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

摘要

Let us consider the boundary-value problem (-u"(x) - λu(x) + g(u(x)) = a sin x + h{top}～(x), x∈ [0, π], u(0) = u(π) = 0, where g : R → R is a continuous and T-periodic function with zero mean value, not identically zero, (λ, a) ∈ R{sup}2 and h{top}～ ∈ C[0, π] with ∫(h{top}～(x)sinsdx (from 0 to π) = 0. If λ{sub}1 denotes the first eigenvalue of the associated eigenvalue problem, we prove that if (λ, a) → (λ{sub}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.