A priori estimates and uniqueness of inflection points for positive solutions of semipositone problems

Abstract

We prove that positive solutions of $-\Delta u = \lambda f(u)$ in $\Omega$ and $u = 0$ on $\partial \Omega$ where $f$ is increasing, concave, and $f(0) < 0$ satisfy $c_{1} \leq {\lambda f(d) \over d} \leq c_{2}$ where $d = \sup u.$ Also, we show that solutions of the above have exactly one inflection point.