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Existence and asymptotic behaviour of ground state solutions for Kirchhoff-type equations with vanishing potentials

  • Dongdong SunEmail author
  • Zhitao Zhang
Article
  • 28 Downloads

Abstract

In this paper, we investigate and obtain the existence and asymptotic behaviour of the positive ground state solutions to Kirchhoff-type equations with vanishing potentials:
$$\begin{aligned} \left\{ \begin{array}{ll} -\left( \varepsilon ^2a+\varepsilon b{\int \limits _{\mathbb {R}^3}}{|\nabla v|}^{2}\right) \Delta v+V(x)v=v^{p} &{}\text {in}~\mathbb {R}^3, \\ v>0,&{}v\in H^{1}(\mathbb {R}^3), \end{array} \right. \end{aligned}$$
where \(\varepsilon >0\) is a small parameter, \(a,b>0\) are constants, \(3<p<5\). The potential V can vanish, i.e. \({\mathcal {Z}}:=\{x\in \mathbb {R}^3|V(x)=0\}\ne \emptyset \).

Keywords

Kirchhoff-type problems Vanishing potentials Schrödinger equation Ground state solution Asymptotic behaviour 

Mathematics Subject Classification

35J60 35J20 35B38 

Notes

Acknowledgements

The authors would like to express sincere thanks to the anonymous referees for their carefully reading the manuscript and valuable comments and suggestions.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of MathematicsQilu Normal UniversityJinanPeople’s Republic of China
  2. 2.Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingPeople’s Republic of China
  3. 3.School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijingPeople’s Republic of China

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