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Existence and Concentration of Positive Solutions for Nonlinear Kirchhoff-Type Problems with a General Critical Nonlinearity

Part of: Elliptic equations and systems Partial differential equations

Published online by Cambridge University Press:  17 July 2018

Jianjun Zhang ,
David G. Costa  and
João Marcos do Ó
Jianjun Zhang
Affiliation:
College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing 400074, PR China (zhangjianjun09@tsinghua.org.cn) Dip. di Scienza e Alta Tecnologia, Università degli Studi dell'Insubria, via Valleggio 11, 22100 Como, Italy
David G. Costa
Affiliation:
Department of Mathematical Sciences, University of Nevada, Las Vegas, PO Box 454020 NV, USA (costa@unlv.nevada.edu)
João Marcos do Ó
Affiliation:
Department of Mathematics, Federal University of Paraíba, 58051-900 João Pessoa-PB, Brazil (jmbo@pq.cnpq.br)
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Abstract

We are concerned with the following Kirchhoff-type equation

$$ - \varepsilon ^2M\left( {\varepsilon ^{2 - N}\int_{{\open R}^N} {\vert \nabla u \vert^2{\rm d}x} } \right)\Delta u + V(x)u = f(u),\quad x \in {{\open R}^N},\quad N{\rm \ges }2,$$
where M ∈ C(ℝ+, ℝ+), V ∈ C(ℝN, ℝ+) and f(s) is of critical growth. In this paper, we construct a localized bound state solution concentrating at a local minimum of V as ε → 0 under certain conditions on f(s), M and V. In particular, the monotonicity of f(s)/s and the Ambrosetti–Rabinowitz condition are not required.

Keywords

Kirchhoff equationsexistence and concentrationcritical growth

MSC classification

Primary: 35B25: Singular perturbations
Secondary: 35B33: Critical exponents 35J61: Semilinear elliptic equations
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 4 , November 2018 , pp. 1023 - 1040
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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