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Monatshefte für Mathematik

, Volume 182, Issue 2, pp 335–358 | Cite as

Existence and concentration of ground state solutions for singularly perturbed nonlocal elliptic problems

  • Dengfeng Lü
Article

Abstract

We consider the following singularly perturbed nonlocal elliptic problem
$$\begin{aligned} -\left( \varepsilon ^{2}a+\varepsilon b\displaystyle \int _{\mathbb {R}^{3}}|\nabla u|^{2}dx\right) \Delta u+V(x)u=\displaystyle \varepsilon ^{\alpha -3}(W_{\alpha }(x)*|u|^{p})|u|^{p-2}u, \quad x\in \mathbb {R}^{3}, \end{aligned}$$
where \(\varepsilon >0\) is a parameter, \(a>0,b\ge 0\) are constants, \(\alpha \in (0,3)\), \(p\in [2, 6-\alpha )\), \(W_{\alpha }(x)\) is a convolution kernel and V(x) is an external potential satisfying some conditions. By using variational methods, we establish the existence and concentration of positive ground state solutions for the above equation.

Keywords

Kirchhoff-type equation Hartree-type nonlinearity  Variational method Ground state solution Concentration phenomena 

Mathematics Subject Classification

35J60 35B25 35B40 

Notes

Acknowledgments

The author is grateful to Professor Shuangjie Peng for his helpful suggestions and careful guidance.

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Copyright information

© Springer-Verlag Wien 2016

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsCentral China Normal UniversityWuhanPeople’s Republic of China
  2. 2.School of Mathematics and StatisticsHubei Engineering UniversityXiaoganPeople’s Republic of China

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