Positive solutions of a nonlinear eigenvalue problem with discontinuous non linearity

We seek non-trivial solutions (u, λ)∊ C1([0, l])×[0,∊), with u(x)≧0 for all x∊[0,1], of the nonlinear eigenvalue problem-u″(x) = λf(u(x)) for x ∊ (0,1) and u(0) = u(1) = 0, where/: [0, ∞) → [0, ∞) is such that f(p) = 0, for p ∊ [0,1), and f(p) = K(p), for p ∊ (1, ∞), and K: [1, ∞) → (0, ∞) is assumed to be twice continuously differentiable. (The value f(1) is only required to be positive.) Existence and multiplicity theorems are given in the cases where f is asymptotically sub-linear and f is asymptotically super-linear. Moreover if strengthened assumptions are made on the growth of the non-linear term f we obtain the precise number of non-trivial solutions for given values of λ ∊ [0, ∞). © 1979, Royal Society of Edinburgh. All rights reserved.