Non-negative solution of a sublinear elliptic problem

Abstract In this paper the existence of solutions, (, u), of the problemll - u= u -a (x) |u|^p-1u \;, array. is explored for 0 1, it is known that thereis an unbounded component of such solutions bifurcating from (₁, 0), where ₁ is the smallest eigenvalue of - in under Dirichlet boundary conditions on. Thesesolutions have u P, the interior of the positive cone. Thecontinuation argument usedwhen p>1 to keep u Pfails if 0 < p < 1. Nevertheless when 0 < p < 1, we are still able to show that there is a component ofsolutions bifurcating from (₁, ), unbounded outside ofa neighborhood of (₁, ), and having u 0. Thisnon-negativity for u cannot be improved as is shown via adetailed analysis of the simplest autonomous one-dimensionalversion of the problem: its set of non-negative solutionspossesses a countable set of components, each of them consistingof positive solutions with a fixed (arbitrary) number of bumps. Finally, the structure of these components is fully described.