  In this paper the existence of solutions, $(\lambda,u)$, of the problem
$$-\Delta u=\lambda u -a(x)|u|^{p-1}u \quad \hbox{in }\Omega, \qquad
  u=0 \quad \hbox{on}\;\;\partial\Omega,$$ 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 $\sigma_1$ is the smallest eigenvalue
of $-\Delta$ in $\Omega$ under Dirichlet boundary conditions on
$\partial\Omega$. These solutions have $u \in P$, the interior of the positive
cone. The continuation argument used when $p>1$ to keep $u \in P$ 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 $(\sigma_1, \infty)$, unbounded
outside of a neighborhood of $(\sigma_1, \infty)$, and having $u \gneq 0$. 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.
