Multiplicity and concentration of solutions for Kirchhoff equations with magnetic field
type: Item
instance of: scholarly article
title: Multiplicity and concentration of solutions for Kirchhoff equations with magnetic field
zbMATH Open document ID: 1475.35137
author: Chao Ji
author: Vicenţiu D. Rădulescu
published in: Advanced Nonlinear Studies
publication date: 2021-09-10T00:00:00Z
review text: The authors study a nonlinear magnetic Kirchhoff equation of the form \[ \begin{cases} -(a \varepsilon^2+b\varepsilon[u]^2_{A/\varepsilon})\Delta_{A/\varepsilon}u+V(x)u=f(|u|^2)u \text{ in } \mathbb{R}^3,\\ u\in H^1(\mathbb{R}^3,\mathbb{C}), \, \varepsilon >0, \, a,b>0 \text{ (constants)}. \end{cases} \] In this problem, \(-\Delta_{A}u= \left(\frac{1}{i}\nabla-A(x)\right)^2u\) is the magnetic Laplace operator, \([u]^2_{A}=\int_{\mathbb{R}^3}|\nabla_A u|^2dx\), \(A:\mathbb{R}^3\to\mathbb{R}^3\) is the magnetic potential, \(V:\mathbb{R}^3\to\mathbb{R}\) and \(f \in C(\mathbb{R},\mathbb{R})\) satisfy certain regularities. Combining variational methods, a penalization technique and the Ljusternik-Schnirelmann theory, the authors establish multiplicity properties of solutions and concentration phenomena for \(\varepsilon\) small.
reviewed by: Calogero Vetro
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 7394114
zbMATH Keywords: Kirchhoff equation
zbMATH Keywords: magnetic field
zbMATH Keywords: concentration
zbMATH Keywords: multiple solutions
zbMATH Keywords: variational methods
MaRDI profile type: Publication
full work available at URL: ans-2021-2130
OpenAlex ID: W3161186853
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DOI: 10.1515/ANS-2021-2130
description: scientific article; zbMATH DE number 7394114