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

Abstract

We deal with the Neumann boundary value problem \begin{equation*} \begin{cases} \, u" + \bigl{(} λa^{+}(t)-μa^{-}(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 $g:[0,1]\to \mathbb{R}$ is a continuous function such that $g(0)=g(1)=0$, $g(s)>0$ for $0<s<1$ and ${\lim }_{s\to 0^{+}}g(s)/s=0$. We prove the existence of three solutions when $\lambda$ and $\mu$ are positive and sufficiently large.

16 pages, 5 PNG figures