On existence and multiplicity of solutions for elliptic equations involving the \(p\)-Laplacian
type: Item
instance of: scholarly article
title: On existence and multiplicity of solutions for elliptic equations involving the \(p\)-Laplacian
zbMATH Open document ID: 1172.35022
author: Xian Wu
author: Jilin Chen
published in: NoDEA. Nonlinear Differential Equations and Applications
publication date: 2009-06-08T00:00:00Z
review text: In this paper, under suitable assumptions on \(f(x,t)\) and \(a(x)\) the authors proved that there are at least three solutions to the following \(p\)-Laplacian eigenvalue problems with Dirichlet and Neumann boundary vaules: \[ \begin{cases} -\Delta_p (u) = \lambda\, f(x,u),&\text{in } \Omega,\\ u = 0, &\text{on } \partial \Omega, \end{cases}\tag{1} \] and \[ \begin{cases} -\Delta _p (u) + a(x)|u|^{p - 2} u = \lambda f(x,u), &\text{in } \Omega, \\ \frac{{\partial u}}{{\partial n}} = 0, &\text{on } \partial \Omega, \end{cases}\tag{2} \] for \(\lambda\) in some interval \(\Lambda\). Moreover, if \(\lambda =1\), by using the KKM principle they proved also some existence results for problems (1) and (2).
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35J70
zbMATH DE Number: 5562669
zbMATH Keywords: \(p\)-Laplacian
zbMATH Keywords: eigenvalue problem
zbMATH Keywords: boundary value problem
zbMATH Keywords: KKM principle
reviewed by: Huansong Zhou
MaRDI profile type: Publication
full work available at URL: s00030-008-8017-9
OpenAlex ID: W2075263788
DOI: 10.1007/S00030-008-8017-9
description: scientific article; zbMATH DE number 5562669