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′1+(u′)2)′=λf(u),x∈(−L,L),u(−L)=u(L)=0, where λλ and LL 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)=euf(u)=eu, f(u)=(1+u)p(p>0)f(u)=(1+u)p(p>0), f(u)=eu−1f(u)=eu−1, f(u)=up(p>0)f(u)=up(p>0), and f(u)=au(a>0)f(u)=au(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)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.