Global bifurcation and exact multiplicity of positive solutions for a positone problem with cubic nonlinearity

Abstract

We study the global bifurcation and exact multiplicity of positive solutions of {u'' (chi) + lambda f(epsilon)(u) = 0, -1 < chi < 1, u(-1) = u(1) = 0, f epsilon(u) = -epsilon u(3) + sigma u(2) - kappa u + rho, where lambda, epsilon > 0 are two bifurcation parameters, and sigma, rho > 0, 0 < kappa <= root sigma rho are constants. We prove the global bifurcation of bifurcation curves for varying epsilon > 0. More precisely, there exists (epsilon) over tilde > 0 such that, on the (lambda, parallel to u parallel to infinity)-plane, the bifurcation curve is S-shaped for 0 < epsilon < (epsilon) over tilde and is monotone increasing for epsilon >= (epsilon) over tilde. Thus we are able to determine the exact number of positive solutions by the values of epsilon and lambda. Our results extend those of Hung and Wang (K.-C. Hung, S.-H. Wang, Global bifurcation and exact multiplicity of positive solutions for a positone problem with cubic nonlinearity and their applications, Trans. Amer. Math. Soc., in press) from kappa <= 0 to kappa <= root sigma rho. (C) 2012 Elsevier Inc. All rights reserved.