Non-negative solutions of a sublinear elliptic problem

In this paper the existence of solutions, (, u), of the problem - u= u -a (x) |u|^p-1u in, u=0 on\;\;, is explored for 0 1, it is known that there is an unbounded component of such solutions bifurcating from (₁, 0), where ₁ is the smallest eigenvalue of - in under Dirichlet boundary conditions on. 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 < p < 1. Nevertheless when 0 < p < 1, we are still able to show that there is a component of solutions bifurcating from (₁, ), unbounded outside of a neighborhood of (₁, ), 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.