A priori bounds and renormalized Morse indices of solutions of an elliptic system
type: Item
instance of: scholarly article
title: A priori bounds and renormalized Morse indices of solutions of an elliptic system
zbMATH Open document ID: 0964.35037
DOI: 10.1016/S0294-1449(00)00110-4
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 2001-07-02T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/78494
review text: Let \(\Omega\) be a smooth bounded domain in \({\mathbb R}^N\), \(N\geq 3\). Assume \(u\) and \(v\) are arbitrary solutions of the system \(-\Delta u=b(x)v^q\), \(-\Delta v=a(x)u^p\), where \(p,q>1\) and \(1/(p+1)+1/(q+1)>1-2/N\). After defining a ``renormalized'' Morse index, the authors prove the following Bahri-Lions type result: Set \[ f(u,v)=\int_\Omega\{\nabla u\cdot\nabla v-a(x)|u|^{p+1}/(p+1)- b(x)|v|^{q+1}/(q+1)\} dx. \] Then, for any integer \(m\geq 1\), there exists a constant \(C_m\) depending on \(a\), \(b\) and \(\Omega\) such that any critical point of \(f\) with lower Morse index \(\mu_-(z_0)\leq m\) satisfies \(\sup_\Omega\{ |u|,|v|\}\leq C_m\).
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 49J35
Mathematics Subject Classification ID: 49K20
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 1487048
zbMATH Keywords: Morse index
zbMATH Keywords: elliptic system
zbMATH Keywords: critical point
author: S. B. Angenent
author: Robert Vandervorst
reviewed by: Vicenţiu D. Rădulescu
MaRDI profile type: Publication
cites work: Solutions of superlinear elliptic equations and their morse indices
cites work: Q5536163
cites work: On the Morse index in variational calculus
cites work: The Spectral Flow and the Maslov Index
cites work: Q5607484
OpenAlex ID: W2041635046
description: scientific article; zbMATH DE number 1487048