Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations

Abstract

Abstract This paper provides a uniqueness result for positive solutions of the Neumann and periodic boundary value problems associated with the ϕ-Laplacian equation ( ϕ ( u ′ ) ) ′ + a ( t ) g ( u ) = 0 , (\phi \left(u^{\prime} ))^{\prime} +a\left(t)g\left(u)=0, where ϕ is a homeomorphism with ϕ(0) = 0, a(t) is a stepwise indefinite weight and g(u) is a continuous function. When dealing with the p-Laplacian differential operator ϕ(s) = ∣s∣ p−2 s with p > 1, and the nonlinear term g(u) = u γ with γ ∈ R \gamma \in {\mathbb{R}} , we prove the existence of a unique positive solution when γ ∈ ]− ∞ \infty , (1 − 2p)/(p − 1)] ∪ ]p − 1, + ∞ \infty [.