Positive solutions for a one-dimensional Sturm-Liouville semipositone superlinear p-Laplacian problem

Abstract

We prove the existence of a positive classical solution for the p-Laplacian equation –(r(t)ϕ(u'))' = –λh(u) + f (t, u), t ∈ (0, 1) with Sturm-Liouville boundary conditions, where ϕ(s) = |s|p‒2 s; p > 1; r : [0, 1] → (0, ∞); f : (0, 1) × [0;∞) → ℝ is a Carathéodory function satisfying a superlinear condition at 0 and 1 involving the principal eigenvalue of –(r(t)ϕ(u'))' h : (0,∞) → (0,∞) is allowed to have infinite semipositone structure at 0, and λ ≥ 0 is a small parameter.