On singular Sturm–Liouville boundary-value problems

Abstract

We consider the existence of positive solutions for the boundary-value problem

(q(t)ϕ(u′))′ + λf(t,u) = 0, r < t < R,

au(r) − bϕ⁻¹ (q(r))u′(r) = 0, cu(R) + dϕ⁻¹(q(R))u′(R) = 0,

where ϕ(u′) = |u′|^p−2u′, p > 1, λ > 0, f is p-superlinear or p-sublinear at ∞ and is allowed to become −∞ at u = 0. Our results unify and extend many known results in the literature.