Ljusternik-Schnirelmann theorem for the generalized Laplacian
type: Item
instance of: scholarly article
title: Ljusternik-Schnirelmann theorem for the generalized Laplacian
zbMATH Open document ID: 0946.35057
author: Matti Tienari
published in: Journal of Differential Equations
publication date: 2000-10-10T00:00:00Z
review text: The author proves a Ljusternik-Schnirelman theorem for the generalized Laplacian \(\Delta_m (u)= \nabla ((m(|\nabla u|)/|\nabla u|) \nabla u)\). Consider the problem (1) \(-\Delta_m(u)= \lambda g(u)\) in a bounded subset \(\Omega \subset {\mathbb R}^N\) with the segment property and \(u=0\) on \(\partial \Omega\). The function \(m\colon [0,\infty[\to [0,\infty[\) is supposed to be increasing and continuous such that \(m(t)=0\) if and only if \(t=0\) and \(m(t)\to\infty\) as \(t\to\infty\); the function \(g\colon {\mathbb R}\to {\mathbb R}\) is odd and satisfies some growth condition. Using a Galerkin approximation, the pseudomonotonicity of \(-\Delta_m\) and a modified usual structure of a complementary system of Orlicz-Sobolev spaces, it is proved that (1) admits an infinite sequence of eigenvalues that tends to infinity.
reviewed by: Youssef Jabri
Mathematics Subject Classification ID: 35P30
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 47J10
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 1437350
zbMATH Keywords: Ljusternik-Schnirelman theorem
zbMATH Keywords: generalized Laplacian
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1437350