On some nonlinear equations involving the \(p\)-Laplacian with critical Sobolev growth
type: Item
instance of: scholarly article
title: On some nonlinear equations involving the \(p\)-Laplacian with critical Sobolev growth
zbMATH Open document ID: 0955.35031
author: Emmanuel Hebey
author: F. Demengel
published in: Advances in Differential Equations
publication date: 2001-02-21T00:00:00Z
review text: The author considers nonlinear elliptic equations \[ -\text{div}(|\nabla|^{p- 2}\nabla u)+ a(x)|u|^{p-2} u= f(x)|u|^{p^*- 2}\quad\text{in }\Omega, \] \[ u\not\equiv 0,\quad u= 0\quad\text{on }\partial\Omega \] in which \(p^*= np/(n- p)\) is the critical Sobolev exponent for the embedding of \(W^{1,p}\) in \(L^p\). Conditions are found to be satisfied by the domain \(\Omega\) and function \(f(x)\) in order that there exist a positive solution \(u\in W^{1,p}_0(\Omega)\cap C^{1,\alpha}(\overline\Omega)\cap C^{1,\alpha}(\Omega)\) to the equation (Theorem 1) as well as conditions under which a modal solution \(u\in W^{1,p}(\Omega)\) exists (Theorem 2). In the proof of both theorems modified veriational techniques are used.
Mathematics Subject Classification ID: 35J70
Mathematics Subject Classification ID: 35B33
Mathematics Subject Classification ID: 35A15
Mathematics Subject Classification ID: 35A05
Mathematics Subject Classification ID: 35B05
zbMATH DE Number: 1449026
zbMATH Keywords: nonlinear elliptic equation
zbMATH Keywords: \(p\)-Laplacian
zbMATH Keywords: positive solutions
zbMATH Keywords: nodal solutions
reviewed by: Bronius Kvedaras
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1449026