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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 $N \geq 2$. Let $f\in C^1([0, \infty), \mathbb{R})$, $f(0)=0$, $f(β) = β, \ f(s)<s\ \text{for}\ s\in (0,β), \ f(s)>s\ \text{for}\ s\in (β, \infty)$ and $f'(β)>λ^{r}_k$. D. Bonheure, B. Noris and T. Weth [Ann. Inst. H. Poincaré Anal. Non Linéaire 29(4) (2012)] proved the existence of nondecreasing, radial positive solutions of the semilinear Neumann problem $$ -Δu+u=f(u) \ \text{in}\ B_1,\ \ \ \ \partial_νu=0 \ \text{on}\ \partial B_1 $$ for $k=2$, and they conjectured that there exists a radial solution with $k$ intersections with $β$ provided that $f'(β) >λ^r_k$ for $k>2$. In this paper, we show that the answer is yes.