Classification and evolution of bifurcation curves for the one-dimensional Minkowski-curvature problem and its applications

In this paper, we study the classification and evolution of bifurcation curves of positive solutions for the one-dimensional Minkowski-curvature problem{−(u′/1−u′2)′=λf(u), in (−L,L),u(−L)=u(L)=0, where λ,L>0, f∈C[0,∞)∩C2(0,∞) and f(u)>0 for u≥0. Furthermore, we show that, for sufficiently large L>0, the bifurcation curve SL may have arbitrarily many turning points. Finally, we apply these results to obtain the global bifurcation diagrams for Ambrosetti-Brezis-Cerami problem, Liouville-Bratu-Gelfand problem and perturbed Gelfand problem with the Minkowski-curvature operator, respectively. Moreover, we will make two lists which show the different properties of bifurcation curves for Minkowski-curvature problems, corresponding semilinear problems and corresponding prescribed curvature problems.