Positive solutions for a one-dimensional Sturm-Liouville semipositone superlinearp-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, whereϕ(s)= |s|p‒2s;p> 1;r: [0, 1] → (0, ∞); f : (0, 1) × [0;∞) →ℝis 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, andλ≥ 0 is a small parameter.