Non-negative solution of a sublinear elliptic problem

Abstract

In this paper the existence of solutions, $(\lambda,u)$,of the problem$$\left\lbrace\begin{array}{ll} -\D u=\l u -a(x)|u|^{p-1}u & \quad \hbox{in }\O,\\ u=0 &\quad \hbox{on}\;\;\p\O, \end{array}\right.$$is explored for $0 < p < 1$. When $p>1$, it is known that thereis an unbounded component of such solutions bifurcating from$(\s_1, 0)$, where $\s_1$ is the smallest eigenvalue of $-\D$ in$\O$ under Dirichlet boundary conditions on $\p\O$. Thesesolutions have $u \in P$, the interior of the positive cone. Thecontinuation argument usedwhen $p>1$ 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 ofsolutions bifurcating from $(\s_1, \infty)$, unbounded outside ofa neighborhood of $(\s_1, \infty)$, and having $u \gneq 0$. 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.