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

Abstract: 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.