Bifurcation curves and exact multiplicity of positive solutions for Dirichlet problems with the Minkowski-curvature equation

In this paper, we consider the bifurcation curves and exact multiplicity of positive solutions of the one-dimensional Minkowski-curvature equation { − ( u ′ 1 − u ′ 2 ) ′ = λ f ( u ) , x ∈ ( − L , L ) , u ( − L ) = 0 = u ( L ) , where λ and L are positive parameters, f ∈ C [ 0 , ∞ ) ∩ C 2 ( 0 , ∞ ) , and f ( u ) > 0 for 0 < u < L . We give the precise description of the structure of the bifurcation curves and obtain the exact number of positive solutions of the above problem when f satisfies f ″ ( u ) > 0 and u f ′ ( u ) ≥ f ( u ) + 1 2 u 2 f ″ ( u ) for 0 < u < L . In two different cases, we obtain that the above problem has zero, exactly one, or exactly two positive solutions according to different ranges of λ . The arguments are based upon a detailed analysis of the time map.