Bifurcation curve for the Minkowski-curvature equation with concave or geometrically concave nonlinearity

We study the bifurcation curve and exact multiplicity of positive solutions in the space C² ( (-L,L) ) ∩ C ( [-L,L] ) for the Minkowski-curvature equation $$ \{ {array}{l} - ( {u(x)}{√{1- ( {u(x)} ) ²}} ) =λ f(u),\ \ -L< x< L, \\ u(-L)=u(L)=0.{array} . $$ where $λ >0$ is a bifurcation parameter, $f∈ C[0,∞ )∩ C²(0,∞ )$ satisfies $f(u)>0$ for $u>0$ and f is either concave or geometrically concave on $(0,∞ )$ . If f is a concave function, we prove that the bifurcation curve is monotone increasing on the $(λ , u ∞)$ -plane. If f is a geometrically concave function, we prove that the bifurcation curve is either ⊂-shaped or monotone increasing on the $(λ , u ∞)$ -plane under a mild condition. Some interesting applications are given.