Positive symmetric solutions of singular semipositone boundary value problems

Abstract

Using the method of upper and lower solutions, we prove that the singular boundary value problem, \[ -u'' = f(u) ~ u^{-\alpha} \quad \textrm{in} \quad (0, 1), \quad u'(0) = 0 = u(1) \, , \] has a positive solution when $0 < \alpha < 1$ and $f : \mathbb{R} \to \mathbb{R}$ is an appropriate nonlinearity that is bounded below; in particular, we allow $f$ to satisfy the semipositone condition $f(0) < 0$. The main difficulty of this approach is obtaining a positive subsolution, which we accomplish by piecing together solutions of two auxiliary problems. Interestingly, one of these auxiliary problems relies on a novel fixed-point formulation that allows a direct application of Schauder's fixed-point theorem.