Proof of the Bonheure-Noris-Weth conjecture on oscillatory radial solutions of Neumann problems

Let $B_1$ be the unit ball in ${\mathbb{R}}^N$ with $N\ge 2$. Let $f\in C^1([0, \infty), $\mathbb{R}$)$,$f(0)=0$,$f(\beta) = \beta, \ f(s)<s
$\text{for}$\ s\in (0,\beta), \ f(s)>s\ $\text{for}$\ s\in (\beta, \infty)$and$f’(\beta)>\lambda^{r}_k$. D. Bonheure, B. Noris and T. Weth [Ann. Inst. H. Poincar'e Anal. Non Lin'eaire 29(4) (2012)] proved the existence of nondecreasing, radial positive solutions of the semilinear Neumann problem $$-\mathrm{\Delta }u+u=f(u)\text{ in }{B}_1,{\mathrm{\partial }}_{\nu }u=0\text{ on }\mathrm{\partial }{B}_1$$ for $k=2$, and they conjectured that there exists a radial solution with $k$ intersections with $\beta$ provided that $f^{\mathrm{\prime }}(\beta )>{\lambda }_k^r$ for $k>2$. In this paper, we show that the answer is yes.