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

Abstract

We are interested in the structure of the positive radial solutions of the supercritical Neumann problem ${\epsilon }^2\Delta u-u+u^p=0$ 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 $\{{\epsilon }_n^{\ast }{\}}_{n=1}^{\infty }$ (${\epsilon }_1^{\ast }>{\epsilon }_2^{\ast }>\cdots \to 0$) such that this problem has infinitely many singular solutions $\{({\epsilon }_n^{\ast },U_n^{\ast }){\}}_{n=1}^{\infty }\subset \mathbb{R}\times (C^2(0,1)\cap C^1(0,1])$ and that the nonconstant regular solutions consist of infinitely many smooth curves in the $(\epsilon ,U(0))$-plane. It is shown that each curve blows up at ${\epsilon }_n^{\ast }$ and if $p_{\mathrm{S}}<p<p_{\mathrm{JL}}$, then each curve has infinitely many turning points around ${\epsilon }_n^{\ast }$. Here, $p_{\mathrm{JL}}$ stands for the Joseph-Lundgren exponent. In particular, the problem has infinitely many solutions if $\epsilon \in \{{\epsilon }_n^{\ast }{\}}_{n=1}^{\infty }$. We also show that there exists $\bar{\epsilon }>0$ such that the problem has no nonconstant regular solution if $\epsilon >\bar{\epsilon }$. The main technical tool is the intersection number between the regular and singular solutions.

arXiv comment37 pages, 1 figure