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Structure of the Positive Radial Solutions for the Supercritical Neumann Problem $\mathbf \varepsilon^2\Delta u-u+u^p=0$ in a Ball

J. Math. Sci. Univ. Tokyo
Vol. 22 (2015), No. 3, Page 685–739.

Miyamoto, Yasuhito
Structure of the Positive Radial Solutions for the Supercritical Neumann Problem $\mathbf \varepsilon^2\Delta u-u+u^p=0$ in a Ball
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Abstract:
We are interested in the structure of the positive radial solutions of the supercritical Neumann problem in a unit ball $\[ \begin{cases} \e^2\left(U''+\frac{N-1}{r}U'\right)-U+U^p=0, & 00, & 0p_S:=(N+2)/(N-2)$, $N\ge 3$. We show that there exists a sequence $\{\e_n^*\}_{n=1}^{\infty}$ ($\e_1^*>\e_2^*>\cdots\rightarrow 0$) such that this problem has infinitely many singular solutions $\{(\e_n^*,U_n^*)\}_{n=1}^{\infty}\subset\R\times (C^2(0,1)\cap C^1(0,1])$ and that the nonconstant regular solutions consist of infinitely many smooth curves in the $(\e,U(0))$-plane. It is shown that each curve blows up at $\e_n^*$ and if $p_S0$ such that the problem has no nonconstant regular solution if $\e>\bar{\e}$. The main technical tool is the intersection number between the regular and singular solutions.
Keywords: Global bifurcation diagram, intersection number, singular solution, Joseph-Lundgren exponent.
Mathematics Subject Classification (2010): Primary 35J25, 25B32; Secondary 34C23, 34C10.
Mathematical Reviews Number: MR3408072
Received: 2013-12-04