Positive solutions for some quasilinear equations with critical and supercritical growth
type: Item
instance of: scholarly article
title: Positive solutions for some quasilinear equations with critical and supercritical growth
zbMATH Open document ID: 1162.35029
author: Giovany M. Figueiredo
author: Marcelo F. Furtado
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2007-04-13T00:00:00Z
review text: The goal of this paper is to study the number of solutions of some quasilinear problems. More precisely, the authors prove the existence and multiplicity of positive solutions for the problem \[ -\text{div}(a(\varepsilon x)|\nabla u|^{p-2}\nabla u)+ u^{p-1}= f(u)+ u^{p^*-1}\text{ in }\mathbb{R}^d,\;u\in W^{1,p}(\mathbb{R}^d), \] where \(\varepsilon> 0\) is a small parameter, \(2\leq p< d\), \(p^*= {dp\over d-p}\), \(a\) is a positive potential and \(f\) is a superlinear function. The authors obtain the existence of a ground state solution and relate the number of positive solutions with the topology of the set where \(a\) attains its minimum. To this end the authors use minimax theorems and Ljusternik-Schnirelmann theory.
reviewed by: Messoud Efendiev
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 47J30
zbMATH DE Number: 5142216
zbMATH Keywords: positive solutions
zbMATH Keywords: critical problems
zbMATH Keywords: supercritical problems
zbMATH Keywords: Ljusternik-Schnirelmann theory
zbMATH Keywords: quasilinear equations
MaRDI profile type: Publication
full work available at URL: j.na.2006.02.012
OpenAlex ID: W2007614093
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DOI: 10.1016/J.NA.2006.02.012
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description: scientific article; zbMATH DE number 5142216