On an integral equation of the form \(u(x)= F(x)+ \int G(x,\xi)u_ +^{(n-2)/2}(\xi)d\xi/ \int u_ +^{(n-2)/2}(\xi)d\xi\) for \(n=2\) and \(n=3\)
type: Item
instance of: scholarly article
title: On an integral equation of the form \(u(x)= F(x)+ \int G(x,\xi)u_ +^{(n-2)/2}(\xi)d\xi/ \int u_ +^{(n-2)/2}(\xi)d\xi\) for \(n=2\) and \(n=3\)
zbMATH Open document ID: 0827.45003
DOI: 10.1007/BF02110785
author: V. P. Maslov
published in: Mathematical Notes
publication date: 1995-07-10T00:00:00Z
review text: The integral equation \[ u(x)= F(x)+ {\int_{u> 0} G(x, \xi) u^{(n- 2)/2}(\xi)d\xi\over \int_{u> 0} u^{(n- 2)/2} (\xi)d\xi} \] is deduced heuristically, where \(x\in \mathbb{R}^n\), \(\xi\in \mathbb{R}^n\), the function \(G(x, \xi)\) plays the role of the potential of the particle interaction, and the function \(F(x)\) the role of an external potential. The most frequent cases in physics are for \(n= 3\), when the attractive or repulsive potential is of the form \({\alpha\over r}\), where \(\alpha> 0\) is a physical constant, and \(r= \sqrt{(x_1- \xi_1)^2+ (x_2- \xi_2)^2+ (x_3- \xi_3)^2}\) is the distance between particles. Under some conditions, the existence and uniqueness of a continuous solution for the above integral equation are studied.
Mathematics Subject Classification ID: 45H05
Mathematics Subject Classification ID: 35C15
zbMATH DE Number: 771993
zbMATH Keywords: integral equation
zbMATH Keywords: potential
zbMATH Keywords: particle interaction
zbMATH Keywords: external potential
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
zbMATH Keywords: continuous solution
reviewed by: Zongyi Hou
MaRDI profile type: Publication
cites work: On the integral equation \(u(x)=F(x)+\int G(x,\xi)u_ +^{k/2}(\xi)d\xi/\int u_ +^{k/2}(\xi)d\xi\)
cites work: SOME SMOOTH ERGODIC SYSTEMS
cites work: Nonlinear scalar field equations. II: Existence of infinitely many solutions
description: scientific article; zbMATH DE number 771993