Non-negative solutions of a sublinear elliptic problem

In this paper, the existence of solutions, (lambda, u), of the problem {-Delta u = lambda u - a(x)|u|(p-1)u in Omega, u = 0 on partial derivative Omega is explored for 0<p<1. When p > 1, it is known that there is an unbounded component of such solutions bifurcating from (sigma(1),0), where sigma 1is the smallest eigenvalue of-Delta in Omega under Dirichlet boundary conditions on partial derivative Omega. These solutions have u is an element of P, the interior of the positive cone. The continuation argument used when p > 1 to keep u is an element of P fails if 0 < p < 1. Nevertheless when 0 < p < 1, we are still able to show that there is a component of solutions bifurcating from (sigma 1,infinity), unbounded outside of a neighborhood of (sigma 1,infinity), and havingu0. 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