Bounded Components of Positive Solutions of Nonlinear Abstract Equations
In this work a general class of nonlinear abstract equations satisfying a generalized strong maximum principle is considered in order to show that any bounded component of positive solutions bifurcating from the curve of trivial states (λ, u) = (λ, 0) at a nonlinear eigenvalue λ = λ₀ must meet the c...
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| Date: | 2005 |
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| Main Authors: | , , |
| Format: | Article |
| Language: | English |
| Published: |
Інститут прикладної математики і механіки НАН України
2005
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| Series: | Український математичний вісник |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Bounded Components of Positive Solutions of Nonlinear Abstract Equations / S. Cano-Casanova, J. Lopez-Gomez, M. Molina-Meyer // Український математичний вісник. — 2005. — Т. 2, № 1. — С. 38-51. — Бібліогр.: 13 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| Summary: | In this work a general class of nonlinear abstract equations satisfying a generalized strong maximum principle is considered in order to show that any bounded component of positive solutions bifurcating from the curve of trivial states (λ, u) = (λ, 0) at a nonlinear eigenvalue λ = λ₀ must meet the curve of trivial states (λ, 0) at another singular value λ₁ ≠ λ₀. Since the unilateral theorems of P. H. Rabinowitz [13, Theorems 1.27 and 1.40] are not true as originally stated (c.f. the counterexample of E. N. Dancer [6]), in order to get our main result the unilateral theorem of J. Lopez-Gomez [11, Theorem 6.4.3] is required. |
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