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A bifurcation problem governed by the boundary condition II
Author(s) -
GarcíaMelián Jorge,
Rossi Julio D.,
Sabina De Lis José C.
Publication year - 2007
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdl001
Subject(s) - mathematics , uniqueness , bounded function , bifurcation , domain (mathematical analysis) , infinity , eigenvalues and eigenvectors , mathematical analysis , boundary (topology) , work (physics) , pure mathematics , nonlinear system , physics , quantum mechanics , thermodynamics
In this work we consider the problem Δ u = a ( x ) u p in Ω , ∂ u ∂ ν = λ u on ∂Ω, where Ω is a smooth bounded domain, ν is the outward unit normal to ∂Ω, λ is regarded as a parameter and 0 < p < 1. We consider both cases where a ( x ) > 0 in Ω or a ( x ) is allowed to vanish in a whole subdomain Ω 0 of Ω. Our main results include existence of non‐negative non‐trivial solutions in the range 0 < λ < σ 1 , where σ 1 is characterized by means of an eigenvalue problem, uniqueness and bifurcation from infinity of such solutions for small λ, and the appearance of dead cores for large enough λ.