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# The Concentration Behavior of Ground States for a Class of Kirchhoff-type Problems with Hartree-type Nonlinearity

Guangze Gu
• School of Mathematics and Statistics, Central South University, Changsha, 410083 Hunan, P. R. China; and Department of Mathematics, University of Texas at San Antonio, San Antonio, 78249 Texas, USA
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/ Xianhua Tang
• Corresponding author
• School of Mathematics and Statistics, Central South University, Changsha, 410083 Hunan, P. R. China
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Published Online: 2019-05-08 | DOI: https://doi.org/10.1515/ans-2019-2045

## Abstract

In this paper, we consider the Kirchhoff equation with Hartree-type nonlinearity

$\left\{\begin{array}{cc}\hfill -& \left({\epsilon }^{2}a+\epsilon b{\int }_{{ℝ}^{3}}{|\nabla u|}^{2}dx\right)\mathrm{\Delta }u+V\left(x\right)u={\epsilon }^{\mu -3}\left({\int }_{{ℝ}^{3}}\frac{K\left(y\right)F\left(u\left(y\right)\right)}{{|x-y|}^{\mu }}dy\right)K\left(x\right)f\left(u\right),\hfill \\ & u\in {H}^{1}\left({ℝ}^{3}\right),\hfill \end{array}$

where $\epsilon >0$ is a small parameter, $a,b>0$, $\mu \in \left(0,3\right)$, $V,K$ are two positive continuous function and F is the primitive function of f which is superlinear but subcritical at infinity in the sense of the Hardy–Littlewood–Sobolev inequality. We show that the equation admits a positive ground state solution for $\epsilon >0$ sufficiently small. Furthermore, we prove that these ground state solutions concentrate around such points which are both the minima points of the potential V and the maximum points of the potential K as $\epsilon \to 0$.

MSC 2010: 35J60; 35B25; 35B40; 35J20

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Revised: 2019-04-02

Accepted: 2019-04-04

Published Online: 2019-05-08

Funding Source: China Scholarship Council

Award identifier / Grant number: 201806370022

Funding Source: Hunan Provincial Innovation Foundation for Postgraduate

Award identifier / Grant number: CX2018B052

Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11571370

This work was supported by China Scholarship Council (201806370022), Hunan Provincial Innovation Foundation for Postgraduate (CX2018B052) and National Natural Science Foundation of China (11571370).

Citation Information: Advanced Nonlinear Studies, ISSN (Online) 2169-0375, ISSN (Print) 1536-1365,

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