for his/her useful comments and suggestions which help to improve and clarify the paper greatly.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Claudianor O. Alves
    • 1
  • Alânnio B. Nóbrega
    • 1
  • Minbo Yang
    • 2
  1. 1.Universidade Federal de Campina GrandeUnidade Acadêmica de MatemáticaCampina GrandeBrazil
  2. 2.Department of MathematicsZhejiang Normal UniversityJinhuaPeople’s Republic of China

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