Multiple positive solutions for a class of Neumann problems

Abstract

We study the existence of multiple positive solutions of the Neumann problem \begin{equation*} \begin{split} -u''(x)&=\lambda f(u(x)), \qquad x\in(0,1),\\ u'(0)&=0=u'(1), \end{split} \end{equation*} where $\lambda$ is a positive parameter, $f\in C([0,\infty),\mathbb{R})$ and for some $\beta>0$ such that $f(0)=0$, $f(s)>0$ for $s\in(\beta,\infty)$, $f(s)\beta)$ is the unique positive zero of $F(s)=\int_0^sf(t)\,dt$. In particular, we prove that there exist at least $2n+1$ positive solutions for $\lambda\in \big(\frac{n^2\pi^2}{f'(\beta)},\infty\big)$, where $n\in \mathbb{N}$. The proof of our main result is based upon the bifurcation and continuation methods