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Let $B_1$ be the unit ball in $\mathbb{R}^N$ with $N \geq 2$. Let $f\inC^1([0, \infty), \mathbb{R})$, $f(0)=0$, $f(\beta) = \beta, \ f(s)<s\\text{for}\ s\in (0,\beta), \ f(s)>s\ \text{for}\ s\in (\beta, \infty)$ and$f'(\beta)>\lambda^{r}_k$.  D. Bonheure, B. Noris and T. Weth [Ann. Inst. H. Poincar\'e Anal. NonLin\'eaire 29(4) (2012)] proved the existence of nondecreasing, radial positivesolutions 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 theyconjectured that there exists a radial solution with $k$ intersections with$\beta$ provided that $f'(\beta) >\lambda^r_k$ for $k>2$. In this paper, weshow that the answer is yes.

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