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We are interested in the structure of the positive radial solutions of thesupercritical Neumann problem $\varepsilon^2\Delta u-u+u^p=0$ on a unit ball in$\mathbb{R}^N$ , where $N$ is the spatial dimension and $p>p_S:=(N+2)/(N-2)$,$N\ge 3$. We show that there exists a sequence$\{\varepsilon_n^*\}_{n=1}^{\infty}$($\varepsilon_1^*>\varepsilon_2^*>\cdots\rightarrow 0$) such that this problemhas infinitely many singular solutions$\{(\varepsilon_n^*,U_n^*)\}_{n=1}^{\infty}\subset\mathbb{R}\times(C^2(0,1)\cap C^1(0,1])$ and that the nonconstant regular solutions consist ofinfinitely many smooth curves in the $(\varepsilon,U(0))$-plane. It is shownthat each curve blows up at $\varepsilon_n^*$ and if$p_{\rm{S}}<p<p_{\rm{JL}}$, then each curve has infinitely many turning pointsaround $\varepsilon_n^*$. Here, $p_{\rm{JL}}$ stands for the Joseph-Lundgrenexponent. In particular, the problem has infinitely many solutions if$\varepsilon\in\{\varepsilon_n^*\}_{n=1}^{\infty}$. We also show that thereexists $\bar{\varepsilon}>0$ such that the problem has no nonconstant regularsolution if $\varepsilon>\bar{\varepsilon}$. The main technical tool is theintersection number between the regular and singular solutions.

Structure of the positive radial solutions for the supercritical Neumann problem $\varepsilon^2\Delta u-u+u^p=0$ in a ball