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

Abstract

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.