Exact multiplicity of positive solutions for the one-dimensional Minkowski-curvature problem with convex-concave nonlinearity

Abstract We establish the bifurcation curves and exact multiplicity of positive solutions for Dirichlet problem with mean curvature operator in the Minkowski spacewhere λ and L are positive parameters, f∈C1[0, ∞) ∩ C2(0, ∞) satisfies f(0) = 0 and f(u) > 0 for u ∈ (0, L), and f is convex-concave on (0, L). By a detailed analysis of the time map, we determine the interval of parameter λ in which the above problem has zero, exactly one or exactly two positive solutions when f satisfies certain conditions.