On singular nonpositone semilinear elliptic problems

Abstract

We prove the existence of a large positive solution for the boundary value problems $$ \begin{alignat}{2} -\Delta u &=\lambda (-h(u)+g(x,u))&\quad& \text{in }\Omega , \\ u &=0 &\quad &\text{on }\partial \Omega , \end{alignat} $$ where $\Omega $ is a bounded domain in ${\mathbb R}^{N}$, $\lambda $ is a positive parameter, $g(x,\cdot)$ is sublinear at $\infty$, and $h$ is allowed to become $\infty $ at $u=0$. Uniqueness is also considered.