Existence of two nontrivial solutions for a class of elliptic eigenvalue problems
type: Item
instance of: scholarly article
title: Existence of two nontrivial solutions for a class of elliptic eigenvalue problems
zbMATH Open document ID: 0963.35058
published in: Archiv der Mathematik
publication date: 2001-06-18T00:00:00Z
review text: Given a bounded domain \(\Omega\) in \({\mathbb R}^n\), \(n \geq 3\) with a regular boundary \(\partial \Omega\), the following family \((P_{\lambda, \mu})\) of elliptic eigenvalue problems is considered: \[ -\Delta u = \lambda (f(u) + \mu g(u)) \quad \text{in }\Omega, \qquad u = 0 \quad \text{on }\partial \Omega \] where \(\lambda\), \(\mu \in \mathbb R\) and \(f, g: {\mathbb R} \to {\mathbb R}\) are continuous. The main result gives sufficient conditions on \(f\) and \(g\) under which \((P_{\lambda, \mu})\) has at least two weak solutions in \(H^1_0(\Omega)\). The proof is based on the mountain pass theorem and suitable inequalities involving \(\lambda\) and \(\mu\).
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35P30
Mathematics Subject Classification ID: 35B30
zbMATH DE Number: 1507826
zbMATH Keywords: eigenvalue problem
zbMATH Keywords: mountain pass theorem
zbMATH Keywords: Palais-Smale condition
zbMATH Keywords: critical point
author: Salvatore A. Marano
author: Dumitru Motreanu
reviewed by: Stanisław Migórski
MaRDI profile type: Publication
full work available at URL: s000130050473
OpenAlex ID: W2082250279
DOI: 10.1007/S000130050473
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description: scientific article; zbMATH DE number 1507826