Taiwanese Journal of Mathematics

SHAPE AND STRUCTURE OF THE BIFURCATION CURVE OF A BOUNDARY BLOW-UP PROBLEM

Shin-Hwa Wang and Yueh-Tseng Liu

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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 $\lambda \gt 0$.

Article information

Source
Taiwanese J. Math., Volume 9, Number 2 (2005), 201-214.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500407796

Digital Object Identifier
doi:10.11650/twjm/1500407796

Mathematical Reviews number (MathSciNet)
MR2142573

Zentralblatt MATH identifier
1085.34023

Subjects
Primary: 34B15: Nonlinear boundary value problems 34C23: Bifurcation [See also 37Gxx]

Keywords
bifurcation curve boundary blow-up problem nonnegative solution sign-changing solution exact multiplicity

Citation

Wang, Shin-Hwa; Liu, Yueh-Tseng. SHAPE AND STRUCTURE OF THE BIFURCATION CURVE OF A BOUNDARY BLOW-UP PROBLEM. Taiwanese J. Math. 9 (2005), no. 2, 201--214. doi:10.11650/twjm/1500407796. https://projecteuclid.org/euclid.twjm/1500407796


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