Shape and structure of the bifurcation curve of a boundary blow-up problem

We study the shape and the structure of the bifurcation curve f(a)(p) (= root lambda) with rho := min(x is an element of(0,1))u(x) of (sign-changing and nonnegative) solutions of the boundary blow-up problem {-u''(x) = lambda f (u(x)), 0 < x < 1, lim u(x) x -> 0(+) = infinity = lim u(x) x -> 1(-) , where lambda is a positive bifurcation parameter and the Lipschitz continuous conacve function f = f(a)(u) = {-vertical bar u vertical bar(p) if u <= -a(1)/(p), -a if - a(1)/(p) < u < a(1)/(p), -vertical bar u vertical bar(p) if u >= a(1)/(p), with constants p > 1 and a > 0. We mainly show that the bifurcation curve Gf(n) (rho) satisfies lim(rho ->+/-infinity) Gf(n)(rho) = 0 and Gf(n) (rho) has a exactly one critical point, a maximum, on (-infinity, infinity). Thus we are able to determine the exact number of (sign-changing and nonnegative) solutions of the problem for each lambda > 0.