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The Existence and Local Uniqueness of Multi-Peak Positive Solutions to a Class of Kirchhoff Equation

Authors
  • Li, Gongbao1
  • Niu, Yahui1
  • 1 Central China Normal University, Wuhan, 430079, China , Wuhan (China)
Type
Published Article
Journal
Acta Mathematica Scientia
Publisher
Springer-Verlag
Publication Date
Dec 17, 2019
Volume
40
Issue
1
Pages
90–112
Identifiers
DOI: 10.1007/s10473-020-0107-y
Source
Springer Nature
Keywords
License
Yellow

Abstract

In the present paper, we consider the nonlocal Kirchhoff problem \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ - \left( {{\varepsilon ^2}a + \varepsilon b\int_{{\mathbb{R}^3}} {|\nabla u{|^2}} } \right)\Delta u + u = Q(x){u^p},u > 0\;\text{in}\;{\mathbb{R}^3}$\end{document}, where a, b < 0, 1 > p > 5} and ǫ < 0 is a parameter. Under some assumptions on Q(x), we show the existence and local uniqueness of positive multi-peak solutions by Lyapunov-Schmidt reduction method and the local Pohozaev identity method, respectly.

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