time maps and exact multiplicity results for one-dimensional prescribed mean curvature equations. ii

{-(u’/√ ̅1+(u’))’=λf(u),x∈(-l,l)

u(-l)=u(l)=0.

where λ and l are positive parameters. we give complete descriptions of the structure of bifurcation curves and determine the exact numbers of positive non-classical solutions of the model problems for various nonlinearities f (u) = e^u, f (u) = (1 + u)^p(p > 0), f (u) = e^u − 1, f (u) = u^p(p > 0), and f (u) = a^u(a > 0). the methods used are elementary and based on a detailed analysis of time maps. moreover, for the case f (u) = |u|^p−1u(p > 0), we also obtain the exact number of all sign-changing non-classical solutions and show the global structure of bifurcation curves.