Sufficient conditions for exact bifurcation curves in Minkowski curvature problems and their applications

In this paper, we continued and extended the work in [1,2,3] by establishing sufficient conditions to determine the exact shape of the bifurcation curve of positive solutions for the Minkowski curvature problem$ \left\{ \begin{array}{l} -\left( u^{\prime }/\sqrt{1-{u^{\prime }}^{2}}\right) ^{\prime } = \lambda f(u), \text{ in }\left( -L, L\right) , \\ u(-L) = u(L) = 0, \end{array} \right. $where $ \lambda, L > 0 $ and $ f\in C^{2}(0, \infty) $. Notice that we allow $ f(0^{+}) = -\infty $. To illustrate the applicability of these results, we presented some examples, such as the diffusive logistic equation with a Holling type-Ⅱ functional response.