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

Abstract

The paper provides a uniqueness result for positive solutions of the Neumann and periodic boundary value problems associated with the $\varphi$-Laplacian equation \begin{equation*} \bigl{(} ϕ(u') \bigr{)}' + a(t) g(u) = 0, \end{equation*} where $\varphi$ is a homeomorphism with $\varphi (0)=0$, $a(t)$ is a stepwise indefinite weight and $g(u)$ is a continuous function. When dealing with the $p$-Laplacian differential operator $\varphi (s)=|s{|}^{p-2}s$ with $p>1$, and the nonlinear term $g(u)=u^{\gamma }$ with $\gamma \in \mathbb{R}$, we prove the existence of a unique positive solution when $\gamma \in ]-\infty ,(1-2p)/(p-1)]\cup ]p-1,+\infty [$.

24 pages, 2 figures