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 $C^{2}\left ( (-L,L)\right ) \cap C\left ( [-L,L]\right ) $ for the Minkowski-curvature equation $$ \left \{ \textstyle\begin{array}{l} -\left ( \dfrac{u^{\prime }(x)}{\sqrt{1-\left ( {u^{\prime }(x)}\right ) ^{2}}}\right ) ^{\prime }=\lambda f(u),\text{\ \ }-L< x< L, \\ u(-L)=u(L)=0.\end{array}\displaystyle \right . $$ where $\lambda >0$ is a bifurcation parameter, $f\in C[0,\infty )\cap C^{2}(0,\infty )$ satisfies $f(u)>0$ for $u>0$ and f is either concave or geometrically concave on $(0,\infty )$ . If f is a concave function, we prove that the bifurcation curve is monotone increasing on the $(\lambda ,\left \Vert u\right \Vert _{\infty })$ -plane. If f is a geometrically concave function, we prove that the bifurcation curve is either ⊂-shaped or monotone increasing on the $(\lambda ,\left \Vert u\right \Vert _{\infty })$ -plane under a mild condition. Some interesting applications are given.