Abstract
We study the shape and the structure of the bifurcation curve f <sub>a</sub> (ρ) (= √λ) with ρ: = min <sub>xε(0,1)</sub> u(x) of (sign-changing and nonnegative) solutions of the boundary blow-up problem {-u″(x) = λf(u(x)), 0 < x < 1, lim x→0 <sup>+</sup> u(x) = ∞ = lim x→1 <sup>-</sup> u(x), where λ is a positive bifurcation parameter and the Lipschitz continuous conacve function f = f <sub>a</sub> (u) = {-|u| <sup>p</sup> if u ≤ -a <sup>1/p</sup> , -a if - a <sup>1/p</sup> < u < a <sup>1/p</sup> , -|u| <sup>p</sup> if u ≥ a <sup>1/p</sup> , with constants p > 1 and a > 0. We mainly show that the bifurcation curve G <sub>fa</sub> (ρ) satisfies lim <sub>ρ→±∞</sub> G <sub>fa</sub> (ρ) = 0 and G <sub>fa</sub> (ρ) has a exactly one critical point, a maximum, on (-∞, ∞). Thus we are able to determine the exact number of (sign-changing and nonnegative) solutions of the problem for each λ > 0.