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2005 SHAPE AND STRUCTURE OF THE BIFURCATION CURVE OF A BOUNDARY BLOW-UP PROBLEM
Shin-Hwa Wang, Yueh-Tseng Liu
Taiwanese J. Math. 9(2): 201-214 (2005). DOI: 10.11650/twjm/1500407796

Abstract

We study the shape and the structure of the bifurcation curve $f_{a}(\rho)$ ($= \sqrt{\lambda}$) with $\rho: = \min_{x \in (0,1)} u(x)$ of (sign-changing and nonnegative) solutions of the boundary blow-up problem \[ \left\{ \begin{array}{l} -u''(x) = \lambda f(u(x)),\; 0 \lt x \lt 1, \\ \lim\limits_{x \to 0^+} u(x) = \infty = \lim\limits_{x \to x^-} u(x), \end{array} \right. \] where $\lambda$ is a positive bifurcation parameter and the Lipschitz continuous concave function \[ f = f_{a}(u) = \begin{cases} -|u|^p & \textrm{if } u \leq -a^{1/p}, \\ -a & \textrm{if } -a^{1/p} \lt u \lt a^{1/p}, \\ -|u|^p & \textrm{if } u \geq a^{1/p}, \end{cases} \] with constants $p \gt 1$ and $a \gt 0$. We mainly show that the bifurcation curve $G_{f_a}(\rho)$ satisfies $\lim_{\rho \rightarrow \pm \infty} G_{f_a}(\rho) = 0$ and $G_{f_a}(\rho)$ has a exactly one critical point, a maximum, on $(-\infty,\infty )$. Thus we are able to determine the exact number of (sign-changing and nonnegative) solutions of the problem for each $\lambda \gt 0$.

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Shin-Hwa Wang. Yueh-Tseng Liu. "SHAPE AND STRUCTURE OF THE BIFURCATION CURVE OF A BOUNDARY BLOW-UP PROBLEM." Taiwanese J. Math. 9 (2) 201 - 214, 2005. https://doi.org/10.11650/twjm/1500407796

Information

Published: 2005
First available in Project Euclid: 18 July 2017

zbMATH: 1085.34023
MathSciNet: MR2142573
Digital Object Identifier: 10.11650/twjm/1500407796

Subjects:
Primary: 34B15, 34C23

Rights: Copyright © 2005 The Mathematical Society of the Republic of China

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