Time maps and exact multiplicity results for one-dimensional prescribed mean curvature equations. II

We consider non-classical solutions of the quasilinear boundary value problem {-(u'/root 1 + (u')(2)) = lambda f(u), x is an element of (-L, L), u(-L) = u(L) = 0, where lambda 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) = vertical bar u vertical bar(p-1)u(p > 0), we also obtain the exact number of all sign-changing non-classical solutions and show the global structure of bifurcation curves. (C) 2011 Elsevier Ltd. All rights reserved.