Morse theory without (PS) condition at isolated values and strong resonance problems
type: Item
instance of: scholarly article
title: Morse theory without (PS) condition at isolated values and strong resonance problems
zbMATH Open document ID: 0956.58007
author: Norimichi Hirano
author: Zhi-Qiang Wang
author: Shujie Li
published in: Calculus of Variations and Partial Differential Equations
publication date: 2001-03-04T00:00:00Z
review text: The authors consider the elliptic boundary value problem \[ -\Delta u = \lambda u + q(x,u) \quad\text{in }\Omega, \qquad u=0 \quad\text{on }\partial\Omega,\tag{*} \] where \(\Omega\subset{\mathbb R}^{N}\) is bounded, \(\lambda\) is an eigenvalue of \(-\Delta\), \(q(x,0)=0\) and \(q(x,t)\to 0\), \(q'_{t}(x,t)\to 0\) as \(|t|\to\infty\). Solutions of (*) correspond to critical points of the functional \(I(u) = {1\over 2}\int_{\Omega}(|\nabla u|^2-\lambda u^2) dx - \int_{\Omega}Q(x,u) dx\) in the Sobolev space \(H^1_{0}(\Omega)\); here \(Q(x,t) := \int_{0}^tq(x,s) ds\). An important role in the analysis of (*) is played by the set \(\Lambda\) of values \(c\) such that for some \(v\in \text{Ker} (-\Delta -\lambda) \setminus \{0\}\), \(-\int_{\Omega}Q(x,tv) dx\to c\) as \(t\to\infty\). If \(\Lambda\) is bounded, then problem (*) is called strongly resonant. In this paper it is shown that if \(\Lambda\) is a finite set, then the functional \(I\) satisfies the Morse inequalities, with an extra part in the Morse polynomial compensating for the loss of the Palais-Smale condition at the levels \(c\in\Lambda\). This result is subsequently employed in order to show that under different sets of hypotheses, (*) contains 2, 3 or 4 nontrivial solutions. Here the set \(\Lambda\) is a singleton if \(\dim\text{Ker} (-\Delta -\lambda) \geq 2\) and consists of 1 or 2 points if \(\dim\text{Ker} (-\Delta -\lambda) = 1\). The results of this paper considerably extend earlier work on strongly resonant problems where it was usually assumed that \(\Lambda\) is a singleton and existence of only 1 nontrivial solution was shown. The fact that strongly resonant problems can be treated by a Morse theory which is very close to the usual one is of independent interest.
reviewed by: Andrzej Szulkin
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 47J30
zbMATH DE Number: 1455253
zbMATH Keywords: elliptic boundary value problem
zbMATH Keywords: Morse inequalities
zbMATH Keywords: multiple solutions
zbMATH Keywords: strong resonance
MaRDI profile type: Publication
full work available at URL: s005260050152
OpenAlex ID: W2009683859
DOI: 10.1007/S005260050152
description: scientific article; zbMATH DE number 1455253