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Zbl 1066.47063
López-Gómez, Julián; Molina-Meyer, Marcela
Bounded components of positive solutions of abstract fixed point equations: Mushrooms, loops and isolas. (English)
[J] J. Differ. Equations 209, No. 2, 416-441 (2005). ISSN 0022-0396

Let $U$ be an ordered Banach space whose positive cone is normal and has nonempty interior. This paper is devoted to the study of the nonlinear abstract equation ${\cal L}(\lambda)u+{\cal R}(\lambda,u)=0$ for $(\lambda ,u)\in \Bbb R\times U$, where ${\cal L}(\lambda)$ is a Fredholm operator of index $0$ and ${\cal R}\in C(\Bbb R\times U;U)$ is compact on bounded sets and $\lim_{u\rightarrow 0}{\cal R}(\lambda,u)/\Vert u\Vert =0$. \par The main result of the present paper concerns the bounded components of positive solutions emanating from $(\lambda,u)=(\lambda,0)$. The proofs are based on refined techniques from modern bifurcation theory.
[Teodora-Liliana Rădulescu (Craiova)]

MSC 2000:: *47J05 Equations involving nonlinear operators (general)
47J15 Abstract bifurcation theory
35B30 Dependence of solutions of PDE on initial and boundary data
35B32 Bifurcation (PDE)
35B50 Maximum principles (PDE)

Keywords: positive solutions; compact solution components; nonlinear abstract equations; strong maximum principle

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