Infinitely many radial solutions of an elliptic system
type: Item
instance of: scholarly article
title: Infinitely many radial solutions of an elliptic system
zbMATH Open document ID: 0652.35030
DOI: 10.1016/S0294-1449(16)30356-0
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 1987-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/78143
review text: The system \[ \Delta u_ 1+f_ 1(u_ 1,u_ 2)=0,\quad \Delta u_ 2+f_ 2(u_ 1,u_ 2)=0,\quad u_ i=u_ i(x),\quad x\in {\mathbb{R}}\quad n,\quad n>1, \] is studied with \(f_ i=\partial F/\partial u_ i\) for some \(F\in C^ 2({\mathbb{R}}^ 2)\). Under some technical assumptions on F two main results are presented: The existence of infinitely many radial solutions and the existence of a radial solution with prescribed winding number K or \(K+1\), \(K\in {\mathbb{N}}\). The proof is split up into three papers of which two are forthcoming. In the paper in question the notion of a winding number - counting how many times a trajectory winds around certain two dimensional phases - is introduced. A family of equations depending on \(\epsilon >0\) is studied and the existence of infinitely many solutions to this perturbed problem is shown. A geometrical proof is given. As \(\epsilon\to 0\) some sequence of the set of solutions with winding number K (assumed here to be given) will converge to a solution of the original problem with winding number K. For this, a priori estimates are proved.
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35A30
Mathematics Subject Classification ID: 35J15
zbMATH DE Number: 4063627
zbMATH Keywords: existence
zbMATH Keywords: infinitely many radial solutions
zbMATH Keywords: winding number
zbMATH Keywords: perturbed problem
zbMATH Keywords: a priori estimates
author: D. Terman
reviewed by: Reinhard Racke
MaRDI profile type: Publication
cites work: Ground states of \(-\Delta u=f(u)\) and the Emden-Fowler equation
cites work: Nonlinear scalar field equations. II: Existence of infinitely many solutions
cites work: On the existence and structure of stationary states for a nonlinear Klein-Gordon equation
cites work: Minimum action solutions of some vector field equations
cites work: Q4180600
cites work: Q5330830
cites work: Existence of solitary waves in higher dimensions
cites work: Traveling Wave Solutions of a Gradient System: Solutions with a Prescribed Winding Number. II
cites work: Q5751269
description: scientific article; zbMATH DE number 4063627