On one-dimensional superlinear indefinite problems involving the ϕ -Laplacian

Abstract

Let Ω : = (a, b) ⊂ R, m∈ L1(Ω) and ϕ: R→ R be an odd increasing homeomorphism. We consider the existence of positive solutions for problems of the form {-ϕ(u′)′=m(x)f(u)inΩ,u=0on∂Ω,where f: [0 , ∞) → [0 , ∞) is a continuous function which is, roughly speaking, superlinear with respect to ϕ. Our approach combines the Guo-Krasnoselskiĭ fixed-point theorem with some estimates on related nonlinear problems. We mention that our results are new even in the case m≥ 0.