  Let $B_1$ be the unit ball in $\mathbb{R}^N$ with $N \geq 2$. 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 $$ -\Delta u+u=f(u) \ \text{in}\
B_1,\ \ \ \ \partial_\nu u=0 \ \text{on}\ \partial B_1 $$ for $k=2$, and they
conjectured that there exists a radial solution with $k$ intersections with
$\beta$ provided that $f'(\beta) >\lambda^r_k$ for $k>2$. In this paper, we
show that the answer is yes.
