Multiplicity results near the principal eigenvalue for boundary‐value problems with periodic nonlinearity
A. Cañada · Mathematische Nachrichten · 2007
Abstract Let us consider the boundary‐value problem equation image where g: ℝ → ℝ is a continuous and T ‐periodic function with zero mean value, not identically zero, (λ, a) ∈ ℝ2 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)