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

TL;DR

This paper proves the conjecture by Bonheure, Noris, and Weth that for certain nonlinear Neumann problems in a unit ball, there exist oscillatory radial solutions with multiple intersections, confirming the existence of solutions with prescribed oscillation properties.

Contribution

The paper confirms the conjecture that radial solutions with multiple intersections exist under specified conditions, extending previous results to all integers k>2.

Findings

Existence of radial solutions with k intersections for all k>2.

Validation of the Bonheure-Noris-Weth conjecture.

Extension of previous results to broader conditions.

Abstract

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.