Mountain pass type solutions for quasilinear elliptic equations
type: Item
instance of: scholarly article
title: Mountain pass type solutions for quasilinear elliptic equations
zbMATH Open document ID: 0959.35057
published in: Calculus of Variations and Partial Differential Equations
publication date: 2001-04-18T00:00:00Z
review text: Let \(\Omega\) be a bounded domain in \({\mathbb R}^N\) and let \(g:\overline{\mathbb R}\times {\mathbb R}\rightarrow {\mathbb R}\) be a continuous function. The paper deals with the problem \(-\text{div} (a(|\nabla u(x)|)\nabla u(x))=g(x,u)\) in \(\Omega\), subject to the Dirichlet boundary condition \(u=0\) on \(\partial\Omega\). The function \(a:{\mathbb R}_+\rightarrow {\mathbb R}\) is such that the mapping \(\phi (s)=sa(|s|)\) is an increasing homeomorphism from \({\mathbb R}\) onto \({\mathbb R}\). Under appropriate conditions on \(\phi\) and \(g\), the authors prove the existence of nontrivial weak solutions of mountain pass type in an Orlicz-Sobolev space. The proof is based on variational arguments and refined estimates. Several interesting examples are also given in the work. The paper is very well written and the techniques developed by the authors can be adapted for the treatment of wide classes of boundary value problems.
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35A15
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 35A05
zbMATH DE Number: 1525159
zbMATH Keywords: weak solution
zbMATH Keywords: mountain pass solution
zbMATH Keywords: quasilinear elliptic equation
zbMATH Keywords: Orlicz-Sobolev space
author: Philippe Clément
author: Marta Garcia-Huidobro
author: Raúl Manásevich
author: Klaus Schmitt
reviewed by: Vicenţiu D. Rădulescu
MaRDI profile type: Publication
DOI: 10.1007/S005260000025
description: scientific article; zbMATH DE number 1525159