Non-negative solutions of a sublinear elliptic problem

In this paper, the existence of solutions, \((\lambda ,u)\) $(\lambda ,u)$ , of the problem \(\begin{aligned} \left\{ \begin{array}{ll} -\Delta u=\lambda u -a(x)|u|^{p-1}u &{} \quad \hbox {in }\Omega ,\\ u=0 &{}\quad \hbox {on}\;\;\partial \Omega , \end{array}\right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}-\mathrm{\Delta }u=\lambda u-{a(x)|u|}^{p-1}u\hfill & \text{in}\mathrm{\Omega },\hfill \\ u=0\hfill & \text{on}\partial \mathrm{\Omega },\hfill \end{array}\right)\end{array}$$ is explored for \(0< p < 1\) $0<p<1$ . When \(p>1\) $p>1$ , it is known that there is an unbounded component of such solutions bifurcating from \((\sigma _1, 0)\) $({\sigma }_1,0)$ , where \(\sigma _1\) ${\sigma }_1$ is the smallest eigenvalue of \(-\Delta \) $-\mathrm{\Delta }$ in \(\Omega \) $\mathrm{\Omega }$ under Dirichlet boundary conditions on \(\partial \Omega \) $\partial \mathrm{\Omega }$ . These solutions have \(u \in P\) $u\in P$ , the interior of the positive cone. The continuation argument used when \(p>1\) $p>1$ to keep \(u \in P\) $u\in P$ fails if \(0< p < 1\) $0<p<1$ . Nevertheless when \(0< p < 1\) $0<p<1$ , we are still able to show that there is a component of solutions bifurcating from \((\sigma _1, \infty )\) $({\sigma }_1,\infty )$ , unbounded outside of a neighborhood of \((\sigma _1, \infty )\) $({\sigma }_1,\infty )$ , and having \(u \gneq 0\) $u⪈0$ . 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.