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

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 [.

Keywords

• uniqueness
• indefinite weight
• positive solutions
• p-laplacian
• boundary value problems
• superlinear functions
• singular equations
• 34b15
• 34b16
• 34b18
• 34c25