Multiple solutions for resonant semilinear elliptic problems in \(\mathbb R^N\)
type: Item
instance of: scholarly article
title: Multiple solutions for resonant semilinear elliptic problems in \(\mathbb R^N\)
zbMATH Open document ID: 1160.35407
author: Gabriel López Garza
author: Adolfo Rumbos
published in: Journal of Mathematical Analysis and Applications
publication date: 2005-04-21T00:00:00Z
review text: The authors study the existence of the nontrivial weak solutions of the problem \[ -\Delta u= h(x)(\lambda u+ g(u)),\quad x\in\mathbb{R}^N,\quad u\in D^{1,2}, \] where \(\lambda\) is a real parameter, \(h: \mathbb{R}^N\to \mathbb{R}\) and \(g: \mathbb{R}\to \mathbb{R}\) are functions satisfying natural assumptions. Here \(D^{1,2}\) denotes the Hilbert space obtained through completion of \(C^\infty_c(\mathbb{R}^N)\), where \(N\geq 3\) with respect to the norm \(\| u\|:= (\int|\nabla u|^2\, dx)^{1/2}\). Existence of multiple solutions, for certain values of \(g'(0)\), is obtained by imposing a generalized Landesman-Lazer type condition. To this end, the authors use the saddle point theorem, the mountain pass theorem, as well as a Morse-index result with a combination a Leray-Schauder index theorem for mountain pass type critical points. The results of this paper are based upon multiplicity results for resonant problems on bounded domains.
reviewed by: Messoud Efendiev
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 47J30
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 2159795
zbMATH Keywords: Resonance
zbMATH Keywords: Landesman-Lazer condition
zbMATH Keywords: Saddle point
zbMATH Keywords: Mountain pass type critical point
zbMATH Keywords: Morse index
zbMATH Keywords: Leray-Schauder index
MaRDI profile type: Publication
full work available at URL: j.jmaa.2004.12.013
OpenAlex ID: W2009046543
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DOI: 10.1016/J.JMAA.2004.12.013
description: scientific article; zbMATH DE number 2159795