Positive solutions for nonlinear problems involving the one-dimensional ϕ -Laplacian

Abstract

Let Ω := (a, b) ⊂ R, m ∈ L1 (Ω) and λ > 0 be a real parameter. Let L be the differential operator given by Lu := −φ (u) + r (x) φ (u), where φ : R → R is an odd increasing homeomorphism and 0 ≤ r ∈ L1 (Ω). We study the existence of positive solutions for problems of the form Lu = λm (x) f (u) in Ω, u = 0 on ∂Ω, where f : [0,∞) → [0,∞) is a continuous function which is, roughly speaking, sublinear with respect to φ. Our approach combines the sub and supersolution method with some estimates on related nonlinear problems. We point out that our results are new even in the cases r ≡ 0 and/or m ≥ 0.