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Advanced Nonlinear Studies

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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
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  • 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

{-(ε2a+εb3|u|2dx)Δu+V(x)u=εμ-3(3K(y)F(u(y))|x-y|μdy)K(x)f(u),uH1(3),

where ε>0 is a small parameter, a,b>0, μ(0,3), 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 ε>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 ε0.

Keywords: Kirchhoff Equation; Hartree-type Nonlinearity; Multiplicity; Concentration; Variational Methods

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

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About the article


Received: 2019-03-10

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, DOI: https://doi.org/10.1515/ans-2019-2045.

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