Nonnegative solution curves of semipositone problems with Dirichlet boundary conditions

We consider the boundary value problem - u ′ ′ ( x ) = λ f ( u ( x ) ) , x ∈ ( - 1 , 1 ) , u ( - 1 ) = 0 = u ( 1 ) , where λ > 0 is a parameter and f ∈ C 2 ( 0 , ∞ ) is monotonically increasing and concave up such that f ( 0 ) < 0 (i.e. is the semipositone). In this paper we study the case p = θ and p ∈ ( θ , + ∞ ) . ( p is the supremum of the nonnegative solution and θ is such that F ( θ ) = ∫ 0 θ f ( s ) d s = 0 . ) We discuss existence and multiplicity results for nonnegative solutions.