Positive solutions of a non-linear eigenvalue problem with discontinuous non-linearity

Synopsis

We seek non-trivial solutions (u,λ)∈C¹([0,1])×[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 f:[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 ƒ(1) is only required to be positive.)

Existence and multiplicity theorems are given in the cases where ƒ is asymptotically sub-linear and ƒ is asymptotically super-linear. Moreover if strengthened assumptions are made on the growth of the non-linear term ƒ we obtain the precise number of non-trivial solutions for given values of λ ∈ [0, ∞).