In this paper, we study global bifurcation diagrams and the exact multiplicity of positive solutions for Minkowski-curvature problem

where \(\lambda >0\) is a bifurcation parameter and \(L>0\) is an evolution parameter. It can be viewed as a variant of the one-dimensional Liouville–Bratu–Gelfand problem. We prove that there exists \(L_{0}>0\) such that the bifurcation curve \(S_{L}\) is S-shaped for \(L>L_{0}\) and is monotone increasing for \(0<L\le L_{0}\). We also study, in the \(\left( L,\lambda ,\left\| u\right\| _{\infty }\right) \)-space, the shape and structure of the bifurcation surface. Finally, we will make a list which shows the different properties of bifurcation curves for Minkowski-curvature problem, corresponding semilinear problem and corresponding prescribed curvature problem.

在本文中，我们研究了全局分叉图和Minkowski曲率问题的正解的精确多重性

其中\（\ lambda> 0 \）是分叉参数，\（L> 0 \）是演化参数。可以将其视为一维Liouville–Bratu–Gelfand问题的变体。我们证明了存在\（L_ {0}> 0 \） ，使得分支曲线\（S_ {L} \）是S形为\（L> L_ {0} \），并且单调增加\（ 0 <L \ le L_ {0} \）。我们还在\（\ left（L，\ lambda，\ left \ | u \ right \ | _ {\ infty} \ right）\）中研究空间，分叉表面的形状和结构。最后，我们将给出一个列表，显示针对Minkowski曲率问题，相应的半线性问题和相应的规定曲率问题的分岔曲线的不同性质。