A global bifurcation theorem for a multiparameter positone problem and its application to the one-dimensional perturbed Gelfand problem

We study the global bifurcation and exact multiplicity of positive solutions for {u ''(x) + lambda f(epsilon)(u) = 0, - 1 < x < 1, u(-1) = u(1) = 0, where lambda > 0 is a bifurcation parameter, epsilon is an element of Theta is an evolution parameter, and Theta (sigma(1),sigma(2) ) is an open interval with 0 <= sigma(1) < sigma(2) < infinity. Under some suitable hypotheses on f(epsilon), we prove that there exists epsilon(0) is an element of Theta such that, on the (lambda,parallel to u parallel to infinity)-plane, the bifurcation curve is S-shaped for sigma(1) epsilon < epsilon(0) and is monotone increasing for epsilon(0) epsilon <= sigma(2). We give an application to prove global bifurcation of bifurcation curves for the one-dimensional perturbed Gelfand problem.