Three solutions for a perturbed Dirichlet problem

In this paper we prove the existence of at least three distinct solutions to the following perturbed Dirichlet problem: { − Δ u = f ( x , u ) + λ g ( x , u ) in  Ω u = 0 on  ∂ Ω , where Ω ⊂ R N is an open bounded set with smooth boundary ∂ Ω and λ ∈ R . Under very mild conditions on g and some assum...

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Bibliographic Details
Published in:Nonlinear analysis 2008-06, Vol.68 (12), p.3879-3883
Main Authors: Cordaro, Giuseppe, Rao, Giuseppe
Format: Article
Language:English
Subjects:
Critical points
Exact sciences and technology
Mathematical analysis
Mathematics
Partial differential equations
Sciences and techniques of general use
Weak solutions
Weakly sequentially lower semicontinuity
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Summary:In this paper we prove the existence of at least three distinct solutions to the following perturbed Dirichlet problem: { − Δ u = f ( x , u ) + λ g ( x , u ) in  Ω u = 0 on  ∂ Ω , where Ω ⊂ R N is an open bounded set with smooth boundary ∂ Ω and λ ∈ R . Under very mild conditions on g and some assumptions on the behaviour of the potential of f at 0 and + ∞ , our result assures the existence of at least three distinct solutions to the above problem for λ small enough. Moreover such solutions belong to a ball of the space W 0 1 , 2 ( Ω ) centered in the origin and with radius not dependent on λ .
ISSN:0362-546X
1873-5215
DOI:10.1016/j.na.2007.04.027