Multiplicity results near the principal eigenvalue for boundary‐value problems with periodic nonlinearity

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)