# Three positive solutions to an indefinite Neumann problem: a shooting method

## TL;DR

This paper proves the existence of three positive solutions for a Neumann boundary value problem with an indefinite weight function, using a shooting method, when parameters are large.

## Contribution

It introduces a novel shooting method approach to establish three solutions for an indefinite Neumann problem with specific weight and nonlinearity conditions.

## Key findings

- Existence of three positive solutions for large parameters mbda and mu.
- Application of shooting method to indefinite boundary value problems.
- Conditions on the nonlinearity g ensuring multiple solutions.

## Abstract

We deal with the Neumann boundary value problem \begin{equation*} \begin{cases} \, u" + \bigl{(} \lambda a^{+}(t)-\mu a^{-}(t) \bigr{)}g(u) = 0, \\ \, 0 < u(t) < 1, \quad \forall\, t\in\mathopen{[}0,T\mathclose{]},\\ \, u'(0) = u'(T) = 0, \end{cases} \end{equation*} where the weight term has two positive humps separated by a negative one and $g\colon \mathopen{[}0,1\mathclose{]} \to \mathbb{R}$ is a continuous function such that $g(0)=g(1)=0$, $g(s) > 0$ for $0<s<1$ and $\lim_{s\to0^{+}}g(s)/s=0$. We prove the existence of three solutions when $\lambda$ and $\mu$ are positive and sufficiently large.