Multiple solutions for asymptotically linear resonant elliptic problems
type: Item
instance of: scholarly article
title: Multiple solutions for asymptotically linear resonant elliptic problems
zbMATH Open document ID: 1112.35077
DOI: 10.12775/TMNA.2003.014
published in: Topological Methods in Nonlinear Analysis
publication date: 2003-10-27T00:00:00Z
review text: The author considers the nonlinear elliptic problem \[ \begin{cases} -\Delta u= g(x,u)\quad &\text{in }\Omega,\\ u= 0\quad\text{on }\partial\Omega,\end{cases}\tag{P} \] where \(\Omega\) is a bounded domain in \(\mathbb R^N\) with smooth boundary \(\partial\Omega\), \(g:\Omega\times \mathbb R\to \mathbb R\) is a function of class \(C^1\) such that \(g(x, 0)= 0\) and which is asymptotically linear at infinity. He defines critical groups starting with homological groups with coefficients in \(\mathbb Z\). He uses some theorems in \textit{K. C. Chang} [Infinite-dimensional Morse theory and multiple solution problems, Birkhäuser, Boston (1993; Zbl 0779.58005)] to compute explicitely these groups in order to prove existence of multiple solutions of (P). He studies resonant and nonresonant cases.
reviewed by: Mohammed Bouchekif
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35B38
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 47J30
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 1997216
zbMATH Keywords: Cerani condition
zbMATH Keywords: multiple solutions
zbMATH Keywords: Morse theory
zbMATH Keywords: double resonance
author: Francisco Odair de Paiva
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1997216