  We are interested in the structure of the positive radial solutions of the
supercritical 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 problem
has 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 of
infinitely many smooth curves in the $(\varepsilon,U(0))$-plane. It is shown
that each curve blows up at $\varepsilon_n^*$ and if
$p_{\rm{S}}<p<p_{\rm{JL}}$, then each curve has infinitely many turning points
around $\varepsilon_n^*$. Here, $p_{\rm{JL}}$ stands for the Joseph-Lundgren
exponent. In particular, the problem has infinitely many solutions if
$\varepsilon\in\{\varepsilon_n^*\}_{n=1}^{\infty}$. We also show that there
exists $\bar{\varepsilon}>0$ such that the problem has no nonconstant regular
solution if $\varepsilon>\bar{\varepsilon}$. The main technical tool is the
intersection number between the regular and singular solutions.
