Non-negative solutions of a sublinear elliptic problem

In this paper, the existence of solutions, (λ,u), of the problem (Formula presented.) is explored for 0<1. When p>1, it is known that there is an unbounded component of such solutions bifurcating from (σ1,0), where σ1 is the smallest eigenvalue of -Δ in Ω under Dirichlet boundary conditions on ∂Ω. These solutions have u∈P, the interior of the positive cone. The continuation argument used when p>1 to keep u∈P fails if 0<1. Nevertheless when 0<1, we are still able to show that there is a component of solutions bifurcating from (σ1,∞), unbounded outside of a neighborhood of (σ1,∞), and having u⪈0. This non-negativity for u cannot be improved as is shown via a detailed analysis of the simplest autonomous one-dimensional version of the problem: its set of non-negative solutions possesses a countable set of components, each of them consisting of positive solutions with a fixed (arbitrary) number of bumps. Finally, the structure of these components is fully described.