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Multiplicity results near the principal eigenvalue for boundary‐value problems with periodic nonlinearity
Author(s) -
Cañada A.
Publication year - 2007
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200410477
Subject(s) - mathematics , eigenvalues and eigenvectors , multiplicity (mathematics) , mathematical analysis , bifurcation , zero (linguistics) , nonlinear system , boundary value problem , principal value , pure mathematics , combinatorics , linguistics , philosophy , physics , quantum mechanics
Let us consider the boundary‐value problemwhere g : ℝ → ℝ is a continuous and T ‐periodic function with zero mean value, not identically zero, ( λ , a ) ∈ ℝ 2 and $ \tilde h $ ∈ C [0, π ] with ∫ π 0 $ \tilde h $ ( x ) sin x dx = 0. If λ 1 denotes the first eigenvalue of the associated eigenvalue problem, we prove that if ( λ , a ) → ( λ 1 , 0), then the number of solutions increases to infinity. The proof combines Liapunov–Schmidt reduction together with a careful analysis of the oscillatory behavior of the bifurcation equation. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)