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

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 $\lambda \gt 0$.