Non-negative solution of a sublinear elliptic problem

Abstract In this paper the existence of solutions, (λ,u),of the problemarrayll - u= u -a(x)|u|ᵖ⁻¹u & in ,\\ u=0 & on\;\;, array.is explored for $0 < p < 1$. When $p>1$, 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.