On principal eigenvalues for quasilinear elliptic differential operators: an Orlicz-Sobolev space setting
type: Item
instance of: scholarly article
title: On principal eigenvalues for quasilinear elliptic differential operators: an Orlicz-Sobolev space setting
zbMATH Open document ID: 0936.35067
author: Vy Khoi Le
author: Raúl Manásevich
author: Marta Garcia-Huidobro
author: Klaus Schmitt
published in: NoDEA. Nonlinear Differential Equations and Applications
publication date: 1999-09-15T00:00:00Z
review text: The existence of principal eigenvalues (i.e., having a positive eigenfunction) for some quasilinear elliptic operators generalizing the \(p\)-Laplacian is proved. More precisely, this result is proved for the eigenvalue problem \(-\text{div} (A(|\nabla u|^2 \nabla u)= \lambda\psi(u)\) in \(D\), \(u=0\) on \(\partial D\), where \(D\) is a smooth bounded domain and the nonlinearities \(A\) and \(\psi\) correspond to the case of very fast-growing nonlinear functions, as for example \(e^{s^2}-1\). It is shown that under suitable assumptions on the nonlinearities the problem can be solved in the framework of eigenvalues to variational inequalities and then a theorem by Kubrusky on the existence of Lagrange multipliers is applied. Some existence results for an associated nonlinear problem are also obtained.
reviewed by: Jesús Hernández
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35P30
Mathematics Subject Classification ID: 35J85
zbMATH DE Number: 1310757
zbMATH Keywords: principal eigenvalue
zbMATH Keywords: quasilinear elliptic problem
zbMATH Keywords: Orlicz-Sobolev spaces
zbMATH Keywords: positive solution
MaRDI profile type: Publication
full work available at URL: s000300050073
OpenAlex ID: W2058851508
DOI: 10.1007/S000300050073
description: scientific article; zbMATH DE number 1310757