Non-negative solutions of a sublinear elliptic problem

Abstract

In this paper, the existence of solutions, \((\lambda ,u)\), of the problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u=\lambda u -a(x)|u|^{p-1}u &{} \quad \hbox {in }\Omega ,\\ u=0 &{}\quad \hbox {on}\;\;\partial \Omega , \end{array}\right. \end{aligned}$$

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.