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

TL;DR

This paper establishes the uniqueness of positive solutions for boundary value problems involving indefinite $ ext{phi}$-Laplacian equations, including specific cases like the $p$-Laplacian with power nonlinearities, under certain conditions.

Contribution

It provides a new uniqueness result for positive solutions of $ ext{phi}$-Laplacian boundary value problems with indefinite weights and nonlinearities.

Findings

Uniqueness of positive solutions for $ ext{phi}$-Laplacian equations with indefinite weights.

Existence of a unique positive solution for $p$-Laplacian with specific nonlinearities.

Conditions on $ ext{gamma}$ for solution uniqueness in power nonlinearities.

Abstract

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