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

Abstract

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 $f^{\prime }(\beta )>{\lambda }_k^r$. 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 $$-\Delta u+u=f(u)\text{in}{B}_1,{\partial }_{\nu }u=0\text{on}\partial B_1$$ for $k=2$, and they conjectured that there exists a radial solution with $k$ intersections with $\beta$ provided that $f^{\prime }(\beta )>{\lambda }_k^r$ for $k>2$. In this paper, we show that the answer is yes.

References in corpus (1)

• Increasing radial solutions for Neumann problems without growth restrictions