Global bifurcation diagrams and multiplicity of positive solutions for the one-dimensional Minkowski-curvature problem with singular nonlinearity

In this paper, we study the global bifurcation diagrams and multiplicity of positive solutions for the one-dimensional Minkowski-curvature equation with singular nonlinearity { − ( u ′ 1 − u ′ 2 ) ′ = λ ( 1 − u ) p , in ( − L , L ) , u ( − L ) = u ( L ) = 0 , where λ, p, L are three positive parameters. We determine the number of positive solutions corresponding to the vary of the values of these parameters based on time-map approach. For this open question, we observe that the global bifurcation diagrams differ significantly between two cases: Case 1. For 0 < L < 1, | u ′| < 1 ensures u < L < 1, resulting in a non-singular problem; Case 2. For L ≥ 1, additional condition u < 1 is needed to avoid the occurrence of singularity. We show that the bifurcation curve of positive solutions either is strictly increasing or S -like shaped in the first case, while the bifurcation curve is more complicated, strictly increasing, S -like shaped, or ⊃-like shaped in the other one.