Existence and uniqueness of positive solutions of semilinear elliptic equations with Coulomb potentials on \({\mathbb{R}}^ 3\)
type: Item
instance of: scholarly article
title: Existence and uniqueness of positive solutions of semilinear elliptic equations with Coulomb potentials on \({\mathbb{R}}^ 3\)
zbMATH Open document ID: 0596.35014
DOI: 10.1007/BF01211596
author: Lorenzo Jeanneret
author: Rafael D. Benguria
published in: Communications in Mathematical Physics
publication date: 1986-01-01T00:00:00Z
review text: The authors discuss the existence and uniqueness of positive classical solutions u(x) of the problem \[ (1)\quad -\Delta u-Z | x|^{-1} u+a(u)=0\quad in\quad {\mathbb{R}}^ 3,\quad u(x)\to 0\quad as\quad x\to \infty. \] In (1), Z is a positive real constant and the function a(.) satisfies some hypothesis. The following important results are obtained. First, if \(u\in E\) is a solution of (1), then (i) u is continuous in \({\mathbb{R}}^ 3\), more precisely \(u\in C^{0,\alpha}\) all \(\alpha <1\); (ii) u is bounded in \({\mathbb{R}}^ 3\) and u(x)\(\to 0\) as \(| x| \to \infty\); (iii) \(u\in H^ 2\); (iv) either \(u\equiv 0\) or \(u>0\) everywhere \(u\in C^ 2\) except at \(x=0\); (v) if u has radial symmetry, it satisfies the following cusp condition at the origin: (2) \(2(d/dr)u(0)+Zu(0)=0.\) Besides, they only consider the radial equation \[ (3)\quad -u''-(2/r)u'- (Z/r)u+a(u)=0,\quad u'=(du/dr),\quad etc. \] It is also proved that there is at most one positive solution of (3), (2) which goes to zero at infinity, under some hypothesis. Finally, they further apply the existence and uniqueness results to the Thomas-Fermi-Dirac-von Weizsäcker equation without electrostatic repulsion.
Mathematics Subject Classification ID: 35G15
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35B05
zbMATH DE Number: 3960036
zbMATH Keywords: semilinear elliptic equation
zbMATH Keywords: radial symmetry
zbMATH Keywords: cusp condition
zbMATH Keywords: positive solution
zbMATH Keywords: Thomas-Fermi-Dirac-von Weizsäcker equation
Wikidata QID: Q59158391
reviewed by: Li Mingzhong
MaRDI profile type: Publication
cites work: Thomas-fermi and related theories of atoms and molecules
cites work: The Thomas-Fermi-von Weizsäcker theory of atoms and molecules
cites work: Semi-classical limit theorems for Hartree-Fock theory
cites work: Analysis of the Thomas-Fermi-von Weizsäcker equation for an infinite atom without electron repulsion
cites work: Uniqueness of positive solutions of semilinear equations in \(R^ n\).
cites work: Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus
cites work: A general rearrangement inequality for multiple integrals
cites work: Q5187597
cites work: Symmetry and related properties via the maximum principle
cites work: Q3923023
description: scientific article; zbMATH DE number 3960036