Properties of the positive solution of a generalized Thomas-Fermi-von Weizsäcker equation
type: Item
instance of: scholarly article
title: Properties of the positive solution of a generalized Thomas-Fermi-von Weizsäcker equation
zbMATH Open document ID: 0733.35012
DOI: 10.1215/S0012-7094-91-06309-X
author: Mark S. Ashbaugh
author: Rafael D. Benguria
author: Cecilia Yarur
published in: Duke Mathematical Journal
publication date: 1991-01-01T00:00:00Z
review text: This paper is devoted to a qualitative study of the positive solutions of a generalized Thomas-Fermi-von Weizsäcker equation: \[ -\Delta u+(cu^ q-\phi)u=-\phi_ 0u \] in \({\mathbb{R}}^ 3\), where \(\phi =V(x)-\int (u^ 2(y)/| x-y|)dy\) is the total electric potential, \(V(x)=\sum^{k}_{i=1}z_ i/| x-R_ i|,\) \(q\geq 1\), \(c\geq 0\). Set \(\sum^{k}_{i=1}z_ i=Z\). For \(q>1\) and \(k=1\) it is proved that \(\int u^ 2(y)dy>Z\) a result previously known for \(q=4/3\) and arbitrary k. The paper also covers the case \(q=1\) where the validity of the above inequality is shown to depend upon c. Moreover in this case an asymptotic analysis of the solution is carried out.
reviewed by: Liviu I. Nicolaescu
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 45K05
Mathematics Subject Classification ID: 35Q40
zbMATH DE Number: 4212561
zbMATH Keywords: L\({}^ 2\)-estimate from below
zbMATH Keywords: positive solutions
zbMATH Keywords: Thomas-Fermi-von Weizsäcker equation
zbMATH Keywords: asymptotic analysis
Wikidata QID: Q59158321
MaRDI profile type: Publication
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full work available at URL: s0012-7094-91-06309-x
OpenAlex ID: W1605214954
description: scientific article; zbMATH DE number 4212561