A curve of positive solutions for an indefinite sublinear Dirichlet problem

Abstract

We investigate the existence of a curve $q\mapsto u_q$, with $q\in (0,1)$, of positive solutions for the problem $(P_{a,q})$: $-\Delta u=a(x)u^q$ in $\Omega$, $u=0$ on $\partial \Omega$, where $\Omega$ is a bounded and smooth domain of ${\mathbb{R}}^N$ and $a:\Omega \to \mathbb{R}$ is a sign-changing function (in which case the strong maximum principle does not hold). In addition, we analyze the asymptotic behavior of $u_q$ as $q\to 0^{+}$ and $q\to 1^{-}$. We also show that in some cases $u_q$ is the ground state solution of $(P_{a,q})$. As a byproduct, we obtain existence results for a singular and indefinite Dirichlet problem. Our results are mainly based on bifurcation and sub-supersolutions methods.