Monatshefte für Mathematik

, Volume 182, Issue 2, pp 335–358

# 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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