A note on the implicit function theorem for quasi-linear eigenvalue problems

We consider the quasi-linear eigenvalue problem − Δ p u = λ g ( u ) subject to Dirichlet boundary conditions on a bounded open set Ω , where g is a locally Lipschitz continuous function. Imposing no further conditions on Ω or g , we show that for λ near zero the problem has a bounded solution which...

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Bibliographic Details
Published inNonlinear analysis Vol. 75; no. 5; pp. 2806 - 2811
Main Author Nittka, Robin
Format Journal Article
LanguageEnglish
Published Elsevier Ltd 01.03.2012
Subjects
Boundary conditions
Compact resolvent
Dirichlet problem
Eigenvalues
Gelfand’s equation
Implicit function theorem
Nonlinearity
Quasi-linear eigenvalue problem
Theorems
secondary
Quasi-linear eigenvalue problem
Implicit function theorem
Gelfand’s equation
Compact resolvent
primary
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Summary:We consider the quasi-linear eigenvalue problem − Δ p u = λ g ( u ) subject to Dirichlet boundary conditions on a bounded open set Ω , where g is a locally Lipschitz continuous function. Imposing no further conditions on Ω or g , we show that for λ near zero the problem has a bounded solution which is unique in the class of all small solutions. Moreover, this curve of solutions parameterized by λ depends continuously on the parameter.
Bibliography:ObjectType-Article-2
SourceType-Scholarly Journals-1
ObjectType-Feature-1
content type line 23
ISSN:0362-546X
1873-5215
DOI:10.1016/j.na.2011.11.023