On semiclassical states for Dirac equations
type: Item
instance of: scholarly article
title: On semiclassical states for Dirac equations
zbMATH Open document ID: 1480.35337
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 2021-05-19T00:00:00Z
review text: This paper is devoted to the study of the existence and concentration of solutions for the stationary Dirac equation \[i\varepsilon\sum_{k=1}^{3}\alpha_{k}\partial_{k}u-a\beta u+V\left( x\right) u=P\left( x\right) f\left( \left\vert u\right\vert \right) u+Q\left( x\right) \left\vert u\right\vert u,\tag{Dirac} \] where \(\varepsilon>0\) is a small parameter, \(a>0\) is a constant, \(V,P,Q\in C\left( \mathbb{R}^{3},\mathbb{R}\right) \) are bounded functions, \(V(x)\) has a negative global minimum and \(P(x)\), \(G(x)\) have a positive global maximum. The aim of this paper is to study how the behavior of the three potentials \(V,P,Q\) affects the existence and concentration (\(\varepsilon\rightarrow0^{+} \)) of semiclassical solutions to (Dirac). The presence of three potentials creates a competition between them: each would try to attract ground states to their minimum and maximum points. First, for any \(\varepsilon>0\) small enough a least energy solution is obtained. Then, the concentration behaviour of these solution as \(\varepsilon\rightarrow0\) is investigated. The concentration position of least solutions is determinated in terms of the potentials \(V,P,Q\): they concentrate at such points \(x\) where \(V(x)\) is small or both \(P(x)\) and \(Q(x)\) are large. Moreover, the exponantial decay estimate of these solutions is deduced.
Mathematics Subject Classification ID: 35Q40
Mathematics Subject Classification ID: 49J35
Mathematics Subject Classification ID: 81Q05
Mathematics Subject Classification ID: 35A01
zbMATH DE Number: 7349839
zbMATH Keywords: nonlinear Dirac equation
zbMATH Keywords: least energy solutions
zbMATH Keywords: concentration
author: A. Benhassine
reviewed by: I. P. Naumkin
MaRDI profile type: Publication
full work available at URL: s00033-021-01541-7
OpenAlex ID: W3157027298
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DOI: 10.1007/S00033-021-01541-7
description: scientific article; zbMATH DE number 7349839