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

Abstract

The paper provides a uniqueness result for positive solutions of the Neumann and periodic boundary value problems associated with the $ϕ$-Laplacian equation \begin{equation*} \bigl{(} ϕ(u') \bigr{)}' + a(t) g(u) = 0, \end{equation*} 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 $γ\in\mathbb{R}$, we prove the existence of a unique positive solution when $γ\in\mathopen{]}-\infty,(1-2p)/(p-1)\mathclose{]} \cup \mathopen{]}p-1,+\infty\mathclose{[}$.